Nonlinear Differential Problems with Smooth and Nonsmooth Constraints (Mathematical Analysis and its Applications) [1 ed.] 0128133864, 9780128133866

Table of contents :
Dedication
About the Author
Preface
Acknowledgment
Introduction
Chapter 1: Elements of Functional Analysis and Operator Theory
1.1 Sobolev Spaces
1.2 Optimization, Subdifferentials and Generalized Gradients
1.3 Monotone and Pseudomonotone Operators
1.4 Notes
Chapter 2: Elements of Regularity Theory and Maximum Principle
2.1 Properties of the Solution Set
2.2 Regularity Theory for Nonlinear Elliptic Equations
2.3 Strong Maximum Principle for Nonlinear Elliptic Equations
2.4 Notes
Chapter 3: Nonlinear Elliptic Eigenvalue Problems
3.1 Eigenvalue Problem for p-Laplacian Under Different Boundary Conditions
3.2 Eigenvalue Problems for (p,q)-Laplacian with Indefinite Weights
3.3 Eigenvalue Problems for Asymptotically Homogeneous Differential Elliptic Operators
3.4 Notes
Chapter 4: Nonlinear Elliptic Equations with General Dependence on the Gradient of the Solution
4.1 Gradient Dependent Problems with p-Laplacian
4.2 Gradient Dependent Problems with (p,q)-Laplacian
4.3 Gradient Dependent Problems Driven by Nonhomogeneous Differential Operators
4.4 Notes
Chapter 5: Constant-Sign and Sign-Changing Solutions for Quasilinear Elliptic Problems
5.1 The Case of Dirichlet Boundary Condition
5.2 The Case of Neumann Boundary Condition
5.3 Problems with Multivalued Terms
5.4 Notes
Chapter 6: Nonlinear Elliptic Systems
6.1 Elliptic Systems Fully Depending on the Gradient of the Solution
6.2 Subsolutions-Supersolutions for Nonlinear Elliptic Systems
6.3 Elliptic Systems with Variational Structure
6.4 Notes
Chapter 7: Singular Quasilinear Elliptic Systems
7.1 The Case of Singularities with Respect to the Solution
7.2 The Case of Singularities with Respect to the Gradient of the Solution
7.3 Notes
Chapter 8: Evolutionary Variational and Quasivariational Inequalities
8.1 Evolutionary Variational Inequalities
8.2 Evolutionary Quasivariational Inequalities
8.3 Notes
Chapter 9: Control Problems for Evolutionary Differential Inclusions
9.1 Semilinear Evolutionary Inclusions with Control Parameter
9.2 Evolutionary Problems Driven by Variational Inequalities
9.3 Notes
Bibliography
Index

Citation preview

Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Mathematical Analysis and its Applications Series

Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Dumitru Motreanu University of Perpignan, Department of Mathematics, 66000 Perpignan, France

Series Editor

Themistocles Rassias

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1800, San Diego, CA 92101-4495, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2018 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-813386-6 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Candice Janco Acquisition Editor: Graham Nisbet Editorial Project Manager: Jaclyn A. Truesdell Production Project Manager: Sreejith Viswanathan Designer: Mark Rogers Typeset by VTeX

Dedication

This book is dedicated to the memory of my wife Mariana Claudia Motreanu Dumitru Motreanu

v

About the Author

Dumitru Motreanu is Professor of Mathematics at the University of Perpignan, in France. Dr. Motreanu received his PhD degree in 1978 from the University of Iasi, Romania. In 1991 he was a recipient of Simeon Stoilov Award from the Romanian Academy of Sciences. His areas of expertise cover partial differential equations and nonlinear analysis. He obtained original results in smooth and nonsmooth variational methods, nonlinear eigenvalue problems, multiple solutions for elliptic equations and systems. He coauthored seven monographs in mathematics and published more than 190 professional articles in prestigious international journals of mathematics, as well as more than 20 chapters in research volumes. He also edited a handbook of nonconvex analysis and three special issues of international journals. His results are cited and used in more than 1900 refereed papers. He serves in the editorial boards of more than 20 academic journals in mathematics.

ix

Preface

The book comprises a wide range of subjects passing from eigenvalue problems for nonlinear elliptic operators to control problems for evolution equations driven by variational inequalities. In the large panel of topics covered in the book, the reader will discover surprisingly strong links between the underlying components. The core of the whole material is represented by the study, through structural differential properties, of nonlinear smooth or nonsmooth problems subject to constraints of different type. The choice of statements and approach reflects the author’s scientific preferences and original contributions during the last five years. The book is organized in nine chapters arranged gradually and having to a large extent an independent existence. Many of the results included here appear for the first time in the book form. The notes given in the final section of every chapter address the evolution of main ideas within a specific theme attempting to make a bridge from classical mathematics to recent research including open problems. The references are carefully selected listing titles strictly tied to the mentioned results. The author is deeply indebted to Dr. Lucas Fresse and Dr. Viorica Venera Motreanu for their huge efforts in reading, correcting and improving the text. The author expresses his gratitude to Prof. Themistocles M. Rassias, the editor of the series “Mathematical Analysis and its Applications” for his kind invitation and generous support. The author thanks the Editorial Project Manager Jackie Truesdell and the Senior Acquiring Editor Graham Nisbet for their outstanding editorial work. Many thanks are also due to Publisher’s Academic Press with Elsevier Inc., specifically, the Project Manager, Reference Content Production, Sreejith Viswanathan for his great assistance.

Dumitru Motreanu Perpignan October 2017 xi

Acknowledgment

I thank my beloved daughter Dr. Viorica Venera Motreanu and Dr. Lucas Fresse for thoroughly checking and greatly improving the text. I thank Prof. Themistocles M. Rassias for kindly inviting me to contribute with a volume in the series entitled “Mathematical Analysis and its Applications” for which he is the Series Editor. I thank my Editors and Publishers with Academic Press of Elsevier Inc., and first of all the Editorial Project Manager Jackie Truesdell, the Senior Acquiring Editor Graham Nisbet, and the Project Manager, Reference Content Production, Sreejith Viswanathan for highly professional assistance and constant support. Dumitru Motreanu November 2017

xiii

Introduction

This book is built on two main ideas regarding the topics under consideration: to exploit the presence of the available differentiability properties (even in a weak sense) of the data in the problem and to figure out the natural (smooth or nonsmooth) constraints that the solutions should satisfy. The interplay between these two essential features is systematically taken into account in every studied subject. In order to give a hint about the meaning of this mutual relationship, let us focus on a simplified but relevant situation. Consider an inclusion problem 0 ∈ A(x, u, ∇u) on a bounded domain  ⊂ RN , with A :  × R × RN → 2R satisfying a (global) growth condition sup |A(x, s, y)| ≤ α(s) + β(y) on  × R × RN for α : R → R and β : RN → R monotone, and seek for a solution u :  → R with a gradient ∇u :  → RN subject to the pointwise constraints on the pattern of subsolution–supersolution and obstacle bound u(x) ≤ u(x) ≤ u(x)

and

|∇u(x)| ≤ u(x), ˜

˜ are in some Lp (), p ≥ 1. Then the with u, u, u˜ such that α(u), α(u), β(u) needed growth for the nonlinear operator A driving the problem is actually local and pointwise, namely ˜ ≡ w(x) sup |A(x, s, y)| ≤ max{α(u(x)), α(u(x))} + β(u(x)) ˜ with w ∈ Lp (). for all x ∈ , s ∈ [u(x), u(x)], |y| ≤ u(x), The book consists of nine chapters upon which we give a brief overview. Chapter 1 provides a concise presentation of selected facts related to Sobolev spaces, differentiability and generalized monotonicity. Chapter 2 discusses regularity properties of solutions and a general strong maximum principle. Chapter 3 treats eigenvalue problems for important nonlinear operators as p-Laplacian, (p, q)-Laplacian, nonhomogeneous differential operators by admitting various xv

xvi Introduction

boundary conditions. Chapter 4 focuses on the nonlinear elliptic problems exhibiting full dependence on the gradient of the solution, which prevents the application of variational methods. Chapter 5 presents results ensuring the existence of solutions with precise sign information, namely positive, negative, and nodal (i.e., sign-changing) solutions for quasilinear elliptic problems, possibly with multivalued terms. Chapter 6 reports on systems of quasilinear elliptic equations paying special attention to the general method of subsolution– supersolution giving rise to the prominent concept of a trapping region. Chapter 7 sets forth existence of positive solutions for systems of quasilinear elliptic equations allowing singularities with respect to the solutions and the gradients of the solutions. Chapter 8 contains various theorems on the solvability of variational and quasilinear inequalities incorporating in a unifying way the elliptic and evolutionary cases. Chapter 9 aims to handle general evolutionary inclusion problems involving control parameters and constraints for the solutions subject to variational inequalities. We close every chapter with a section where one finds notes on related developments, open problems with partial answers, and historic comments.

Chapter 1

Elements of Functional Analysis and Operator Theory 1.1 SOBOLEV SPACES We start by presenting some basic elements of integration theory in a Banach space. Let I be an open interval in R of Lebesgue measure |I | and let X be a real Banach space with the norm  · , dual space X ∗ , and duality bracket ·, · between X and X ∗ . We denote by Cc (I, X) the Banach space of continuous functions from I to X with compact support. Definition 1.1. The function f : I → X is measurable if there exists a sequence {fn } ⊂ Cc (I, X) such that fn (t) → f (t) in X for a.e. t ∈ I . A measurable function f : I → X is called integrable if there exists a sequence {fn } ⊂ Cc (I, X) such that  fn (t) − f (t) dt = 0. (1.1) lim n→∞ I

Proposition 1.2. If the function f : I → X is integrable, then there exists a  unique point in X denoted I f (t) dt (called the integral of f over I ) such that for every sequence {fn } ⊂ Cc (I, X) satisfying (1.1) one has   lim fn (t) dt = f (t) dt. (1.2) n→∞ I

I

  Proof. It is straightforward to show from (1.1) that I fn (t) dt is a Cauchy sequence in X, so the limit in (1.2) exists. Furthermore, again through (1.1), a direct verification ensures that the limit in (1.2) does not depend on the sequence {fn } chosen in (1.1). A powerful criterion of integrability is provided in the following result. Theorem 1.3 (Bochner’s theorem). A measurable function f : I → X is integrable if and only if f  : I → R is integrable. Moreover, in the case of integrability, it holds    f (t) dt ≤ f (t) dt. (1.3) I

I

Nonlinear Differential Problems with Smooth and Nonsmooth Constraints. DOI: 10.1016/B978-0-12-813386-6.00001-8 Copyright © 2018 Elsevier Inc. All rights reserved.

1

2 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Proof. Assume that f : I → X is integrable. If the sequence {fn } ⊂ Cc (I, X) fulfills (1.1), then the integrability of f  : I → R follows readily by considering the sequence {fn } ⊂ Cc (I, R). Conversely, by the measurability of f : I → X and the integrability of f  : I → R, we can find a sequence {fn } ⊂ Cc (I, X) with fn (t) → f (t) in X for a.e. t ∈ I and a sequence {gn } ⊂ Cc (I, R) with gn → f  in L1 (I ). Setting hn =

gn fn fn  +

1 n

,

by Lebesgue’s dominated convergence theorem we infer that  hn (t) − f (t) dt = 0, lim n→∞ I

so f : I → X is integrable. Inequality (1.3) follows because    hn (t) dt ≤ hn (t) dt I

I

and we pass to the limit as n → ∞. We recall the Hahn–Banach separation theorem. For its proof we refer to [31, Theorems 1.6, 1.7]. Theorem 1.4. Let A and B be two nonempty, disjoint, and convex subsets of a normed space X. (i) Assume that one of them is open. Then there exist f ∈ X ∗ \ {0} and α ∈ R such that f, x ≤ α, ∀x ∈ A, and f, y ≥ α, ∀y ∈ B. (ii) Assume that the set A is closed and the set B is compact. Then there exist f ∈ X ∗ \ {0} and real numbers α < β such that the strict separation holds f, x ≤ α, ∀x ∈ A, and f, y ≥ β, ∀y ∈ B. Two important consequences of Theorem 1.4 (ii) are needed in the sequel. Corollary 1.5. A linear subspace S of a normed space X is dense in X if and only if the following holds: f ∈ X ∗ with f, x = 0 for all x ∈ S implies f = 0. Proof. The necessity part follows from the continuity of f . For the sufficiency, suppose there is x0 ∈ X \ S. Then Theorem 1.4 provides f ∈ X ∗ \ {0} with the property f, x < f (x0 ), ∀x ∈ S.

Elements of Functional Analysis and Operator Theory Chapter | 1

3

Replacing x by λx, with λ ∈ R, enables us to deduce that f vanishes on S, hence f = 0 in view of our hypothesis. This is impossible because the above inequality implies f (x0 ) > 0 (take x = 0). The reached contradiction achieves the proof. Corollary 1.6. Let f : I → X be an integrable function on an open interval I and let K be a closed convex subset of a real Banach space X such that f (t) ∈ K for all t ∈ I . Then it holds  1 f (t) dt ∈ K. (1.4) |I | I Proof. Arguing by contradiction, assume that (1.4) is not true. Then Theorem 1.4 provides g ∈ X ∗ \ {0} and ε > 0 satisfying  g, x + ε < g, f (t) dt, ∀x ∈ K. I

In particular, for a.a. t ∈ I , we get  g, f (t) + ε < g,

f (t) dt. I

Integrating over I leads to a contradiction, which completes the proof. At this point we focus on certain spaces of integrable functions. Definition 1.7. Given an open interval I in R, a real Banach space X, and a number p ∈ [1, +∞], it is said that a function f : I → X (in fact, its equivalence class with respect to equality a.e.) belongs to Lp (I, X) if f is measurable and f  ∈ Lp (I ). When endowed with the norm  f Lp (I,X) =

1 f (t)p dt

p

for 1 ≤ p < +∞,

I

f L∞ (I,X) = ess sup f (t), t∈I

Lp (I, X) (for 1 ≤ p ≤ +∞) becomes a Banach space. The notation f ∈ p Lloc (I, X) stands for the fact that f ∈ Lp (J, X) for every open subinterval J of I with J compact and contained in I . In order to pass to Sobolev spaces, we need weak differentiability properties. Denote by Cck (I, X), with 1 ≤ k ≤ ∞, the Banach space of functions of class C k from I to X with compact support. If X = R, we just denote Cck (I ).

4 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Lemma 1.8. Let f, g ∈ L1loc (I, X) satisfy   ϕ(t)g(t) dt = − ϕ (t)f (t) dt for all ϕ ∈ Cc∞ (I ). I

I

Then for every t0 ∈ I there corresponds x0 ∈ X (actually, f is continuous on I and x0 = f (t0 )) such that  t f (t) = x0 + g(s) ds for all t ∈ I . t0

Proof. Without loss of generality we may suppose that I = R and g ∈ L1 (R, X). Then we can approximate gn → g in L1 (R, X) with a sequence {gn } ⊂ Cc∞ (R, X). For a ϕ ∈ Cc∞ (R), the integration by parts yields   ϕ(t)g(t) dt = lim ϕ(t)gn (t) dt n→∞ R R  t  

ϕ (t) gn (s) ds dt = − lim n→∞ R



=−

R

ϕ (t)



t0 t

g(s) ds

 dt.

t0

Through the hypothesis, we find     t ϕ (t) f (t) − g(s) ds dt = 0. R

t0

Taking into account that ϕ ∈ Cc∞ (R) is arbitrary, the conclusion ensues. We are ready to discuss the Sobolev spaces of functions from I to X. Definition 1.9. Given an open interval I in R, a real Banach space X and a number p ∈ [1, +∞], it is said that f ∈ W 1,p (I, X) if f ∈ Lp (I, X) and there exists g ∈ Lp (I, X) (necessarily unique, see Lemma 1.8) such that   ϕ(t)g(t) dt = − ϕ (t)f (t) dt for all ϕ ∈ Cc1 (I ). I

I

We denote g = f =

df dt

. Endowed with the norm

f W 1,p (I,X) = f Lp (I,X) + f Lp (I,X) , W 1,p (I, X) is a Banach space. If p = 2, H 1 (I, X) := W 1,2 (I, X) is a Hilbert space. We point out a useful characterization of the elements of the space W 1,p (I, X).

Elements of Functional Analysis and Operator Theory Chapter | 1

5

Theorem 1.10. For any f ∈ Lp (I, X), the following properties are equivalent: (i) f ∈ W 1,p (I, X);  (ii) f (t) = f (s) +

t

g(τ ) dτ for a.a. s, t ∈ I ,

s

with some g ∈ Lp (I, X) (actually, g = f , see Definition 1.9). Proof. The equivalence can be easily established on the basis of Lemma 1.8. Corollary 1.11. The continuous embedding W 1,p (I, X) → Cb,u (I , X) holds, where the notation Cb,u (I , X) means the Banach space of bounded and uniformly continuous functions from I to X. In particular, one has W 1,p (I, X) → L∞ (I, X). If p > 1, the Hölder regularity can be achieved W 1,p (I, X) → C

p−1 p

(I , X).

Proof. Using Theorems 1.3 and 1.10, for every f ∈ W 1,p (I, X) it turns out that  t    | f (t) − f (s) | ≤ f (t) − f (s) ≤  f (τ ) dτ  for a.a. s, t ∈ I . s

(1.5) The second inequality in (1.5) shows the uniform continuity of f on I , whereas by means of the first inequality in (1.5) we can infer that f  ∈ W 1,p (I ), so f is bounded on I . If p > 1, (1.5) in conjunction with Hölder’s inequality implies f (t) − f (s) ≤ |t − s|

p−1 p

f Lp (I,X) for a.a. s, t ∈ I ,

whence the conclusion. Remark 1.12. If 1 ≤ p, q < +∞ and if X → Y is a continuous embedding of Banach spaces, then Cc∞ (I , X) is dense in Lp (I, X) ∩ W 1,q (I, Y ) (cf. [32, Proposition A.2.46]). For higher order Sobolev spaces of functions from I to X we proceed inductively. Given an open interval I in R, a real Banach space X, a number p ∈ [1, +∞], and an integer m > 1, we set W m,p (I, X) = {f ∈ W m−1,p (I, X) : f ∈ W m−1,p (I, X)}.

(1.6)

Corollary 1.13. For every integer m ≥ 1 and number p ∈ [1, +∞], W m,p (I, X) is a Banach space and one has the continuous embedding W m,1 (I, X) → C m−1 (I , X), and W m,p (I, X) → C

m−1, p−1 p

(I , X) provided p > 1.

Proof. The result follows inductively from (1.6) and Corollary 1.11.

6 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

We pass to Sobolev spaces of functions defined on multidimensional domains. Let be an open subset of RN with Lebesgue measure | |. For every multiindex α = (α1 , . . . , αN ) of length |α| = α1 + · · · + αN , where α1 , . . . , αN are nonnegative integers, the weak derivative D α u of a locally integrable function u ∈ L1loc ( ) is the distribution defined by D α u, ϕ = (−1)|α|



u(x)D α ϕ(x) dx for all ϕ ∈ Cc∞ ( ).

Important particular cases are the (weak) partial derivatives ∂i u = 1, . . . , N , and the (weak) gradient ∇u = (∂1 u, . . . , ∂N u).

∂u ∂xi

for i =

Definition 1.14. Given an open set in RN , a nonnegative integer m, and a number p ∈ [1, +∞], it is said that u ∈ W m,p ( ) if u ∈ Lp ( ) and D α u ∈ Lp ( ) for every multiindex α with |α| ≤ m. With respect to the norm uW m,p ( ) =



D α uLp ( ) ,

|α|≤m

W m,p ( ) is a Banach space. The closure of Cc∞ ( ) in W m,p ( ), with p < m,p +∞, is the Banach space W0 ( ). When p = 2, H m ( ) := W m,2 ( ) and H0m ( ) := W0m,2 ( ) are Hilbert spaces. We mention a significant tool to generate elements of the Sobolev space W 1,p ( ). For more details we refer to [146,78]. Theorem 1.15 (Marcus–Mizel). If F : R → R is Lipschitz continuous with F (0) = 0 and u ∈ W 1,p ( ), then for the composition F ◦ u there hold F ◦ u ∈ W 1,p ( ) and ∇(F ◦ u)(x) = F (x)∇u(x) for a.e. x ∈ . Moreover, if p < +∞, the Nemytskii operator NF : W 1,p ( ) → W 1,p ( ) given by NF (u) = F ◦ u is continuous. Corollary 1.16. For every u ∈ W 1,p ( ), there hold u+ := max{u, 0} ∈ W 1,p ( ), u− := max{−u, 0} ∈ W 1,p ( ), and



∇u if u > 0, −∇u if u < 0, + − ∇u = ∇u = 0 if u ≤ 0, 0 if u ≥ 0. Proof. It suffices to prove the assertion for u+ because u− = u+ − u. Theorem 1.15 can be applied with the function F (t) = t + , which gives u+ ∈ W 1,p ( ). In order to compute its gradient, for every ε > 0 we set 1

Fε (t) = ((t + )2 + ε 2 ) 2 for all t ∈ R.

Elements of Functional Analysis and Operator Theory Chapter | 1

7

Then Theorem 1.15 yields ∇(Fε ◦ u)(x) =

(u(x))+ ∇u(x) for a.a. x ∈ . Fε (u(x))

Letting ε → 0 provides the result. 1,p

A characteristic feature of the Banach space W0 ( ) is set forth. Theorem 1.17 (Poincaré’s inequality). If the open set ⊂ RN is bounded in at least one direction, then there exists a constant C = C(p, ) > 0 such that 1,p

uLp ( ) ≤ C∇uLp ( ) for all u ∈ W0 ( ). Proof. On the basis of the assumption, without loss of generality we may admit 1,p that ⊂ RN−1 × (0, a), with some a > 0. In view of the definition of W0 ( ), it is sufficient to argue with u ∈ Cc1 ( ). Writing x = (x , xn ) ∈ RN−1 × (0, a), it is seen that  xN ∂u |u(x)|p = p |u(x , s)|p−2 u(x , s) (x , s) ds ∂xN 0    a  ∂u  |u(x , s)|p−1  (x , s) ds. ≤p ∂xN 0 The integration over , combined with Hölder’s inequality and Fubini’s theorem, enables us to find   p p−1  ∂u   uLp ( ) ≤ pa uLp ( )   ∂x  p , N L ( ) which entails the result. Remark 1.18. The conclusion of Theorem 1.17 remains valid if the measure | | is finite (for the proof see, e.g., [116, pp. 34–35]). Poincaré’s inequality 1,p ensures that ∇uLp ( ) is an equivalent norm on W0 ( ). In order to clarify the relationship of the Sobolev spaces with other function spaces we need regularity conditions on . Definition 1.19. An open set in RN is said to be of class C m , with an integer m ≥ 1, if for every point x ∈ ∂ there exist a neighborhood U of x in RN and a C m -diffeomorphism  of U onto {x = (x , xN ) ∈ RN−1 × (−1, 1) : |x | < 1} such that (U ∩ ) = {x = (x , xN ) ∈ RN−1 × (0, 1) : |x | < 1}, (U ∩ ∂ ) = {x = (x , 0) : |x | < 1}.

8 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Theorem 1.20 (Sobolev’s and Rellich–Kondrachov embedding theorems). Assume that the open set ⊂ RN is bounded and of class C m , with an integer m ≥ 1, and let 1 ≤ p < +∞. (i) If mp < N , one has the continuous embedding W m,p ( ) → Lq ( ) for Np Np each q ≤ N−mp , which is compact provided q < N−mp . m,p (ii) If mp = N , one has the compact embedding W ( ) → Lq ( ) for each q < +∞. (iii) If mp > N , one has the continuous embedding W m,p ( ) → C k,α ( ), with k the biggest integer such that k ≤ m − Np and 0 ≤ α ≤ m − k − Np , which is compact provided α < m − k −

N p.

The above assertions remain valid without assuming that is of class C m when m,p W m,p ( ) is replaced by W0 ( ). Remark 1.21. One can relax the regularity condition on required in Theorem 1.20 (see Adams [1]). We end this section with the characterization of the dual space W0 ( )∗ ,

denoted W −1,p ( ) with p1 + p1 = 1 (see, e.g., [31, p. 291]). If p = 2, we denote H −1 ( ) := H01 ( )∗ . 1,p

Theorem 1.22. Let be an open subset of RN and 1 ≤ p < +∞. Then ξ ∈

W −1,p ( ) if and only if there exist f0 , f1 , . . . , fN ∈ Lp ( ) such that  f0 (x)v(x) dx +

ξ, v =

N 

fi (x)

i=1

∂v 1,p (x) dx for all v ∈ W0 ( ). ∂xi

In addition, it holds ξ W

−1,p

( )

=

N

fi Lp ( ) .

i=0

In the case where is bounded, one can choose f0 = 0.

1.2 OPTIMIZATION, SUBDIFFERENTIALS AND GENERALIZED GRADIENTS This section deals with functionals on a Banach space X, i.e., real-valued functions on X. The first case that we consider is of a linear and continuous functional f : X → R, which means that f ∈ X ∗ . We start with the classical description of X ∗ when X is a Hilbert space, which is known as Riesz representation theorem. Theorem 1.23. Let H be a real Hilbert space endowed with the scalar product 1/2 (·, ·)H and associated Euclidean norm uH = (u, u)H . For every f ∈ H ∗

Elements of Functional Analysis and Operator Theory Chapter | 1

9

there corresponds a unique uf ∈ H such that f, v = (uf , v)H for all v ∈ H .

(1.7)

f H ∗ = uf H .

(1.8)

Moreover, it holds

Proof. If f = 0, the proof is achieved by taking uf = 0. Suppose that f = 0. Then we can find w ∈ H , w = 0, to have the orthogonal direct sum decomposition H = ker(f ) ⊕ Rw.

(1.9)

Precisely, there is w ∈ ker(f )⊥ := {v ∈ H : (u, v)H = 0, ∀u ∈ ker(f )} with f (w) = 0 (so, w = 0) because f = 0. If v ∈ ker(f )⊥ , we note that v−

f (v) w ∈ ker(f ) ∩ ker(f )⊥ , f (w)

(v) w = 0, which ensures that v ∈ Rw. Then (1.9) follows from the thus v − ff (w) equality ker(f )⊥ = Rw. Let us check that for every f ∈ H ∗ the vector uf = f (w)2 w satisfies the wH

(v) w ∈ ker(f ) whenever v ∈ H , the requirements. Indeed, observing that v − ff (w) orthogonality of w to ker(f ) in (1.9) results in   f (w) f (v) f (v) (uf , v)H = (w, v − w + w)H f (w) f (w) w2H f (w) f (v) = (w, w)H = f, v 2 f (w) wH

for all v ∈ H , so (1.7) holds true. Using (1.7) and Cauchy–Schwarz inequality allows obtaining the estimate f H ∗ = sup |(uf , v)H | ≤ uf H . vH ≤1

On the other hand, the expression of uf renders uf H =

|f (w)| ≤ f H ∗ . wH

Altogether, (1.8) ensues. The uniqueness of uf follows readily from (1.7), which completes the proof.

10 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Corollary 1.24. If H is a Hilbert space, then the dual H ∗ is a Hilbert space. Proof. Applying Theorem 1.23 and with the notation therein, for every f, g ∈ H ∗ we set (f, g)H ∗ = (uf , ug )H . Hence we obtain a scalar product on H ∗ . Moreover, from (1.8) it turns out that f H ∗ = (f, f )H ∗ = (uf , uf )H = uf H , ∀f ∈ H ∗ . 1/2

1/2

Since the normed space (H,  · H ) is complete, the same follows for the normed space (H ∗ ,  · H ∗ ), which completes the proof. We indicate how Theorem 1.23 can be directly used to study linear boundary value problems whose prototype is discussed below. Corollary 1.25. Let be an open subset of RN and let f ∈ H −1 ( ). Then the Dirichlet problem

− u + u = f in , (1.10) u = 0 on ∂ has a unique weak solution u ∈ H01 ( ), that is,   (∇u, ∇v)RN dx + uv dx = f (v), ∀v ∈ H01 ( ).

(1.11)



Proof. Since equality (1.11) reads as (u, v)H 1 ( ) = f (v), ∀v ∈ H01 ( ), 0

the conclusion regarding the unique solvability of problem (1.10) is readily obtained from Theorem 1.23. Remark 1.26. A careful examination of the proof of Theorem 1.23 reveals that the unique solution of problem (1.10) is of the form u=

f (w) w2 1

w

H0 ( )

for w ∈ ker(f )⊥ = {v ∈ H01 ( ) : (z, v)H 1 ( ) = 0 for all z ∈ ker(f )}, 0

with f (w) = 0 provided f = 0. We review a few basic facts regarding differentiability. For a comprehensive theory and applications, we refer to [13].

Elements of Functional Analysis and Operator Theory Chapter | 1

11

Definition 1.27. A function F : U → R defined on an open subset U of a normed space X is said to admit the directional derivative (or differential) at u ∈ U in the direction v ∈ X if there exists the limit dF (u; v) := lim

t→0+

F (u + tv) − F (u) ∈ R. t

Example 1.28. Let U be an open subset of X = Rn endowed with the standard ∂F scalar product. The partial derivatives ( ∂x (u))1≤i≤n of a function F : U → R i at u ∈ U are the directional derivatives (dF (u; ei ))1≤i≤n , where {e1 , . . . , en } stands for the canonical basis of Rn . Definition 1.29. A function F : U → R defined on an open subset U of a normed space X is said to be Gâteaux differentiable (or derivable) at u ∈ U if there exists l ∈ X ∗ such that dF (u; v) := l(v)

for all v ∈ X.

We denote F (u) := l (called the (Gâteaux) differential or derivative of F at u). Definition 1.30. A function F : U → R defined on an open subset U of a normed space X is said to be Fréchet differentiable (or derivable) at u ∈ U if there exists l ∈ X∗ such that lim

h→0

F (u + h) − F (u) − l(h) = 0. hX

We denote F (u) := l (called the (Fréchet) differential or derivative of F at u), which is justified by the next statement. Proposition 1.31. If the function F : U → R is Fréchet differentiable at u, then it is Gâteaux differentiable at u and the differentials coincide. Proof. It suffices to insert h = tv in Definition 1.30, with v ∈ X and t > 0, to get the desired conclusion. A partial converse of Proposition 1.31 is available. Proposition 1.32. If the function F : U → R is Gâteaux differentiable on a neighborhood V of u and the differential F : V → X ∗ is continuous at u, then F is Fréchet differentiable at u. Proof. Without loss of generality we may suppose that V = U . For every h ∈ X, based on Definition 1.29 we may write  1 d F (u + th) dt − F (u)h F (u + h) − F (u) − F (u)h = 0 dt  1 (F (u + th) − F (u))h dt. = 0

Now it is straightforward to derive that F is Fréchet differentiable at u.

12 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

In the case of a Hilbert space, the differential is expressed through the gradient. Proposition 1.33. Let U be an open subset of a Hilbert space X endowed with the scalar product (·, ·)X . If the function F : U → R is Gâteaux differentiable at u ∈ U , then there exists a unique vector ∇F (u) ∈ X (called the gradient of F at u) such that F (u), v = (∇F (u), v)X

for all v ∈ X.

Proof. One applies Theorem 1.23 on the Hilbert space X taking f = F (u). Example 1.34. (i) If U is an open subset of X = Rn endowed with the standard scalar product and F : U → R is Gâteaux differentiable at u ∈ U , then   ∂F (u) . ∇F (u) = ∂xi 1≤i≤n (ii) For the function F (u) = 12 u2X on the Hilbert space X, the gradient of F at every point u ∈ X is ∇F (u) = u. Remark 1.35. The differentiability notions in Definitions 1.27, 1.29, and 1.30 can similarly be introduced for mappings with values in a Banach space in place of scalar values. Propositions 1.31 and 1.32 are valid for vector-valued mappings (see, e.g., [13]). The implicit function theorem as stated below will be very useful in the following. Theorem 1.36. Let E, X, Y be Banach spaces, let F : D → Y be a mapping of class C k with k ≥ 1 defined on an open subset D of E × X and let a point (u0 , x0 ) ∈ D such that F (u0 , x0 ) = 0 and the partial differential Fx (u0 , x0 ) is an isomorphism of X onto Y . Then there exist a neighborhood U of u0 in E, a neighborhood V of x0 in X, and a unique mapping g : U → V of class C k such that (i) (ii) (iii) (iv)

g(u0 ) = x0 ; F (u, g(u)) = 0, ∀u ∈ U ; if F (u, x) = 0 with (u, x) ∈ U × V , then x = g(u); g (u) = −Fx (u, g(u))−1 ◦ Fu (u, g(u)), ∀u ∈ U .

Proof. Let us define G : D → E × Y by G(u, x) = (u, F (u, x)), ∀(u, x) ∈ D.

Elements of Functional Analysis and Operator Theory Chapter | 1

13

It is clear that G is of class C k and its differential at (u0 , x0 ) is expressed as follows: G (u0 , x0 )(u, x) = (u, F (u0 , x0 )(u, x)), ∀(u, x) ∈ E × X. Using the hypothesis, it is straightforward to check that G (u0 , x0 ) is an isomorphism, which permits applying the inverse function theorem. Since G(u0 , x0 ) = (u0 , 0), we find that there exist a neighborhood U of u0 in E and a neighborhood V of x0 in X such that G|U ×V is a C k -diffeomorphism onto a neighborhood of (u0 , 0) in E × Y . Denote the inverse Q = (G|U ×V )−1 on that neighborhood and set Q(v, y) = (h(v, y), ϕ(v, y)). In view of the definition of Q, on a neighborhood of (u0 , 0) in E × Y we have (v, y) = G(Q(v, y)) = (h(v, y), F (h(v, y), ϕ(v, y)), so Q(v, y) = (v, ϕ(v, y))

(1.12)

F (v, ϕ(v, y)) = y.

(1.13)

g(v) = ϕ(v, 0), ∀v ∈ U.

(1.14)

and

Now let us define

From (1.13) and (1.14) we infer that F (v, g(v)) = 0, thus assertion (ii) holds. On the basis of (1.12), (1.13) we derive that (u0 , x0 ) = Q(G(u0 , x0 )) = Q(u0 , F (u0 , x0 )) = Q(u0 , 0) = (u0 , ϕ(u0 , 0)) = (u0 , g(u0 )), whence (i). In order to show (iii), let (u, x) ∈ U × V satisfy F (u, x) = 0. Using again (1.13) we note that (u, x) = Q(G(u, x)) = Q(u, F (u, x)) = Q(u, 0) = (u, ϕ(u, 0)) = (u, g(u)), which confirms that (iii) is valid. Finally, differentiating F (u, g(u)) = 0 implies Fu (u, g(u)) + Fx (u, g(u))g (u) = 0. This enables us to get the formula in (iv) on a possibly smaller neighborhood U of u0 by making use of the hypothesis that Fx (u0 , x0 ) is an isomorphism and the set of isomorphisms of X onto Y is an open subset of L(X, Y ).

14 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Next we focus on the minimization problems with constraints in the form of an equality, that is, inf J (v),

v∈S

(1.15)

where J : X → R is a function of class C 1 on a Banach space X and the set of constraints S is described by S = {v ∈ X : F (v) = 0},

(1.16)

with a function F : X → R of class C 1 . By a solution of problem (1.15) (with S in (1.16)) we mean any point u0 ∈ S satisfying J (u0 ) = inf J (v). v∈S

We seek necessary conditions of optimality for problem (1.15), (1.16), which gives rise to the first result on the Lagrange multiplier rule. The main tool is the implicit function ensured by Theorem 1.36. Theorem 1.37. Assume that 0 is a regular value of F , that is, F (u) = 0 for every u ∈ S, with S in (1.16). Then for every solution u0 ∈ S of problem (1.15) there exists some λ ∈ R (called Lagrange multiplier associated to u0 ) such that J (u0 ) = λF (u0 ).

(1.17)

Proof. Let u0 ∈ S be a solution of problem (1.15), (1.16). Since F (u0 ) = 0, there is w ∈ X with F (u0 )w = 0. Then, denoting X0 = ker F (u0 ), we have the direct sum splitting X = X0 ⊕ Rw.

(1.18)

Indeed, for any u ∈ X it is seen that u − F (u0 )u ∈ X0 , whereas X0 ∩ Rw = {0}, which proves (1.18). Motivated by (1.18), we introduce the function G : X0 × R → R defined by G(v, t) = F (u0 + v + tw), ∀(v, t) ∈ X0 × R.

(1.19)

From (1.19) we obtain that G is a C 1 function, G(0, 0) = F (u0 ) = 0 and Gt (0, 0) = F (u0 )widR . Therefore Theorem 1.36 can be applied, ensuring the existence of a neighborhood V of 0 in X0 , a neighborhood I of 0 in R, and a unique function g : V → I of class C 1 with the properties: g(0) = 0,

(1.20)

Elements of Functional Analysis and Operator Theory Chapter | 1

∀(v, t) ∈ V × I, G(v, t) = 0 ⇔ t = g(v),

g (0) = −(Gt (0, 0))

−1

−1

◦ Gv (0, 0) = −(F (u0 )w)

F (u0 )|X0 = 0.

15

(1.21) (1.22)

We define J˜ : V → R by J˜(v) = J (u0 + v + g(v)w), ∀v ∈ V . It turns out that J˜ is of class C 1 on V and attains its minimum at 0 because (1.20), (1.21), (1.16), and (1.19) entail J˜(v) = J (u0 + v + g(v)w) ≥ J (u0 ) = J˜(0), ∀v ∈ V , thereby J˜ (0) = 0. On the other hand, from (1.22) it is seen that J˜ (0) = J (u0 )(idX0 + g (0)w) = J (u0 )|X0 , which renders J (u0 )|X0 = 0. The obtained result reads as ker F (u0 ) ⊂ ker J (u0 ). If x ∈ X, one has x − (F (u0 )x)w ∈ ker F (u0 ) = X0 , so x − (F (u0 )x)w ∈ ker J (u0 ), or J (u0 )x = J (u0 )(w)F (u0 )x, ∀x ∈ X. Consequently, (1.17) holds by choosing λ = J (u0 )(w), thus the proof is complete. We look again at the minimization problem with constraints (1.15) for a function J : X → R on a normed space X, but this time the constraints are given in the form of a system of inequalities. We provide a second version of Lagrange multiplier rule adapted to this type of constraints. Theorem 1.38. Assume that X is a Banach space, J, Fi : X → R are Gâteaux differentiable functions (i = 1, . . . , m) and u ∈ X is locally a solution of the minimization problem inf

Fi (u)≤0, i=1,...,m

J (u)

16 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

such that there exists vu ∈ X satisfying Fi (u) + Fi (u)vu < 0 for all i = 1, . . . , m. Then there exist λ1 , . . . , λm ∈ R called Lagrange multipliers such that m (i) J (u) + λi Fi (u) = 0; (ii) (iii)

m

i=1

λi Fi (u) = 0; i=1 λi ≥ 0 for all i = 1, . . . , m.

Proof. Consider in the product space Rm × R = Rm+1 the subsets A = {(x, s) = (x1 , . . . , xm , s) ∈ Rm × R : xi < 0 for i = 1, . . . , m, s < 0}, B = {(y, t) = (y1 , . . . , ym , t) ∈ Rm × R : there exists v ∈ X with Fi (u) + Fi (u)v ≤ yi for all i = 1, . . . , m, J (u)v ≤ t}. We see that A and B are convex sets, A is open, (0, 0) ∈ B. Let us show that the sets A and B are disjoint. Arguing by contradiction, suppose that there is (x, t) ∈ A ∩ B, which implies the existence of v ∈ X such that Fi (u) + Fi (u)v < 0 for all i = 1, . . . , m, and J (u)v < 0. By the Gâteaux differentiability of the functions Fi (i = 1, . . . , m) and J (see Definition 1.29), we know that Fi (u + τ v) = (1 − τ )Fi (u) + τ (Fi (u) + Fi (u)v + ai (τ )) with ai (τ ) → 0 as τ → 0+ and J (u + τ v) = (1 − τ )J (u) + τ (J (u) + J (u)v + b(τ )) with b(τ ) → 0 as τ → 0+ . We are thus led to Fi (u + τ v) < 0 for τ > 0 small, i = 1, . . . , m and J (u + τ v) < J (u) for τ > 0 small, contradicting the optimality of the solution u. Hence the sets A and B are disjoint.

Elements of Functional Analysis and Operator Theory Chapter | 1

17

Therefore the separation in Theorem 1.4 can be applied, which ensures the existence of (ξ, r) = (ξ1 , . . . , ξm , r) ∈ Rm+1 \ {(0, 0)} and α ∈ R such that m

ξi xi + rs ≤ α ≤

i=1

m

ξi yi + rt

i=1

whenever (x, s) = (x1 , . . . , xm , s) ∈ A and (y, t) = (y1 , . . . , ym , t) ∈ B. If we set (x, s) = (y, t) = (0, 0), it follows that α = 0, so we have m

ξi xi + rs ≤ 0 ≤

i=1

m

ξi yi + rt

(1.23)

i=1

whenever (x, s) = (x1 , . . . , xm , s) ∈ A and (y, t) = (y1 , . . . , ym , t) ∈ B. Inserting (x, s) = (0, −1) ∈ A in (1.23), we find that r ≥ 0. Moreover, for xi = −1, xj = 0 if j = i and s = 0, from (1.23) it follows that ξi ≥ 0 for i = 1, . . . , m.

(1.24)

r > 0.

(1.25)

We claim that

Indeed, if r = 0 then (1.23) becomes m

ξi xi ≤ 0 ≤

i=1

m

ξi yi

(1.26)

i=1

with ξ = 0 because (ξ, r) = (0, 0). On the other hand, the hypothesis regarding vu ∈ X leads to m

ξi (Fi (u) + Fi (u)vu ) < 0.

i=1

Since this contradicts (1.26), the claim in (1.25) is proven. Thanks to (1.25) we may set λi =

ξi for i = 1, . . . , m. r

In view of (1.24), assertion (iii) holds true. Using (1.25), we can express (1.23) in the form m i=1

λi xi + s ≤ 0 ≤

m i=1

λi yi + t

(1.27)

18 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

whenever (x, s) = (x1 , . . . , xm , s) ∈ A and (y, t) = (y1 , . . . , ym , t) ∈ B. Let us insert in (1.27), for every w ∈ X, yi = Fi (u) + Fi (u)w, with i = 1, . . . , m, and t = J (u)w, which is possible in view of the definition of the set B. This yields J (u)w +

m

λi (Fi (u) + Fi (u)w) ≥ 0 for all w ∈ X.

(1.28)

i=1

Taking w = 0 in (1.28) gives m

λi Fi (u) ≥ 0.

i=1

Then by (iii) and since u is admissible for the minimization problem, we infer that assertion (ii) is fulfilled. If we combine (1.28) and (ii), then we derive

J (u)w +

m

λi Fi (u)w ≥ 0 for all w ∈ X.

i=1

Changing w with −w, this is just (i). The proof is thus complete. In the following we review some elements of the subdifferentiability and critical point theory for locally Lipschitz functions that will be required in the sequel. A function ϕ : X → R on a Banach space (X,  · ) is called locally Lipschitz if for every u ∈ X there correspond a neighborhood U of u and a constant L > 0 such that |ϕ(v) − ϕ(w)| ≤ Lv − w,

∀ v, w ∈ U.

The generalized directional derivative of ϕ at u ∈ X along the direction v ∈ X is defined as ϕ 0 (u; v) := lim sup w→u τ →0+

ϕ(w + τ v) − ϕ(w) . τ

The generalized gradient of ϕ at u ∈ X is defined to be the set

∂ϕ(u) := u∗ ∈ X ∗ : u∗ , v ≤ ϕ 0 (u; v) ∀ v ∈ X . The next proposition collects some useful properties of the preceding notions. Proposition 1.39. If ϕ : X → R is locally Lipschitz, then: (i) ∂ϕ(u) is nonempty, convex, closed, and weak∗ compact for all u ∈ X; ∗ (ii) the multifunction ∂ϕ : X → 2X is upper semicontinuous for the weak∗ topology on X ∗ ;

Elements of Functional Analysis and Operator Theory Chapter | 1

19

(iii) for all y, z ∈ X, one has ϕ 0 (y; z) = max{y ∗ , z : y ∗ ∈ ∂ϕ(y)}; (iv) if ϕ ∈ C 1 (X), then ∂ϕ(u) = {ϕ (u)} for all u ∈ X. A key result in this area is Lebourg’s mean value theorem (see [38, Theorem 2.177]). Proposition 1.40. If ϕ : X → R is locally Lipschitz and u, v ∈ X, then there exist τ ∈ (0, 1) and w ∗ ∈ ∂ϕ(τ u + (1 − τ )v) such that ϕ(v) − ϕ(u) = w ∗ , v − u. We turn to a brief introduction to nonsmooth critical point theory for locally Lipschitz functions. In particular, it covers the smooth (i.e., C 1 ) critical point theory. For more details we refer to [51,169]. Definition 1.41. We say that u ∈ X is a critical point of a locally Lipschitz function ϕ : X → R if 0 ∈ ∂ϕ(u). We denote K(ϕ) := {u ∈ X : u is a critical point of ϕ} and, for any c ∈ R, Kc (ϕ) := {u ∈ K(ϕ) : ϕ(u) = c}

and

ϕ c := {u ∈ X : ϕ(u) ≤ c} .

Definition 1.42. A locally Lipschitz function ϕ : X → R is said to satisfy the (nonsmooth) Palais–Smale condition (for short, (P S)) if every sequence {un } in X such that {ϕ(un )} is bounded and ϕ 0 (un ; v − un ) + εn v − un  ≥ 0, ∀ v ∈ X, n ∈ N, with a sequence {εn } ⊂ (0, +∞) such that εn → 0 as n → ∞, has a convergent subsequence in X. The following nonsmooth version of the deformation theorem given in [51] permits the extension of the minimax principles from the smooth critical point theory to the nonsmooth critical point theory. Theorem 1.43. If the locally Lipschitz function ϕ : X → R on the reflexive Banach space X satisfies condition (P S) and c ∈ R verifies Kc (ϕ) = ∅, then for every ε0 > 0 there exist ε ∈ (0, ε0 ) and a homeomorphism η : X → X such that (i) η(u) = u for all u ∈ ϕ c+ε0 \ ϕ c−ε0 ; (ii) η(ϕ c+ε0 ) ⊂ ϕ c−ε0 . The following minimax result is the nonsmooth version of the mountain pass theorem (see [51]).

20 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Theorem 1.44. If the locally Lipschitz function ϕ : X → R on a reflexive Banach space X satisfies condition (P S) and there exist u0 , u1 ∈ X and 0 < r < u0 − u1  such that max{ϕ(u0 ), ϕ(u1 )} < σr :=

inf

u∈∂Br (u0 )

ϕ(u),

then for  := {γ ∈ C([−1, 1], X) : γ (−1) = u0 , γ (1) = u1 } , c := inf max ϕ(γ (t)), γ ∈ t∈[−1,1]

one has c ≥ σr and Kc (ϕ) = ∅. Proof. The inequality c ≥ ηr is the direct consequence of the definition of c because every path in  crosses the sphere ∂Br (u0 ). Now, arguing by contradiction, suppose that Kc (ϕ) = ∅. Setting ε0 = σr − max{ϕ(u0 ), ϕ(u1 )}, we note that the imposed hypothesis ensures ε0 > 0. Then Theorem 1.43 provides a number ε ∈ (0, ε0 ) and a homeomorphism η : X → X satisfying assertions (i) and (ii) in Theorem 1.43. By the definition of c, we can choose γ ∈  such that ϕ(γ (t)) ≤ c + ε, ∀t ∈ [−1, 1]. Since ϕ(u0 ) + ε0 ≤ σr ≤ c and ϕ(u1 ) + ε0 ≤ σr ≤ c, we find from property (i) in Theorem 1.43 that η(u0 ) = u0 and η(u1 ) = u1 . Therefore η(γ (−1)) = u0 and η(γ (1)) = u1 , so η ◦ γ ∈ . Then property (ii) in Theorem 1.43 asserts c ≤ max ϕ(η(γ (t))) ≤ c − ε. t∈[−1,1]

The obtained contradiction completes the proof. We close this section by formulating a nonsmooth version of the second deformation theorem, which is a special case of [94, Theorem 2.1.1]. Theorem 1.45. If the locally Lipschitz function ϕ : X → R satisfies the (P S) condition in Definition 1.42 and there are real numbers a < b such that Ka (ϕ) is a finite set consisting only of local minimizers of ϕ while Kc (ϕ) = ∅ for all c ∈ ]a, b], then there exists a continuous function h : [0, 1] × ϕ b → ϕ b such that

Elements of Functional Analysis and Operator Theory Chapter | 1

21

(i) h(0, u) = u and h(1, u) ∈ Ka (ϕ) for all u ∈ ϕ b ; (ii) ϕ(h(t, u)) ≤ ϕ(u) for all (t, u) ∈ [0, 1] × ϕ b .

1.3 MONOTONE AND PSEUDOMONOTONE OPERATORS First, we discuss significant classes of linear unbounded operators having monotonicity properties. Definition 1.46. A linear operator A : D(A) → X defined on a vector subspace D(A) (called the domain of A) of a Banach space X is said to be unbounded if sup

u∈D(A), uX ≤1

AuX = +∞.

Example 1.47. Let X be the Banach space of continuous, bounded functions f : R → R endowed with the norm f X = sup |f (t)|. t∈R

Let D(A) be the vector subspace of X formed by the differentiable functions f whose derivative f belongs to X and introduce A : D(A) → X by Af = f . Notice that A is unbounded in the sense of Definition 1.46 because, for instance, one has for the functions fn (t) = sin(nt) that fn X = 1 and Afn X = n. Definition 1.48. A linear (possibly unbounded) operator A : D(A) → X on a Banach space X is said to be accretive if u + λAuX ≥ uX , ∀u ∈ D(A), ∀λ > 0.

(1.29)

An accretive operator A : D(A) → X is said to be m-accretive if for every v ∈ X and for every λ > 0 the equation u + λAu = v

(1.30)

has a solution u ∈ D(A) (denoted Jλ v). Lemma 1.49. Assume that the linear operator A : D(A) → X is m-accretive. Then the following is true: (i) The solution u = Jλ v ∈ D(A) of Eq. (1.30) is unique. (ii) The map Jλ = (I + λA)−1 : X → X (called the resolvent of A) is linear and continuous, with Jλ L(X) ≤ 1, where I denotes the identity map. Proof. The assertions are direct consequences of inequality (1.29). Definition 1.50. The Yosida approximation of an m-accretive operator A : D(A) → X on a Banach space X is the linear operator Aλ := AJλ : X → X.

22 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

The following statement lists useful properties of Yosida approximation. Lemma 1.51. Assume that the linear operator A : D(A) → X is m-accretive. Then: 1 (i) Aλ = (I − Jλ ). λ (ii) The linear map Aλ : X → X is continuous, with Aλ L(X) ≤ λ2 . (iii) Aλ |D(A) = Jλ A. Proof. (i) Eq. (1.30) reads as Jλ u + λAJλ u = u, ∀u ∈ X.

(1.31)

Taking into account Definition 1.50, we obtain the given expression of Aλ . (ii) This results easily from part (i) and Lemma 1.49 (ii). (iii) From (1.31) and the linearity of A we derive AJλ u + λA(AJλ u) = Au, ∀u ∈ D(A). In view of the uniqueness in Lemma 1.49 (i), we can infer that Jλ Au = AJλ u for all u ∈ D(A), which completes the proof. The next result describes the approximation of the identity map by the resolvent Jλ and of the linear unbounded operator A by the Yosida approximation Aλ . Proposition 1.52. Assume that the linear operator A : D(A) → X is m-accretive. Then: (i) Jλ u − uX ≤ λAu for all u ∈ D(A). (ii) If D(A) is dense in X, then Jλ u → u in X as λ → 0+ whenever u ∈ X. (iii) If D(A) is dense in X, then Aλ u → Au in X as λ → 0+ whenever u ∈ D(A). (iv) If D(A) is dense in X, then Jλ u → u in D(A) endowed with the graph norm  · D(A) =  · X + A · X

(1.32)

as λ → 0+ whenever u ∈ D(A). Proof. (i) By (1.31), Lemma 1.51 (iii), and Lemma 1.49 (ii), for every u ∈ D(A) we have u − Jλ uX = λJλ AuX ≤ λJλ L(X) AuX ≤ λAuX . (ii) One uses (i) and the density of D(A) in X. (iii) Through (ii) and Lemma 1.51 (iii), for every u ∈ D(A) it turns out that Aλ u − AuX = Jλ (Au) − AuX → 0 as λ → 0+ . (iv) It suffices to combine (1.32), (ii), and (iii).

Elements of Functional Analysis and Operator Theory Chapter | 1

23

In Hilbert spaces we retrieve the linear monotone operators. Proposition 1.53. Assume that X is a Hilbert space with the scalar product (·, ·)X . Then: (i) The linear operator A : D(A) → X is accretive if and only if it is monotone, that is, (Au, u)X ≥ 0, ∀u ∈ D(A).

(1.33)

(ii) If the linear operator A : D(A) → X is m-accretive, then D(A) is dense in X. Proof. (i) Assume that the linear operator A : D(A) → X is accretive. Then, for every u ∈ D(A), (1.29) implies u2X ≤ λu + λAu2X = u2X + 2λ(Au, u)X + λ2 Au2X , which readily leads to (1.33). Conversely, suppose (1.33). Then, for every u ∈ D(A) we are able to get u2X ≤ u2X + λ(Au, u)X = (u + λAu, u)X ≤ uX u + λAuX . Thus we obtain (1.29). (ii) Combining Corollary 1.5 and Theorem 1.23, it suffices to show that (u, v)X = 0 for all v ∈ D(A) ensures u = 0. Indeed, setting v = Jλ (u), we see that (u, Jλ u)X = 0. Then, due to (1.31) and (1.33), it follows that Jλ (u) = 0. Invoking again (1.31), we obtain u = 0 as desired. An important example of m-accretive operator in partial differential equations is provided by the following result. Theorem 1.54. The negative Laplacian operator − : H01 ( ) → H −1 ( ) on an open subset ⊂ RN defined by  (1.34) − u, v = (∇u, ∇v)RN dx for all v ∈ H01 ( )

is m-accretive with a dense domain H01 ( ). Proof. We have to check Definition 1.48 with X = H −1 ( ), D(A) = H01 ( ) and A = − . First, we note that an obvious modification of Corollary 1.25 ensures that Eq. (1.30) has a solution Jλ v ∈ H01 ( ) unique for each v ∈ H −1 ( ) (Corollary 1.25 deals with the case λ = 1). It remains to verify that the linear operator − : H01 ( ) → H −1 ( ) is accretive. To this end we make use of Proposition 1.53 (i) taking advantage that

24 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

H −1 ( ) = H01 ( )∗ is a Hilbert space (see Corollary 1.24). Precisely, we must show that (− u, u)H −1 ( ) ≥ 0, ∀u ∈ H01 ( ).

(1.35)

On the basis of the definition of the scalar product on the dual of a Hilbert space as done in Corollary 1.24 and using (1.11), for every u ∈ H01 ( ) we have (− u, u)H −1 ( ) = (J1 (− u), J1 u)H 1 ( ) = (J1 u, u)H 1 ( ) − J1 u2H 1 ( ) . 0

0

0

(1.36) On the other hand, the definition of the scalar product on H01 ( ), (1.10) and (1.34) entail that (J1 u, u)H 1 ( ) = (J1 u − (J1 u), u)L2 ( ) = u2L2 ( )

(1.37)

0

and in addition ⎞2

⎛ ⎜ J1 u2H 1 ( ) = u2H −1 ( ) = ⎝ 0

⎛ ≤⎝

sup

v∈H01 ( ),

⎟ (u, v)L2 ( ) ⎠

sup

v∈H01 ( ),

vH 1 ( ) ≤1 0

⎞2

(u, v)L2 ( ) ⎠ ≤ u2L2 ( ) .

(1.38)

vL2 ( ) ≤1

From (1.36), (1.37), and (1.38), we see that (1.35) holds true, which ends the proof. We also mention some facts related to pseudomonotone operators. This is a class of nonlinear operators exhibiting a monotonicity property in a generalized sense. Let X be a Banach space and X ∗ its topological dual. By ·, · we denote the duality brackets of the pair (X ∗ , X), by  the weak convergence, while → stands for the strong convergence. A map A : X → X ∗ is said to be pseudomonotone if for every sequence {xn } ⊂ X such that xn  x in X and lim sup A(xn ), xn − x ≤ 0, n→+∞

one has A(x), x − y ≤ lim inf A(xn ), xn − y for all y ∈ X. n→+∞

X∗

A map A : X → is said to satisfy the (S)+ -property if for every sequence {xn } ⊂ X such that xn  x in X and lim sup A(xn ), xn − x ≤ 0, n→+∞

Elements of Functional Analysis and Operator Theory Chapter | 1

25

one has xn → x in X. The fundamental theorem for pseudomonotone operators (see [38, Theorem 2.99]) reads as follows. Theorem 1.55. Let X be a reflexive Banach space, A : X → X ∗ a pseudomonotone, bounded, coercive operator, and b ∈ X ∗ . Then a solution of the equation Au = b exists. Let C01 ( ) = {u ∈ C 1 ( ) : u = 0 on ∂ }. In what follows it is worth to know that the cone of nonnegative functions C01 ( )+ = {u ∈ C01 ( ) : u ≥ 0 in } has a nonempty interior in the Banach space C01 ( ) = {u ∈ C 1 ( ) : u = 0 on ∂ } given by int (C01 ( )+ ) = {u ∈ C01 ( ) : u(x) > 0, ∀x ∈ , and

∂u (x) < 0, ∀x ∈ ∂ }, ∂ν (1.39)

where ν = ν(x) is the outer unit normal at x ∈ ∂ (see [169, p. 220]). Finally, we cite from [169, Proposition 3.5] a result describing basic prop1,p erties of the p-Laplacian on the spaces W 1,p ( ) and W0 ( ) on an open set ⊂ RN . Proposition 1.56. The nonlinear map A : W 1,p ( ) → W 1,p ( )∗ , with 1 < p < +∞, defined by  A(u), v := (|∇u|p−2 (∇u, ∇v)RN + |u|p−2 uv) dx for all u, v ∈ W 1,p ( )

is bounded, continuous, coercive, strictly monotone (so, maximal monotone and pseudomonotone) and satisfies the (S)+ -property. The same properties are sat 1,p isfied by the nonlinear map − p : W0 ( ) → W −1,p ( ), with 1 < p < +∞, defined by  1,p |∇u|p−2 (∇u, ∇v)RN dx for all u, v ∈ W0 ( ). − p u, v :=

We need some facts regarding the spectrum of the operator − p on Consider the nonlinear weighted eigenvalue problem driven by the p-Laplacian:

− p u = λm(x)|u|p−2 u in , (1.40) u=0 on ∂ ,

1,p W0 ( ).

where m ∈ L∞ ( )+ , m = 0. Problem (1.40) admits a (possibly nonexhaustive if p = 2) sequence of eigenvalues 0 < λ1 (m) < λ2 (m) ≤ · · · → +∞ which are known as Lusternik–Schnirelman eigenvalues, where λ1 (m) and λ2 (m) are the first two eigenvalues of problem (1.40).

26 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Proposition 1.57. If m ∈ L∞ ( )+ \ {0}, then λ1 (m) is simple (in the sense that the associated eigenfunctions together with 0 constitute a one-dimensional vector subspace), isolated, and the corresponding eigenfunctions have constant sign in . The eigenfunctions associated with any eigenvalue λ > λ1 (m) are nodal (i.e., sign changing). If m = 1 then λ1 := λ1 (1) admits the following variational characterization: p

λ1 =

inf

1,p

u∈W0 ( )\{0}

∇uLp ( ) p

uLp ( )

,

with the infimum being attained at an eigenfunction which can be chosen such that uˆ 1 ∈ int (C01 ( )+ ) and uˆ 1 p = 1. If m, m ∈ L∞ ( )+ \ {0}, m ≥ m , and m = m , then λ1 (m) < λ1 (m ). Denoting by λ2 := λ2 (1) the second eigenvalue of the operator − p on 1,p W0 ( ), i.e., of (1.40), we quote from [60] a useful variational characterization of λ2 . Proposition 1.58. It holds λ2 = inf

p

max ∇γ (t)Lp ( )

γ ∈0 t∈[−1,1]

  1,p with 0 := γ ∈ C([−1, 1], S) : γ (±1) = ±uˆ 1 , where S := u ∈ W0 ( ) :

1,p uLp ( ) = 1 is endowed with the induced W0 ( )-topology.

1.4 NOTES Section 1.1 is devoted to fundamental facts regarding the functional setting for the entire monograph. The Sobolev spaces that are considered here consist of functions exhibiting derivatives in the weak (or distributional sense). These spaces are formed either of vector-valued functions defined on a real-line interval, which will be employed for evolutionary partial differential equations, or of real-valued functions defined on a domain in some Euclidean space, which will be employed for elliptic partial differential equations. The integrability in the sense of Bochner and weak differentiability are outlined. For a detailed study we refer to [31,32,36,38,115,204]. From the vast theory of Sobolev spaces we selected the essential elements necessary in the sequel. For a variety of topics in this direction we indicate [1,31, 32,38,103,169,228]. Theorem 1.20 can be proven by transporting the problem on RN through the extension operator P : W m,p ( ) → W m,p (RN ), which exists because is supposed to be of class C m . The development of this approach can be found in [31]. Section 1.2 deals with elements of differentiability and subdifferentiability theories, as well as optimization and smooth and nonsmooth critical point theories. For us the very beginning lies in Riesz representation theorem (stated as

Elements of Functional Analysis and Operator Theory Chapter | 1

27

Theorem 1.23). If in Theorem 1.23 one identifies f ≡ uf for every f ∈ H , one obtains the identification H = H ∗ of the Hilbert space H with its dual H ∗ . This is a relevant example of determination of the dual of a Banach space. Another

relevant example is offered by Theorem 1.22 where the dual W −1,p ( ) of the 1,p Sobolev space W0 ( ) is explicitly described. Other important descriptions of

p dual spaces are Lp ( )∗ = Lp ( ), where p = p−1 with 1 < p < +∞, and 1 ∗ ∞ L ( ) = L ( ) (see, e.g., [31]). The differentiability theories that are object of treatment here are those in the sense of Fréchet and Gâteaux, which are both of prime interest. A key result related to differentiability is the implicit function theorem (Theorem 1.36). Important information concerning differentiability is available in [13,116,152]. The differentiability framework is implemented to study optimization. Specifically, we stress the necessary conditions of optimality of Lagrange multiplier rule type. This is set forth in two separate situations, according to description of the set of constraints through equalities or inequalities. Theorem 1.37 formulates the Lagrange multiplier rule for an equality constraint. The extension of Theorem 1.37 allowing to have finitely many equality constraints in the minimization problem (1.15) is as follows (see [116]). Theorem 1.59. Assume that X is a Banach space, F1 , . . . , Fm ∈ C 1 (X) are such that F1 (u), . . . , Fm (u) are linearly independent in X ∗ whenever u ∈ S := {v ∈ X : F1 (v) = . . . = Fm (v) = 0} and J : U → R is a C 1 function on a neighborhood U of S in X. If u0 ∈ S satisfies J (u0 ) = inf J (v), v∈S

then there exist λ1 , . . . , λm ∈ R (called Lagrange multipliers associated to u0 ) such that J (u0 ) = λ1 F1 (u0 ) + · · · + λm Fm (u0 ). Our result regarding Lagrange multiplier rule for minimization with constraints described by inequalities is stated in Theorem 1.38. It is inspired by Ponstein [202]. The final part of Section 1.2 is a brief exposition of critical point theory. The setting is that of nonsmooth critical point theory for locally Lipschitz functions, which incorporates the smooth (i.e., C 1 ) critical point theory. The nonsmooth critical point theory for locally Lipschitz functions is due to Chang [51]. It relies on the notion of generalized gradient, which was introduced by Clarke (see [55]). An essential result for calculus with generalized gradients is given in Proposition 1.40. The nonsmooth deformation result formulated as Theorem 1.43 is Theorem 3.1 in [51]. It represents the core of the construction of

28 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

the critical point theory for locally Lipschitz functionals. A second deformation result is the object of Theorem 1.45. Efficient tools of finding critical points (or, equivalently, solutions of variational inclusions, in particular equations) are the minimax theorems. Amongst them, a leading place is occupied by the famous mountain pass theorem, which is shown in a nonsmooth form for locally Lipschitz functionals in Theorem 1.44. A recent linking theorem for locally Lipschitz functions can be found in [56]. Another important nonsmooth critical point theory was constructed by Szulkin [221] for the class of nonsmooth functionals that can be expressed as the sum of a C 1 function and a convex, lower semicontinuous ≡ +∞ function. A unification of the nonsmooth critical point theories of Chang and Szulkin was achieved in Motreanu–Panagiotopoulos [174, Chapter 3] covering the class of functions that can be expressed as the sum of a locally Lipschitz function and a convex, lower semicontinuous ≡ +∞ function. The aim of Section 1.3 is twofold: on the one hand, to create the passage to evolution problems, and on the other hand, to link with classes of linear and nonlinear operators. In this respect, the central and far reaching notion is that of monotone operator, and the various generalizations, which is at the basis of the study of evolutionary equations and differential inclusions. In order to cast the principal features, we focus on linear, but unbounded operators, with monotonicity property. Namely, we provide a brief treatment of m-accretive operators, with the fundamental example of the negative Laplacian − on H01 ( ). For the general theory of nonlinear maximal monotone operators and applications to evolution problems, we refer to [21,30–32,115,217,228,236,238]. Section 1.3 is also concerned with a powerful generalization of maximal monotone operators defined on the whole space, namely the pseudomonotone operators. The surjectivity result stated as Theorem 1.55 constitutes an efficient tool to prove existence of solutions. For relevant additional information, we address to [38, 169,228,238]. A striking example of pseudomonotone operator is the negative p-Laplacian − p for 1 < p < +∞. The only linear case is for p = 2. The complete description of the spectrum of − p when p = 2 is an open problem. One knows only what is called the beginning of its spectrum, basically the characterization of the first two eigenvalues, which is discussed in Propositions 1.57 and 1.58. For important developments and proofs, we refer to [2,14,38,60,67,69,75,89,131,149, 169,186,211,216].

Chapter 2

Elements of Regularity Theory and Maximum Principle 2.1 PROPERTIES OF THE SOLUTION SET First we present an abstract result from the transversality theory that will be used in the sequel. Let U be an open set of a separable reflexive Banach space X, let E be another separable Banach space, and consider a function I : U × E → R with the properties: (H1 ) There exists the partial derivative Iu : U × E → X ∗ of I with respect to u ∈ U and this is a C r mapping with r ≥ 1; (H2 ) There exists the second order partial derivative Iuu (u, a) : X → X ∗ of I with respect to u ∈ U at any point (u, a) ∈ U × E with Iu (u, a) = 0

(2.1)

and it is a Fredholm operator of index zero, that is, dim ker Iuu (u, a) = codim Im Iuu (u, a) < +∞. Lemma 2.1. Assume that I : U × E → R verifies hypotheses (H1 ) and (H2 ). Then the next conditions are equivalent: (H3 ) For every point (u, a) ∈ U × E satisfying (2.1), the kernel ker(Iu ) (u, a) splits X × E, and the kernels of the linear operators given by the second order partial derivatives Iuu (u, a) : X → X ∗ and Iua (u, a) : X → E ∗ fulfill ker Iuu (u, a) ∩ ker Iua (u, a) = {0};

(2.2)

(H3 ) 0 ∈ X ∗ is a regular value of Iu : U × E → X ∗ , i.e., for each (u, a) ∈ U × E solving (2.1), the derivative (Iu ) (u, a) : X × E → X ∗ is surjective and its kernel splits. Proof. If (u, a) satisfies (2.1), then (Iu ) (u, a) is surjective if and only if for every v ∈ X ∗ there is w ∈ E such that v − Iua (u, a)(·, w) ∈ Im (Iuu (u, a)).

(2.3)

Nonlinear Differential Problems with Smooth and Nonsmooth Constraints. DOI: 10.1016/B978-0-12-813386-6.00002-X Copyright © 2018 Elsevier Inc. All rights reserved.

29

30 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

The range of the self-adjoint operator Iuu (u, a) : X → X ∗ is determined by Im (Iuu (u, a)) = Im (Iuu (u, a)) = {z ∈ X ∗ : z(x) = 0 for all x ∈ ker Iuu (u, a)}.

(2.4)

The first equality in (2.4) is valid because Iuu (u, a) is a Fredholm operator by hypothesis (H2 ), while the second equality is a classical orthogonality relation (see, e.g., [31, Corollary 2.18]). By assumption (H2 ), ker Iuu (u, a) possesses a   finite basis {ei }m i=1 . Then, from (2.3) and (2.4), it turns out that (Iu ) (u, a) is surjective if and only if for every v ∈ X ∗ , there is an element w ∈ E such that Iua (u, a)(ei , w) = v(ei ) for all i ∈ {1, . . . , m}. The last equality is equivalent to the surjectivity of the linear operator w ∈ E → (Iua (u, a)(ei , w))1≤i≤m ∈ Rm , which in turn is equivalent to the nonexistence of a nonzero vector (α1 , . . . , αm ) ∈ Rm with m 

αi Iua (u, a)(ei , w) = 0 for all w ∈ E

(2.5)

i=1

(see Corollary 1.5). We have thus shown the equivalence between the surjectivity of (Iu ) (u, a) and the linear independence of the linear continuous forms on E {Iua (u, a)(ei , ·)}m i=1 .

(2.6)

Now, writing (2.5) as m 

Iua (u, a)

 αi ei , w = 0 for all w ∈ E,

i=1

the independence of the forms in (2.6) is seen to be valid if and only if property (2.2) holds true. The equivalence between (H3 ) and (H3 ) is proven. We recall a few things that are necessary in the theorem below. A residual set in a metric space means a countable intersection of open dense subsets. The Baire’s category theorem ensures that in a complete metric space any residual set is dense. A critical point u ∈ X of a C 1 function f : X → R on a Banach space X is a point at which the differential vanishes, i.e., f  (u) = 0 (see Definition 1.41). A critical point u ∈ X of a C 2 function f : X → R is called nondegenerate if the second derivative f  (u) : X → X ∗ is an isomorphism. A regular value of a C 1 map f : X → Y between Banach spaces X and

Elements of Regularity Theory and Maximum Principle Chapter | 2

31

Y is a point y ∈ Y such that y does not belong to the range of f or for every x ∈ X with f (x) = y one has that the derivative f  (x) : X → Y is surjective and its kernel splits X. A C 1 map f : X → Y between Banach spaces X and Y is called a Fredholm operator if the derivative f  (x) : X → Y at each x ∈ X is a linear Fredholm operator, that is, the kernel is finite-dimensional and the range is finite-codimensional. It was shown in [219] that if f : X → Y is a Fredholm operator, then its index exists being defined by ind f = dim ker f  (x) − codim Im f  (x) independently of x ∈ X. A fundamental result is the Sard–Smale theorem in [219] (see also [239, pp. 829–830]), which asserts that if a C k map f : X → Y between Banach spaces X and Y is a Fredholm operator with k > max{ind f, 0}, then the set of regular values of f is residual in Y . The next theorem provides generically essential properties of the solution set for problems in variational form (in such a case, the weak solutions coincide with the critical points of the associated Euler functionals). Theorem 2.2. Let U be an open subset of a separable Hilbert space X, let E be a separable Banach space, and let I : U × E → R be a function satisfying assumptions (H1 ), (H2 ), (H3 ). Then G = {a ∈ E : I (·, a) : X → R has only nondegenerate critical points} (2.7) is a residual set in E, hence dense in E. Moreover, if in addition it holds (H4 ) If (un , an ) ∈ U × G satisfies (2.1) for all n and if an → a ∈ G, then {un } has a convergent subsequence in U , then the critical points of I (·, a) are finitely many and depend smoothly on the parameter a ∈ G in the following sense: for every connected component G0 of G, there exists a finite collection {gi }1≤i≤nG0 of C r mappings defined in the neighborhood of G0 in E and taking values in U such that {u ∈ U : Iu (u, a) = 0} = {gi (a)}1≤i≤nG0

for all a ∈ G0 .

(2.8)

Moreover, gi (a) = gj (a) for all a ∈ G0 , i = j . In particular, the critical points of I (·, a) are finitely many and their number is equal to nG0 which is constant whenever a ∈ G0 . Proof. Lemma 2.1 ensures that M := (Iu )−1 (0) is a C 1 submanifold of U × E. Its tangent space at a point (u, a) ∈ M is expressed as T(u,a) M = Iu (u, a)−1 (0).

(2.9)

32 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Denote by P : M → E the restriction to M of the projection of U × E onto E, that is, P (u, a) = a

for all (u, a) ∈ M.

From (2.9) we can infer the equivalence a ∈ E is a regular value of P : M → E ⇐⇒ 0 ∈ X ∗ is a regular value of Iu (·, a) : U → X ∗ .

(2.10)

Hypothesis (H2 ) leads to P : M → E is a Fredholm mapping of index zero.

(2.11)

Assertion (2.11) allows us to apply the Sard–Smale theorem to the C 1 mapping P : M → E, which is possible because index P = 0 < 1. We infer that G = {a ∈ E : a is a regular value of P : M → E}

(2.12)

is a residual set in the Banach space E. The Baire’s category theorem yields the density of the set G in E. Now, on the basis of (2.10) and assumption (H3 ), we can prove that the residual set G in (2.12) coincides with the set introduced in (2.7), which establishes the first part of Theorem 2.2. For the second part of Theorem 2.2, we note from (2.12) that {x ∈ U : (x, a) ∈ M}, when a ∈ G, forms a manifold of dimension equal to index P = 0, so the critical points of I (·, a), for a ∈ G, are isolated. Fix a ∈ G and consider the set of critical points Xa := {u ∈ X : Iu (u, a) = 0} of I (·, a). We have just shown that Xa is a discrete space. Let us prove that the set Xa is compact. If {un } is a sequence with un ∈ Xa , we can apply assumption (H4 ) with an = a for all n, so {un } contains a strongly convergent subsequence. Therefore Xa is discrete and compact, thus it is a finite set. For the final part of the theorem, fix a connected component G0 of G. Claim 1. For every (u, a) ∈ U × G0 satisfying (2.1), there is an open subset Vu,a ⊂ E containing G0 and a C r mapping gu,a : Vu,a → U satisfying (i) gu,a (a) = u ; (ii) gu,a (y) is a critical point of I (·, y) for all y ∈ Vu,a ; (iii) for every y ∈ Vu,a , there are neighborhoods Vy ⊂ Vu,a of y and Wy ⊂ U of gu,a (y) such that gu,a (z) is the unique critical point of I (·, z) in Wy for all z ∈ Vy . By (2.6), we know that Iuu (u, a) : X → X is a linear isomorphism. The implicit function theorem (see Theorem 1.36) yields the existence of a maximal open subset Vu,a ⊂ E containing a and a C r mapping gu,a : Vu,a → U satisfying (i)–(iii). It remains to verify that Vu,a contains G0 . In view of the connectedness of G0 , it suffices to check that Vu,a ∩ G0 is closed in G0 . If

Elements of Regularity Theory and Maximum Principle Chapter | 2

33

a0 ∈ Vu,a ∩ G0 , there is {an } ⊂ Vu,a ∩ G0 such that an → a0 in E, and set un = gu,a (an ). By (H4 ), up to a subsequence, we may assume that un → u0 for some u0 ∈ U , thus Iu (u0 , a0 ) = 0. Since a0 ∈ G0 , the implicit function theorem yields neighborhoods V0 ⊂ E of a0 , W0 ⊂ U of u0 , and a C r mapping g0 : V0 → W0 such that g0 (y) is the unique critical point of I (·, y) in W0 whenever y ∈ V0 . Up to considering V0 smaller if necessary, we may assume that Vu,a ∩ V0 is connected. Using property (iii) of the mapping gu,a , we see that the set {y ∈ Vu,a ∩ V0 : g0 (y) = gu,a (y)} is open and closed in Vu,a ∩ V0 , and nonempty (since it contains an for n large enough), whence g0 |Vu,a ∩V0 = gu,a |Vu,a ∩V0 . Then the maximality of Vu,a implies a0 ∈ Vu,a , so Claim 1 holds true. Claim 2. If (u, a), (u , a  ) ∈ U × G0 satisfy (2.1), then we have either gu,a = gu ,a  on G0 or gu,a (y) = gu ,a  (y) for all y ∈ G0 . Conditions (i)–(iii) of Claim 1 imply that the set {y ∈ G0 : gu,a (y) = gu ,a  (y)} is open and closed in G0 . Claim 2 then follows from the connectedness of the set G0 . Set {gi }i∈J := {gu,a : (u, a) ∈ U × E satisfy (2.1)}. By Claim 2 we know that gi = gj whenever i = j , whereas by the finiteness of the solutions of (2.1) (with a fixed a) we have that J is finite. Claim 3. For every a ∈ G0 , the map J → {u ∈ U : Iu (u, a) = 0}, i → gi (a) is bijective. The injectivity follows from Claim 2. Moreover, given u ∈ U a critical point of I (·, a), we get gu,a (a) = u (by Claim 1). Choosing i ∈ J such that gu,a = gi , we note that u = gi (a), whence the surjectivity. Claim 3 is verified. The construction of the finite family {gi } of C r mappings on the connected component G0 of G as required in (2.8) is completed. We produce two applications of Theorem 2.2. The first one concerns the homoclinic solutions. 2 Let be given a mapping L ∈ C(R, RN ) such that, for t ∈ R, L(t) is a symmetric matrix, which is bounded and positive definite on RN uniformly on R, that is, |L(t)| ≤ c0 and L(t)u · u ≥ c1 |u|2

for all u ∈ RN and all t ∈ R,

with constants c0 , c1 > 0. Hereafter the symbol | · | stands for the Euclidean norm in any space RM (for M ≥ 1) and “ · ” stands for the corresponding scalar product (also denoted by (·, ·)RM ). We are also given a potential V : R × RN → R and a function f : R → RN . With these data, we consider the second-order

34 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Hamiltonian system ⎧ ⎪ ⎨ −q¨ + L(t)q = Vx (t, q) + f (t) for all t ∈ R, q(t) → 0 as t → ±∞, ⎪ ⎩ q(t) ˙ → 0 as t → ±∞,

(2.13)

where Vx denotes the partial derivative of V (t, x) with respect to the second variable x ∈ RN . The solutions of (2.13) are called homoclinics to 0. Recall from Definition 1.9 and Theorem 1.10 that X = H 1 (R, RN ) consists of the absolutely continuous functions from R to RN whose derivative defined almost everywhere is square integrable. The separable Hilbert space X is endowed with the scalar product (·, ·)X introduced as +∞ (q, u)X = (q˙ · u˙ + L(t)q · u) dt for all q, u ∈ X. −∞

The associated norm on X is denoted by  · X . Let us set Y := L2 (R, RN ), which is a separable Hilbert space endowed with the usual norm (see Definition 1.7). We suppose that the potential V : R × RN → R in (2.13) is a function (t, x) → V (t, x) for which V , Vx , Vxx are continuous, Vxx bounded, L(t)x · x − Vx (t, x) · x ≥ α|x|2 ,

∀t ∈ R, ∀x ∈ RN ,

(2.14)

with a constant α > 0, and V (t, 0) = 0,

Vx (t, 0) = 0, lim

|t|→+∞ x→0

Vxx (t, 0) = 0,

Vxx (t, x) = 0.

∀t ∈ R, (2.15)

Example 2.3. A simple example of mappings L(t) and V (t, x) satisfying all 2 our assumptions is L(t)x = x and V (t, x) = − 14 e−t arctan(|x|4 ) for all t ∈ R and x ∈ RN . Our result on problem (2.13) ensures that generically the solution set consists of finitely many homoclinic solutions. It also provides that these solutions are nondegenerate, they depend smoothly on f ∈ L2 (R, RN ), and the famous hypothesis (∗) in the theory of homoclinics (see, e.g., [35]) is generically satisfied for problem (2.13). Theorem 2.4. Under the previous conditions for L(t) and V (t, x), there exists a dense subset A in Y such that for every f ∈ A problem (2.13) has finitely many nondegenerate homoclinic solutions (in the sense that they are nondegenerate critical points of the Euler functional corresponding to (2.13)). Moreover, the set of solutions to (2.13) depends smoothly on f ∈ A in the following sense: for

Elements of Regularity Theory and Maximum Principle Chapter | 2

35

every connected component A0 of A, there is a finite collection {gi }1≤i≤nA0 of C 1 mappings defined in the neighborhood of A0 in Y and taking values in X such that for any f ∈ A0 the solutions of (2.13) are exactly {gi (f )}1≤i≤nA0 . The number nA0 of solutions to (2.13) is constant whenever f ∈ A0 . Proof. The idea is to apply Theorem 2.2. To this end, we define the function I : X × Y → R by I (q, f ) =

|q| ˙ 2 + L(t)q · q − V (t, q) − f · q dt

+∞ 1 

2 −∞ for all (q, f ) ∈ X × Y,

(2.16)

that is, I (·, f ) is the Euler functional associated to problem (2.13) whenever f ∈ Y . The function I in (2.16) is well defined because, by using the mean value theorem twice, the continuous embedding H 1 (R, RN ) ⊂ L2 (R, RN ), and the boundedness of Vxx , one finds a constant C > 0 such that   +∞   V (t, q) dt  ≤ sup |Vxx (t, x)| q2Y ≤ C q2X for all q ∈ X.  −∞

(t,x)∈R×RN

Let us verify condition (H1 ) for U = X and E = Y . Fix q ∈ X, f ∈ Y , and ε > 0. Using again the mean value theorem for V (t, ·) and Vx (t, ·), we have for each ϕ ∈ X, with some λ = λ(t) ∈ [0, 1], the inequalities   +∞   (V (t, q + ϕ) − V (t, q) − Vx (t, q) · ϕ) dt   −∞ +∞ |Vx (t, q + λϕ) − Vx (t, q)| |ϕ| dt ≤ −∞

≤

sup (t,x)∈R×RN

|Vxx (t, x)| ϕ2Y ≤ Cϕ2X .

It turns out that I (·, f ) is Fréchet differentiable and its differential is expressed as +∞  q˙ · ϕ˙ + L(t)q · ϕ − Vx (t, q) · ϕ − f · ϕ dt for all ϕ ∈ X. Iq (q, f )ϕ = −∞

(2.17) In order to establish the continuity of the differential Iq : X × Y → X ∗ , let qn → q in X and fn → f in Y as n → ∞. Then, through the mean value theorem and Cauchy–Schwarz inequality, one obtains   sup 

ϕX ≤1

+∞

−∞

  (Vx (t, qn ) · ϕ − Vx (t, q) · ϕ) dt 

36 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

≤

sup

+∞

sup

ϕX ≤1 −∞ (t,x)∈R×RN

|Vxx (t, x)| |qn − q| |ϕ| dt ≤ C qn − qX ,

with a constant C > 0. Then, by (2.17), the continuity of Iq follows. We show the existence of the second order partial Fréchet derivative Iqq of the functional I defined by (2.16). Let ε > 0 and (q, f ) ∈ X × Y be fixed. From (2.15), one finds r > 0 and δ > 0 to fulfill |Vxx (t, x)| < ε

whenever |t| > r and |x| < δ.

(2.18)

Since q belongs to X = H 1 (R, RN ), there is r > 0 large enough so that |q(t)|
r

(2.19)

(see Corollary 1.11). It is well known that the mapping r q ∈ X → Vx (t, q) dt ∈ RN −r

is Fréchet differentiable. Thus, there exists η > 0 with the property  r     Vx (t, q + ψ) · ϕ − Vx (t, q) · ϕ − (Vxx (t, q) · ϕ) · ψ dt  ≤ εϕX ψX  −r

(2.20) for all ϕ, ψ ∈ X with ψX ≤ η. Due to the continuous embedding X ⊂ L∞ (R, RN ) (see Corollary 1.11), for η > 0 smaller if necessary, we may assume that ψL∞ (R,RN ) ≤ 2δ whenever ψX ≤ η. Then the mean value theorem, Cauchy–Schwarz inequality, and (2.18), (2.19) yield     (Vx (t, q + ψ) − Vx (t, q)) · ϕ dt  ≤ sup |Vxx (t, x)| ϕY ψY (2.21)  |t|>r

|t|>r |x| 0. Again by (2.18), (2.19), we obtain     (Vxx (t, q) · ϕ) · ψ dt  ≤ εCϕX ψX for all ϕ, ψ ∈ X. (2.22)  |t|>r

Combining (2.20)–(2.22), it is seen that  +∞    Vx (t, q + ψ) · ϕ − Vx (t, q)ϕ − (Vxx (t, q) · ϕ) · ψ dt   −∞

≤ ε(1 + 2C)ϕX ψX

Elements of Regularity Theory and Maximum Principle Chapter | 2

37

for all ϕ ∈ X and ψ ∈ X with ψX ≤ η. Therefore, there exists the second order partial derivative Iqq (q, f ), which is equal to Iqq (q, f )(ϕ, ψ) =

+∞ 

−∞

ϕ˙ · ψ˙ + L(t)ϕ · ψ − (Vxx (t, q) · ϕ) · ψ dt

(2.23)

for all ϕ, ψ ∈ X. We claim that Iqq is continuous on X × Y . Indeed, let qn → q in X and fn → f in Y as n → ∞. Fix an ε > 0, and let r > 0, δ > 0 satisfy (2.18) and (2.19). Since q ∈ L∞ (R, RN ) and {qn } converges uniformly to q, there is an r > 0 such that |qn (t)| < δ whenever |t| > r and n is large enough.

(2.24)

The uniform continuity of Vxx on [−r, r] × N , for a compact neighborhood N of q([−r, r]) in RN , and the convergence qn → q in L∞ (R, RN ) imply |Vxx (t, qn (t)) − Vxx (t, q(t))| ≤ ε, ∀t ∈ [−r, r], for n large enough.

(2.25)

By (2.18), (2.19), (2.24), (2.25), we infer that |Vxx (t, qn (t)) − Vxx (t, q(t))| ≤ 2ε

for all t ∈ R,

with any sufficiently large n. This results in  +∞     sup  Vxx (t, qn (t)) − Vxx (t, q(t)) · ϕ(t) · ψ(t) dt  ≤ 2ε, ϕX ≤1 ψX ≤1

−∞

with any large number n. Then from (2.23), the continuity of Iqq is proven. Clearly, the second order partial derivative Iqf (q, f ) exists and is given by Iqf (q, f )(ϕ, g) = −

+∞

−∞

g · ϕ dt

for all f, g ∈ Y, all q, ϕ ∈ X.

(2.26)

The continuity of Iqf on X × Y follows readily from (2.26). According to what was said before, it is true that Iq ∈ C 1 (X × Y, X ∗ ).

(2.27)

By (2.27) we know that condition (H1 ) is satisfied. The next step in the proof is to check condition (H2 ). Let q ∈ X, f ∈ Y and ε > 0. Choose r > 0 and δ > 0 such that (2.18) and (2.19) hold true. Take a bounded sequence {ϕn } in X. The embedding H 1 ([−r, r], RN ) ⊂ L2 ([−r, r], RN )

38 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

is compact, so we may assume that along a relabeled subsequence {ϕn } converges in L2 ([−r, r], RN ).

(2.28)

On the other hand, by (2.18), (2.19), and the boundedness of the sequence {ϕn } in X, it follows that  +∞    sup  (Vxx (t, q) · (ϕn − ϕm )) · ψ dt  (2.29) ψX ≤1

≤



−∞

sup

 |Vxx (t, x)| sup

r

ψX ≤1 −r

(t,x)∈R×RN

|ϕn − ϕm | |ψ| dt + Cε, ∀n, m,

with a constant C > 0. By (2.28) and (2.29), we deduce that along a subsequence   +∞ Vxx (t, q) · ϕn dt −∞

is a Cauchy sequence in X ∗ , so strongly convergent in X ∗ . Hence the operator +∞ ϕ ∈ X → Vxx (t, q) · ϕ dt ∈ X ∗ −∞

is compact. Therefore, by (2.23), Iqq (q, f ) as a linear operator from X to X ∗ is the sum of an isomorphism and a compact map. This implies that Iqq (q, f ) is a Fredholm operator of index zero (see, e.g., [31, p. 169]). Therefore assumption (H2 ) is verified. Now we prove that condition (H3 ) holds. The kernel ker(Iq ) (q, f ) splits in X × Y because X and Y are Hilbert spaces. To verify the second part of (H3 ), we show that ker Iqf (q, f ) = {0} for all (q, f ) ∈ X × Y . Let (q, f ) ∈ X × Y and ϕ ∈ ker Iqf (q, f ).

(2.30)

From (2.26) and (2.30) we derive that +∞ g · ϕ dt = 0 for all g ∈ Y, −∞

which directly yields ϕ = 0, whence (H3 ). Consequently, we are allowed to apply Theorem 2.2 to the functional I on X × Y . A dense subset A in Y is found by setting A = G, with G as in (2.7). Hence, for every f ∈ A, the functional I (·, f ) : X → R defined in (2.16) has only nondegenerate critical points, equivalently nondegenerate solutions to problem (2.13). Thus, for every f ∈ A, problem (2.13) possesses an at most countable discrete set of homoclinic solutions, which establishes the first part of Theorem 2.4.

Elements of Regularity Theory and Maximum Principle Chapter | 2

39

Now we show that condition (H4 ) is fulfilled. Let (qn , fn ) ∈ X × A be such that Iq (qn , fn ) = 0 for all n, and fn → f ∈ A. The equality Iq (qn , fn ) = 0 reads as +∞  q˙n · ϕ˙ + L(t)qn · ϕ − Vx (t, qn ) · ϕ − fn · ϕ dt = 0 for all ϕ ∈ X. −∞

(2.31) Setting ϕ = qn in (2.31), by assumption (2.14) we can see that the sequence {qn } is bounded in X. Then along a relabeled subsequence one has that qn q in X = H 1 (R, RN ) and, by the compactness of the embedding H 1 (R, RN ) ⊂ L2 (R, RN ), that qn → q in L2 (R, RN ) for some q ∈ H 1 (R, RN ). Choosing ϕ = qn − q in (2.31) ensures lim qn 2X = q2X .

n→∞

Since qn q and X is a Hilbert space, we conclude that qn → q in X, which proves condition (H4 ). Finally, we obtain the second part of Theorem 2.4 from Theorem 2.2, which gives that for every f ∈ A the number of solutions of system (2.13) is finite. It is constant, say equal to nA0 , on a connected component A0 of A. Moreover, Theorem 2.2 provides the description of the solution set of (2.13) with a = f ∈ A0 in the form {gi (f )}1≤i≤nA0 , where gi , 1 ≤ i ≤ nA0 , are C 1 mappings on a neighborhood of A0 in A. This completes the proof. Our second application of Theorem 2.2 is to give verifiable criteria for obtaining a Morse function whose critical points be exactly the solutions of the semilinear elliptic boundary value problem  − u = p(x, u) + f (x) in , (2.32) u=0 on ∂, for a bounded domain  in RN with a C 2 boundary ∂. Here f ∈ L2 () is regarded as a parameter, whilst p :  × R → R is given. The following theorem addresses this question. Theorem 2.5. (a) Assume that the function p :  × R → R in (2.32) satisfies the following conditions: (i) p ∈ C 1 ( × R); (ii) there exist constants c1 , c2 ≥ 0 such that the partial derivative pt (x, t) satisfies |pt (x, t)| ≤ c1 + c2 |t|s−1 for all (x, t) ∈  × R, with some s ∈ (1, N+2 N−2 ).

40 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Then the set G = {f ∈ L2 () : I (·, f ) : H01 () → R has only nondegenerate

(2.33)

critical points}, defined with the function I : H01 () × L2 () → R expressed as   1 |∇u|2 − P (x, u) − f u dx for all (u, f ) ∈ H01 () × L2 (), I (u, f ) =  2 (2.34) where ∇u stands for the gradient of u, is residual, hence dense, in L2 (). (b) If we further suppose that the function p :  × R → R is bounded, then the residual set G in (2.33) consists of functions possessing only finitely many critical points. (c) If in addition to hypotheses (i) and (ii) we assume the Ambrosetti– Rabinowitz condition (AR) there are constants μ > 2, q ≥ 0, and M ∈ R such that p(x, t)t − μP (x, t) ≥ M for all (x, t) ∈  × R with |t| ≥ q, where P (x, t) :=

t 0

p(x, τ ) dτ , then the residual set G in (2.33) coincides with

G = {f ∈ L2 () : I (·, f ) : H01 () → R is a Morse function}.

(2.35)

Proof. (a) Assumptions (i) and (ii) imply that the function I introduced in (2.34) is differentiable of class C 2 (see [207, p. 94]). Consequently, condition (H1 ) of Theorem 2.2 is verified. The first order partial derivative Iu (u, f ) at any point (u, f ) ∈ H01 () × L2 () has the expression Iu (u, f )(v) = ((∇u, ∇v)RN − p(x, u)v − f v) dx for all v ∈ H01 (). 

(2.36) Hence the critical points of I (·, f ) coincide with the weak solutions of (2.32). Differentiating in (2.36) with respect to u, one obtains Iuu (u, f )(v, w) = ((∇v, ∇w)RN − pt (x, u)vw) dx for all v, w ∈ H01 (). 

(2.37) We claim that the continuous linear operator K : H01 () → H01 () given by pt (x, u)vw dx for all v, w ∈ H01 () (Kv, w)H 1 () = 0



Elements of Regularity Theory and Maximum Principle Chapter | 2

41

is compact for each weak solution u of problem (2.32). By the assumption imposed on the exponent s in hypothesis (ii), we have s
0 are constants. It turns out that {Kvn } contains a convergent subsequence in H01 (), thus the compactness of K ensues. Writing (2.37) in the form Iuu (u, f ) = idH 1 () − K, 0

(2.38)

it follows from (2.38) and the compactness of K that Iuu (u, f ) : H01 () → H01 () is a Fredholm operator of index zero, whence condition (H2 ) is satisfied. Differentiating in (2.36) with respect to f (and identifying Iuf and If u ) we find for the linear operator Iuf (u, f ) : H01 () → L2 () that Iuf (u, f )(v, h) = − hv dx for all (v, h) ∈ H01 () × L2 (). 

42 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Hence ker Iuf (u, f ) = 0, which shows that condition (H3 ) holds. Theorem 2.2 applies in the case of the function I of (2.34) enabling us to conclude part (a) of the statement of the theorem. (b) We need to show equality (2.35). To this end, we have to verify hypothesis (H4 ) in Theorem 2.2 for the function I : H01 () × L2 () → R. Let (un , fn ) ∈ H01 () × G, with G in (2.7), satisfy Iu (un , fn ) = 0 for all n, and fn → f ∈ G, that is, the convergence in L2 (). Therefore we have ((∇un , ∇v)RN − p(x, un )v − fn v) dx = 0 for all v ∈ H01 (). (2.39) 

Setting v = un , since the function p was supposed to be bounded, the sequence {un } is bounded in H01 (). Then along a relabeled subsequence one has that un u in H01 () and, by the compactness of the embedding H01 () ⊂ L2 (), that un → u in L2 () for some u ∈ H01 (). Inserting ϕ = un − u in (2.39) ensures lim un 2H 1 () = u2H 1 () ,

n→∞

0

0

which proves that (H4 ) holds. so un → u in (c) From assumption (AR) we derive that the functional I (·, f ) : H01 () → R satisfies the Palais–Smale condition for every f ∈ L2 (), that is, if {I (un , f )} is bounded and Iu (un , f ) → 0, then the sequence {un } is strongly convergent in H01 (). For such a reasoning we refer to [207, pp. 10–11]. Consequently, (2.33) implies (2.35), which completes the proof. H01 (),

2.2 REGULARITY THEORY FOR NONLINEAR ELLIPTIC EQUATIONS In this section, we consider regularity results for the weak solutions of certain nonlinear elliptic problems, which include as a particular case problems driven by the p-Laplacian. Such regularity results are established in two steps. First, we prove that the weak solution is in L∞ () and then using this boundedness property, we show the Hölder continuous differentiability up to the boundary for the weak solutions. We do this for the Neumann problem which is more involved. Similar, and in fact simpler, proofs hold for the Dirichlet case. Let  be a bounded domain in RN with a C 2 boundary ∂. We consider the following nonlinear elliptic problem:  −div a(x, ∇u(x)) = f (x, u(x)) in , (2.40) ∂u on ∂. ∂na = 0

Elements of Regularity Theory and Maximum Principle Chapter | 2

43

Here a :  × RN → RN is a continuous map such that, for every x ∈ , a(x, ·) is strictly monotone on RN and continuously differentiable on RN \ {0}. The hypotheses allow in the map div a(x, ∇u(x)) be incorporated as a special case the p-Laplacian p defined by

p u = div (|∇u(x)|p−2 ∇u(x)) for all u ∈ W 1,p (). However, in contrast to the p-Laplacian, the differential operator in (2.40) need not be (p − 1)-homogeneous. Also, f (x, s) is a Carathéodory function (i.e., for all s ∈ R, x → f (x, s) is measurable on  and, for a.a. x ∈ , s → f (x, s) ∂u (x) = is continuous) with subcritical growth in the second variable. Also, ∂n a (a(x, ∇u(x)), ν(x))RN for x ∈ ∂, where ν(x) is the outward unit normal on the boundary ∂. The precise hypotheses on the maps a(x, y) and f (x, s) are the following: H(a) a(x, y) = h(x, |y|)y for all (x, y) ∈  × RN , where h(x, t) > 0 for all (x, t) ∈  × (0, +∞), and (i) a ∈ C( × RN , RN ) ∩ C 1 ( × (RN \ {0}), RN ); (ii) there exist c1 > 0 and 1 < p < +∞ such that Dy a(x, y) ≤ c1 |y|p−2 for every x ∈  and y ∈ RN \ {0}; (iii) there exists c0 > 0 such that (Dy a(x, y)ξ, ξ )RN ≥ c0 |y|p−2 |ξ |2 for every x ∈  and y ∈ RN \ {0}, ξ ∈ RN ; (iv) there exist c2 > 0 such that Dx a(x, y) ≤ c2 (1 + |y|p−1 ) for every x ∈  and y ∈ RN \ {0}. In what follows we consider the function G(x, y) defined by ∇y G(x, y) = a(x, y) for all (x, y) ∈  × RN , and G(x, 0) = 0 for all x ∈  (the potential which corresponds to a(x, y)). Remark 2.6. Let g(x, t) = h(x, t)t for all x ∈  and all t ≥ 0. Hypotheses H(a) imply the one-dimensional estimate c0 t p−2 ≤ gt (x, t) ≤ c1 t p−2 for all x ∈ , t > 0. t We set G0 (x, t) = 0 g(x, s) ds for all (x, t) ∈  × [0, +∞). Clearly, G0 (x, ·) is strictly convex and increasing on [0, +∞). Let G(x, y) = G0 (x, |y|). Then G(x, 0) = 0, G(x, ·) is strictly convex for all x ∈ , and ∇y G(x, y) = (G0 )t (x, |y|)

y y = g(x, |y|) = h(x, |y|)y = a(x, y) |y| |y|

44 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

for all (x, y) ∈  × (RN \ {0}). Therefore, the potential G is uniquely defined. Since G(x, ·) is convex and ∇y G(x, y) = a(x, y) for all (x, y) ∈  × RN , it turns out that (a(x, y), y)RN ≥ G(x, y) for all (x, y) ∈  × RN .

(2.41)

Using H(a) and (2.41), we can easily establish the next result. Lemma 2.7. If hypotheses H(a) (i)–(iii) hold, then (a) for all x ∈ , y → a(x, y) is maximal monotone and strictly monotone; c1 |y|p−1 ; (b) for all (x, y) ∈  × RN , |a(x, y)| ≤ p−1 c0 (c) for all (x, y) ∈  × RN , (a(x, y), y)RN ≥ |y|p . p−1 An immediate consequence of Lemma 2.7 is the following estimate for G. Corollary 2.8. If H(a) (i)–(iii) hold, then for all (x, y) ∈  × RN we have c0 c1 |y|p ≤ G(x, y) ≤ |y|p . p(p − 1) p(p − 1) Example 2.9. Many interesting operators fit the setting of hypotheses H(a). Given θ ∈ C 1 () and θ (x) > 0 for all x ∈ , the following mappings satisfy hypotheses H(a): (e1 ) a(x, y) = θ (x)|y|p−2 y, for p > 1 (corresponds to the weighted p-Laplacian); (e2 ) a(x, y) = θ (x)[|y|p−2 y + ln(1 + |y|p−2 )y], p ≥ 2; (e3 ) for 1 < τ ≤ p ≤ q and τ = 2, ⎧   ⎨ θ (x) |y|p−2 y + |y|q−2 y if |y| ≤ 1,

a(x, y) = ⎩ θ (x) |y|p−2 y + q−2 |y|τ −2 y − q−τ y if |y| > 1; τ −2 τ −2

p−2 y (e4 ) a(x, y) = θ (x) |y|p−2 y + c |y| 1+|y|p (which corresponds to the weighted sum of the p-Laplacian and a generalized mean curvature operator), with 4p 0 < c < 4p(p − 1) if 1 < p < 2 and 0 < c < (p−1) 2 if p ≥ 2. The hypotheses on f (x, s) are: H(f ) f :  × R → R is a Carathéodory function such that |f (x, s)| ≤ a(x) ˆ + c|s|r−1 for a.e. x ∈  and all s ∈ R,  with aˆ ∈ L∞ ()+ , c > 0, 1 ≤ r < p ∗ =

Np N−p

+∞

if N > p , if N ≤ p .

Elements of Regularity Theory and Maximum Principle Chapter | 2

45

For the sake of simplicity, we denote  · q :=  · Lq () for all 1 ≤ q ≤ +∞. First we establish the boundedness of the weak solutions for problem (2.40). Proposition 2.10. Assume H(a) (i)–(iii) and H(f ). Let p¯ ∗ = p ∗ if N > p and p¯ ∗ > r if N ≤ p. If u ∈ W 1,p () is a weak solution of (2.40), then u ∈ L∞ () and u∞ ≤ M = M(c0 , c1 , c, a, ˆ N, p, ||, up¯ ∗ ). Proof. Since W 1,p () → W 1,q () for all 1 ≤ q < p, we only need to check the case p < N . Writing u = u+ − u− with u+ = max{u, 0} ∈ W 1,p (), u− = max{−u, 0} ∈ 1,p W (), it is sufficient to prove the result for u+ because one can similarly proceed for u− . For every M > 0, we set vM (x) = min{u+ , M}. Let l ≥ 0. Using hypothesis H(f ), it is straightforward to get   pl+1 pl+r f (x, u)vM dx ≤ C 1 + u+ pl+r , 

with a constant C > 0. A direct computation based on Lemma 2.7 (c) shows that   (pl + 1)c0 1 pl+1 ∇(v l+1 )p dx. (a(x, ∇u), ∇(vM ))RN dx ≥ M p p − 1 (l + 1)   Then (2.40) ensures     pl + 1  pl+r v l+1 p 1,p ≤ C 1 + pl + 1 1 + u+ pl+r , M W () p p (l + 1) (l + 1)

(2.42) ∗

with a new constant C > 0. Due to the embedding W 1,p () → Lp () and letting M → +∞, from (2.42) it follows that p ∗ (l+1)

1 + u+ p∗ (l+1) ≤ C

(l + 1)p (pl + 1)

∗



p∗ p

1 + u+ pl+r pl+r

 p∗ p

,

(2.43)

with C > 0. Next we construct an increasing sequence qi → +∞ as i → +∞ as follows: ∗ q0 = p ∗ and qi+1 = (qi − r + p) pp for i ≥ 0. For l = qip−r , (2.43) becomes 1 + u+ qi+1 i+1 ≤ C q

(qi − r + p)p

∗

p p∗ (qi − r + 1)

p∗ p



1 + u+ qii q

pp∗

.

(2.44)

Set Xi = ln(1 + u+ qii ). Then (2.44) leads to q

Xi+1 ≤ p ∗ ln[C(qi − r + p)] +

p∗ Xi , p

(2.45)

46 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints ∗

with a constant C > 0. Since qi − r + p < ( pp )i p ∗ , from (2.45) we derive Xi+1 ≤ C1 + iC2 +

p∗ Xi , p

with positive constants C1 , C2 , which results in Xi+1 ≤ C1

i  ∗ j  p j =0

p

 ∗ j  ∗ i+1 i  p p + C2 (i − j ) + X0 . p p j =0

It turns out that u+ qi ≤ M, for some M = M(c0 , c1 , c, a, ˆ N, p, ||, up∗ ). Since qi → +∞ as i → +∞, we conclude that u∞ ≤ M. Remark 2.11. The result of Proposition 2.10 is also true for the Dirichlet problem. The proof is similar via an iterated estimate as above. In the literature this method is usually called Moser’s iteration technique. We indicate a useful regularity result up to the boundary which also establishes continuous dependence for the solution with respect to the reaction term of the nonlinear elliptic equation. Proposition 2.12. Let fn :  × R → R be a Carathéodory function satisfying |fn (x, t)| ≤ D(1 + |t|r−1 )

for a.e. x ∈ , t ∈ R

with some positive constant D independent of n and r ∈ [p, p ∗ ), where p ∗ = +∞ if N ≤ p, p ∗ = pN/(N − p) if N > p. Assume that An :  × RN → RN is a map satisfying H(a) with positive constants c1 , c0 and c2 independent of n. If un is a solution for −div An (x, ∇u) = fn (x, u)

in ,

u=0

on ∂,

1,p

and {un } is bounded in W0 (), then there exist a subsequence {unl } of {un } and u0 ∈ C01 () such that unl → u0 in C01 () as l → ∞. 1,p

Proof. Since {un } is bounded in W0 (), we may assume that {un } weakly 1,p converges to some u0 in W0 () by choosing a subsequence. We can show that there exists C > 0 depending only on ||, p, N , D, c0 , c1 and sup un W 1,p () n

0

such that un ∞ ≤ C by the Moser iteration process (refer to [156, Theorem C] or Proposition 2.10 and Remark 2.11). Since D, c1 and c0 are independent of n, {un ∞ } is bounded. Therefore, the regularity result in [129] guaran1,γ tees that there exist γ ∈ (0, 1) and M > 0 of n such that un ∈ C0 () and

Elements of Regularity Theory and Maximum Principle Chapter | 2

47

un C 1,γ () ≤ M (where we use the fact that c2 is independent of n). Since the 0

1,γ

inclusion of C0 () into C01 () is compact, {un } converges to u0 in C01 () 1,p (note that un u0 in W0 ()). Remark 2.13. The result of Proposition 2.12 is also true for the Neumann problem.

2.3 STRONG MAXIMUM PRINCIPLE FOR NONLINEAR ELLIPTIC EQUATIONS Consider a bounded domain  ⊂ RN with C 2 boundary ∂, a map A :  × R × RN → RN for which we set A(x, t, ξ ) = ( a1 (x, t, ξ ), . . . , aN (x, t, ξ ) ) ∈ RN for all (x, t, ξ ) ∈  × [0, +∞) × RN , and a function f :  × [0, +∞) × RN → R. We are concerned with the inequality problem consisting in finding u :  → R with u ∈ C 1 () such that −div A(x, u, ∇u) ≥ f(x, u, ∇u)

in ,

(2.46)

which is understood in the distributional sense. We will prove a general strong maximum principle for this problem (in particular, for nonlinear elliptic equations). The precise conditions that we impose on the data in (2.46) are listed below. We formulate the required hypotheses on the mapping A in (2.46). H(A) Given 1 < q < +∞, there exist positive constants C1 , C2 , C3 , C4 and 0 < t0 ≤ 1 such that (i) ai ∈ C( × [0, +∞) × RN , R) ∩ C 1 ( × (0, +∞) × (RN \ {0}), R), i = 1, . . . , N ; (ii) ai (x, t, 0) = 0 for all (x,  t) ∈  × [0, +∞), and 1 ≤ i ≤ N ; (iii) Dy A(x, t, y) ≤ C1 t q−1 (1 + | log t|) + |y|q−2 for all x ∈ , 0 ≤ t ≤ t0 , y ∈ RN with 0 < |y| ≤ t0 ; (iv) Dy A(x, t, y)ξ · ξ ≥ C2 |y|q−2 |ξ |2 for all x ∈ , 0 ≤ t ≤ t0 , y ∈ RN with N; 0 < |y| ≤ t0 and ξ ∈ R  q−1 (v) |∂xi ai (x, t, y)| ≤ C3 t (1 + | log t|) + |y|q−1 (1 + | log |y| |) for all x ∈ , 0 < t ≤ t0 , 0 <  |y| ≤ t0 , and 1 ≤ i ≤ N ; (vi) |∂t ai (x, t, y)| ≤ C4 t q−1 (1 + | log t|) + |y|q−1 (1 + | log |y| |) for all x ∈ , 0 < t ≤ t0 , 0 < |y| ≤ t0 , and 1 ≤ i ≤ N . Here the expression t q−1 | log t| is interpreted as equal 0 at t = 0. The following consequence of hypothesis H(A) will be useful.

48 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Lemma 2.14. Assume condition H(A) and let d1 > d0 > 0. If 1 < q ≤ 2, then (A(x, t, ξ ) − A(x, t, η)) · (ξ − η) ≥

C2 2−q

d1

|ξ − η|2

for all x ∈ , t ∈ [0, t0 ] and η, ξ ∈ RN with |ξ |, |η| < d1 . If q > 2, then q−1

(A(x, t, ξ ) − A(x, t, η)) · (ξ − η) ≥

C2 d 0 |ξ − η|2 2q−1 (d0 + d1 )

for all x ∈ , t ∈ [0, t0 ] and η, ξ ∈ RN with d0 < |ξ | < d1 and |η| < d1 . Proof. Case 1 < q ≤ 2. Let η, ξ ∈ RN with |ξ |, |η| < d1 . If θ ξ + (1 − θ )η = 0 for every θ ∈ [0, 1], noting q − 2 ≤ 0 and |θ ξ + (1 − θ )η| < d1 , by hypothesis (iv) in H(A) we obtain (A(x, t, ξ ) − A(x, t, η)) · (ξ − η) 1 = Dy A(x, s, η + s(ξ − η))(ξ − η) · (ξ − η) ds 0



≥ C2

1

|η + s(ξ − η)|q−2 |ξ − η|2 ds ≥

0

C2 2−q d1

|ξ − η|2

(2.47)

whenever x ∈  and t ∈ [0, t0 ]. Similarly, if η = 0 and ξ = 0, then q−2

A(x, s, ξ )ξ ≥ C2 d1

|ξ |2 .

Now we examine the case where 0 belongs to the interior of the line segment [ξ, η] with ξ = 0, η = 0. Let 0 < θ0n < θ1n < 1 such that θ ξ + (1 − θ )η = 0 for every θ ∈ [0, θ0n ] ∪ [θ1n , 1] and θ1n − θ0n ≤ 1/n. Set yin := θin ξ + (1 − θin )η, i = 0, 1, thus ξ −η=

ξ − y1n y0n − η y1n − y0n = = n . 1 − θ1n θ0n θ1 − θ0n

From (2.47) and (2.48), we have (A(x, t, ξ ) − A(x, t, y1n )) · (ξ − η) 1 = (A(x, t, ξ ) − A(x, t, y1n )) · (ξ − y1n ) 1 − θ1n C4 (1 − θ1n ) C2 n 2 |ξ − y | = |ξ − η|2 , ≥ 1 2−q 2−q (1 − θ1n )d1 d1

(2.48)

Elements of Regularity Theory and Maximum Principle Chapter | 2

49

(A(x, t, y0n ) − A(x, t, η)) · (ξ − η) 1 = n (A(x, t, y0n ) − A(x, t, η)) · (y0n − η) θ0 C2 θ n C2 ≥ 2−q |y0n − η|2 = 2−q0 |ξ − η|2 . d1 θ0n d1 By the monotonicity of A in the third variable, (2.48) and θ1n − θ0n → 0, it turns out that (A(x, t, ξ ) − A(x, t, η)) · (ξ − η) = (A(x, t, ξ ) − A(x, t, y1n )) · (ξ − η) + (A(x, t, y1n ) − A(x, t, y0n )) · (ξ − η) + (A(x, t, y0n ) − A(x, t, η)) · (ξ − η) C2 ≥ 2−q (θ0n + (1 − θ1n ))|ξ − η|2 d1 1 + n (A(x, t, y1n ) − A(x, t, y0n )) · (y1n − y0n ) θ1 − θ0n C2 C2 ≥ 2−q (θ0n + (1 − θ1n ))|ξ − η|2 ≥ 2−q |ξ − η|2 (1 − o(1)). d1 d1 Case q > 2. Let η, ξ ∈ RN with d0 < |ξ | < d1 , |η| < d1 , and set θ0 = 1 −

d0 2(d0 + d1 )

and y0 = θ0 ξ + (1 − θ0 )η.

Because of |ξ | > d0 and |η| < d1 , one has |θ ξ + (1 − θ )η| ≥ θ d0 − (1 − θ )d1 = (d0 + d1 )θ − d1 d0 ≥ for every θ ∈ [θ0 , 1]. 2 We see that |tξ + (1 − t)y0 | = | (θ0 + t (1 − θ0 )) ξ + (1 − θ0 − t (1 − θ0 )) η| ≥ for every t ∈ [0, 1]. Noting that ξ − η = (y0 − η)/θ0 = (ξ − y0 )/(1 − θ0 ), by the mean value theorem with some s ∈ (0, 1) we find (A(x, t, ξ ) − A(x, t, η)) · (ξ − η) 1 = (A(x, t, y0 ) − A(x, t, η)) · (y0 − η) θ0 1 Dy A(x, t, sξ + (1 − s)y0 )(ξ − y0 ) · (ξ − y0 ) + 1 − θ0

d0 2

50 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints q−2

q−2

C2 d C 2 d0 |ξ − y0 |2 = q−2 ≥ q−2 0 (1 − θ0 )|ξ − η|2 2 (1 − θ0 ) 2 q−1

=

C2 d0 |ξ − η|2 , q−1 2 (d0 + d1 )

thanks to the monotonicity of A in the third variable and assumption H(A) (iv). We formulate the following hypotheses on the function fin problem (2.46). H(f) f :  × [0, +∞) × RN → R is a continuous function such that there exists a constant λ > 0 for which f(x, s, ξ ) ≥ −λs q−1 (1 + | log s|) − λ|ξ |q−1 (1 + | log |ξ | |)

(2.49)

for all x ∈ , 0 < s ≤ t0 , 0 < |ξ | ≤ t0 , where t0 is as in H(A). The below function will play an essential role in our approach h(t) := t q−1 (1 + | log t|) for t ≥ 0.

(2.50)

Notice that the function h is increasing on [0, t1 ], where t1 := e(q−2)/(q−1) . Lemma 2.15. For any t2 ∈ (0, t1 ) and t3 > t1 , there exists h0 > 0 such that h(|ξ |) − h(|η|) ≥ −h0 |ξ − η| for all ξ, η ∈ RN with t1 ≥ |ξ | ≥ t2 and |η| ≤ t3 . Proof. Since h is increasing on [0, t1 ] (see (2.50)), if t1 ≥ |ξ | ≥ |η| holds, then h(|ξ |) − h(|η|) ≥ 0. In the case of t2 ≤ |ξ | < |η| ≤ t3 , the mean value theorem gives h(|ξ |) − h(|η|) = h (t)(|ξ | − |η|) for some t ∈ (|ξ |, |η|). Setting h0 := max{|h (t)| : t2 ≤ t ≤ t3 } > 0, we obtain h(|ξ |) − h(|η|) ≥ −h0 (|η| − |ξ |) ≥ −h0 |η − ξ |. We first prove a boundary point result, which represents the hard part of the proof of our strong maximum principle regarding problem (2.46). Proposition 2.16. Assume that  is a bounded domain in RN and x1 ∈ ∂ is such that the interior-ball condition is fulfilled at x1 . Let A and f satisfy the hypotheses H(A) and H(f), respectively, with  in place of . If u ∈ C 1 ( ∪ {x1 }) verifies (2.46) in  , u > 0 in  and u(x1 ) = 0, then ∂u(x1 )/∂ν < 0 holds, where ν stands for the outer unit normal to ∂.

Elements of Regularity Theory and Maximum Principle Chapter | 2

51

Proof. The interior-ball condition of  at x1 provides x2 ∈  and T0 > 0 such that the open ball B(x2 , 2T0 ) fulfills B(x2 , 2T0 ) ⊂  and ∂B(x2 , 2T0 ) ∩ ∂ = {x1 }. The assumptions u ∈ C 1 ( ∪ {x1 }) and u(x1 ) = 0 guarantee that for x2 and T0 we can suppose that 0 < u < min{t0 , e(q−2)/(q−1) /2} and |∇u| ≤ K1 in B(x2 , 2T0 ),

(2.51)

with some constant K1 > 0. Denote K0 = N (C3 + C4 K1 + 1)

(2.52)

 C2 , T < min T0 , 4(λ + K0 )

(2.53)

and fix T > 0 satisfying 

where the constants C2 , C3 , C4 are those in hypotheses H(A). Setting the convex combination   T T x3 = 1 − x1 + x2 , T0 T0 we have B(x3 , 2T ) ⊂ B(x2 , 2T0 ) ⊂ 

and

∂B(x3 , 2T ) ∩ ∂ = {x1 }.

Notice that min{u(x) : |x − x3 | = T } > 0, which is true because u > 0 in  . Let us consider Y := {x ∈  : T < |x − x3 | < 2T } and vk (t) :=

1 ekt − 1 k 2 ekT − 1

(2.54)

for all t ∈ [0, T ] and a constant k > 0, as well as the function w(x) := w(r) := vk (2T − r) = vk (t) for x ∈ Y , with r = |x − x3 | and t = 2T − r. We show that for sufficiently large k the function w verifies −div A(x, u, ∇w) + (λ + K0 )h(w) + λh(|∇w|) ≤ K0 h(u)

in Y,

(2.55)

with u ≥ w on ∂Y , where h and K0 are introduced in (2.50) and (2.52), and λ > 0 is the constant in assumption H(f). Direct computation enables us to check that N  ∂ − div A(x, u, ∇w) = − (ai (x, u, ∇w)) ∂xi i=1

52 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

= −

N  ∂ai i=1

∂xi

(x, u, ∇w) −

N  ∂ai i=1

∂t

(x, u, ∇w)

∂u ∂xi

w  Dy A(x, u, ∇w)ei · ei r i=1    w w − − Dy A(x, u, ∇w)(x − x3 ) · (x − x3 ) r2 r3 N

−

in Y,

where {ei }1≤i≤N denotes the usual orthonormal basis in RN . Notice that w  (r) = −vk (t) < 0 and w  (r) = vk (t) = kvk (t) > 0. By taking k sufficiently large, w ∈ C ∞ (Y ) satisfies u ≥ w on ∂Y (note that min u > 0), ∂Y

0 < w = vk ≤ min{t0 , t1 /2}

and 0 < |w  | = vk ≤ min{t0 , t1 /2}

(2.56)

in Y . Since 0 < u < t0 and |∇u| ≤ K1 in Y (see (2.51)), through hypothesis H(A) we obtain − div A(x, u, ∇w) ≤ N C3 (h(u) + h(|∇w|)) + N C4 (h(u) + h(|∇w|))K1  w w  − N C1 (h(u) + |∇w|q−2 ) − w  − C2 |∇w|q−2 r r    v  N C1  q−1 = N C3 + C4 K1 + C1 k h(u) + N C3 + C4 K1 h(vk ) + (vk ) r r  1 − C2 (vk )q−1 k + in Y. r For a sufficiently large k, we may assume that 0 < C1

  vk v C1 1 1 ≤ C1 k ≤ 1 + kT ≤ 1 in Y r T T k e −1

(note that T ≤ r ≤ 2T ). Hence, −div A(x, u, ∇w) ≤ K0 (h(u) + h(vk )) +

N C2  q−1 − C2 (vk )q−1 k (vk ) T

in Y

holds in view of (2.52). We get − div A(x, u, ∇w) + (λ + K0 )h(w) + λh(|∇w|) ≤ K0 h(u) + (λ + K0 )(h(vk ) + h(vk )) − C2 (vk )q−1 k +

N C2  q−1 (vk ) T

in Y.

Applying Lemma 2.17 given below with a sufficiently large k shows that w satisfies (2.55).

Elements of Regularity Theory and Maximum Principle Chapter | 2

53

By means of (2.55), we can prove that if k is sufficiently large then u ≥ w in Y .

(2.57)

Admit for a moment that (2.57) is valid. Note that ν(x1 ) = −(x3 − x1 )/2T . Then, relying on (2.57), in conjunction with u ∈ C 1 ( ∪ {x1 }) and u(x1 ) = 0, we obtain −2T

∂u u(x1 + s(x3 − x1 )) − u(x1 ) (x1 ) = lim ∂ν s s→0+ w(x1 + s(x3 − x1 )) − w(x1 ) = 2T vk (0) > 0, ≥ lim s s→0+

so the desired conclusion is achieved. The proof of (2.57) is given subsequently, which will complete the proof. Lemma 2.17. For sufficiently large k, the functions vk introduced in (2.54) satisfy −C4 k(vk )q−1 + (λ + K0 )(h(vk ) + h(vk )) +

N C2  q−1 0. Y

Y

Set (l) := {x ∈ Y : (V ≥) (w − u)(x) > l} for 0 < l < V and (V ) := {x ∈ Y : (w − u)(x) = V }.

(2.60)

It follows from the subsequent Lemma 2.18 that |(V )| = 0, that is, the set (V ) is of Lebesgue measure zero. Fix r > 1 satisfying r ≤ N/(N − 2) if N ≥ 3. Note that H(f), (2.46) and (2.50) imply −div A(x, u, ∇u) ≥ −λh(u) − λh(|∇u|)

in Y.

Since 0 < min |∇w| ≤ max |∇w| ≤ t1 /2, Y

Y

Lemma 2.15 provides h0 > 0 such that h(|∇w|) − h(|∇u|) ≥ −h0 |∇w − ∇u|

in Y.

Moreover, we note that h(w) > h(u)

on (l)

for any 0 < l < V , because h is increasing in [0, t1 ]. Using ϕ := (w − u − l)+ ∈ W01,∞ (Y )

(2.61)

Elements of Regularity Theory and Maximum Principle Chapter | 2

55

for 0 < l < V as test function (note that u ≥ w on ∂Y ), by (2.55), (2.61), Hölder’s and Sobolev’s inequalities, we obtain (A(x, u, ∇w) − A(x, u, ∇u)) · ∇(w − u − l)+ dx Y ≤ − (λ + K0 ) (h(w) − h(u))(w − u − l)+ dx Y − λ ( h(|∇w|) − h(|∇u|) ) (w − u − l)+ dx Y |∇w − ∇u| (w − u − l) dx = λh0 |∇ϕ|ϕ dx ≤ λh0 (l)

Y

≤ λh0 ∇ϕL2 (Y ) ϕL2r (Y ) |(l)|

(r−1)/2r

≤ λh0 C∗ ∇ϕ2L2 (Y ) |(l)|(r−1)/2r ,

where C∗ > 0 is a constant determined by the embedding of W01,2 (Y ) into L2r (Y ). On the other hand, Lemma 2.14 provides a constant C > 0 such that (A(x, u, ∇w) − A(x, u, ∇u)) · ∇(w − u − l)+ dx Y |∇w − ∇u|2 dx = C∇ϕ2L2 (Y ) . ≥C {w≥u+l}

This amounts to saying that C ≤ λh0 C∗ |(l)|(r−1)/2r because ϕ ≡ 0 for any 0 < l < V . Thus we reach a contradiction taking into account that |(l)| → 0 as l → V , which concludes the proof of (2.57). Lemma 2.18. There holds |(V )| = 0, where (V ) is the set defined by (2.60). Proof. Arguing by contradiction, we suppose that |(V )| > 0. Let ρn ∗ χ be the convolution of ρn and χ , where {ρn (x)} is a sequence of mollifiers and χ(x) = 1 if x ∈ (V ), χ(x) = 0 if x ∈ (V ). Note that supp (ρn ∗ χ) ⊂ Y for sufficiently large n because u ≥ w on ∂Y and (V )(⊂ Y ) is compact. Then it follows from (2.55) and (2.61) that A(x, u, ∇w) · ∇(ρn ∗ χ) dx Y ≤ (K0 h(u) − (λ + K0 )h(w) − λh(|∇w|))ρn ∗ χ dx Y

and



A(x, u, ∇u) · ∇(ρn ∗ χ) dx ≥ −λ

Y

(h(u) + h(|∇u|))ρn ∗ χ dx. Y

56 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

These lead to

(A(x, u, ∇w) − A(x, u, ∇u)) · ∇(ρn ∗ χ) dx ≤ (λ + K0 ) (h(u) − h(w))ρn ∗ χ dx Y + λ (h(|∇u|) − h(|∇w|))ρn ∗ χ dx,

o(1) =

Y \(V )

Y

taking into account in the last inequality that ∇u = ∇w on (V ). Since h is increasing in [0, t1 ] (note that u + V ≤ t1 by (2.56) and (2.51)), a contradiction arises letting n → ∞ in (h(u) − h(u + V )) dx + o(1) 0 > (λ + K0 ) (V ) = (λ + K0 ) (h(u) − h(w))ρn ∗ χ dx + λ (h(|∇u|) − h(|∇w|))ρn ∗ χ dx Y

Y

because of w = u + V and ∇u = ∇v on (V ). This completes the proof. Taking advantage of Proposition 2.16, we can establish our strong maximum principle regarding inequality (2.46). Theorem 2.19. Assume H(A) and H(f). If u ∈ C 1 () satisfies u ≥ 0 in , u ≡ 0 in , and inequality (2.46), then u > 0 in  and ∂u/∂ν < 0 on ∂ hold. Proof. We set [u > 0] := {x ∈  : u(x) > 0}. By hypothesis we know that [u > 0] =  ∅. Since  is connected, it suffices to show that ∂[u > 0] ∩  = ∅. By way of contradiction, assume that ∂[u > 0] ∩  = ∅. Then there exists a point x1 ∈ ∂[u > 0] ∩  such that [u > 0] fulfills the interior-ball condition at x1 . It is so because the set of x ∈ ∂[u > 0] such that [u > 0] satisfies the interior-ball condition at x is dense in ∂[u > 0] (refer to [156, Appendix A]). Choose x2 ∈ [u > 0] and T0 > 0 such that B(x2 , T0 ) ⊂ [u > 0] and ∂B(x2 , T0 ) ∩ ∂[u > 0] = {x1 }. Because u > 0 in B(x2 , T0 ), u(x1 ) = 0 and u satisfies (2.46) in B(x2 , T0 ), applying Proposition 2.16 with  = B(x2 , T0 ) renders that ∇u(x1 ) = 0. However, x1 ∈  is a global minimizer of u, so ∇u(x1 ) = 0. This contradiction ensures that ∂[u > 0] ∩  = ∅, thus u > 0 in . Since ∂ is of class C 2 , the property ∂u/∂ν < 0 on ∂ results from Proposition 2.16. The proof is complete.

2.4 NOTES Section 2.1 takes into account the dependence of the solution sets with respect to parameters. It exploits powerful topological tools as, for instance, the transversality theory. Theorem 2.2 is taken from [162,163]. It is inspired by [214,215] regarding generic results for problems depending on parameters and

Elements of Regularity Theory and Maximum Principle Chapter | 2

57

applications to elliptic boundary value problems. A version of Theorem 2.2 on differentiable manifolds was given in [161]. It is worth mentioning that one can derive directly from Theorem 2.2 the density of Morse functions on a finitedimensional manifold (see [161]). Moreover, with a suitable modification in the argument, Theorem 2.2 leads to the results of Marino and Prodi [148] of approximation by Morse functions on an infinite-dimensional Riemannian manifold (see [162]). Theorem 2.4 provides homoclinics which are nondegenerate in the sense that they are nondegenerate critical points of the Euler functional associated to problem (2.13). Theorem 2.4 can be found in [163]. The property of a solution to be nondegenerate is an important qualitative information, guaranteeing in particular that the solution is isolated. In studying (2.13) one cannot make use of results dealing with the Cauchy problem for ordinary differential equations because the formulation in (2.13) does not involve an initial condition. Another essential feature of (2.13) is the lack of compactness, mainly caused by the fact that t runs in an unbounded set, that is reflected, for instance, in the failure of the Palais–Smale condition for Euler functional associated to (2.13). These difficulties are overcome by developing specific variational methods for the homoclinic problem (2.13), for which we refer to [35,50,59,230]. An open problem in such a context is to find verifiable conditions which guarantee that the celebrated hypothesis (∗) is fulfilled for (2.13) (see, e.g., [35]). Theorem 2.4 postulates that for f belonging to a dense (actually, residual) set G in L2 (R, RN ), problem (2.13) possesses only finitely many solutions, which in addition are nondegenerate. It also provides a partial answer to assertion (∗) expressing its generic validity. We drop out some requirements considered before (see [35,59]). Theorem 2.4 points out the smooth dependence of the solutions q of (2.13) with respect to f ∈ L2 (R, RN ). The abstract result presented in Theorem 2.2 is applied to deduce in Theorem 2.5 that the Morse functions related to Eq. (2.32) exist generically with respect to f ∈ L2 () that is regarded as a parameter. Theorem 2.5 can be found in [162]. Specifically, we show that under natural hypotheses upon the nonlinearity p(x, u) the Euler functional corresponding to (2.32) (see, e.g., [207, p. 61]) is generically a Morse function, which means that it has only nondegenerate critical points and satisfies the Palais–Smale condition. It allows applying different results where the nondegeneracy of solutions is a basic assumption. For a comprehensive discussion in this direction, we refer to the related work of Mawhin [151]. Regularity information concerning the dependence of the solutions of (2.32) with respect to f ∈ L2 () is also available. Section 2.2 contains regularity results for nonlinear elliptic equations, in particular applicable for problems involving the p-Laplacian. It is worth mentioning that the principal part of the considered problems is driven by nonlinear differential operators that are not required to satisfy any homogeneity condition. Proposition 2.10 represents a convenient criterion to show that the solutions are uniformly bounded. It is taken from Miyajima–Motreanu–Tanaka [156, The-

58 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

orem C]. The proof is in the spirit of the method known as Moser iteration technique. The boundedness of solutions is extremely important because it is a fundamental condition to establish the regularity up to the boundary for the solutions. In the case of quasilinear elliptic problems with Dirichlet or Neumann boundary conditions, the regularity up to the boundary was obtained by Lieberman [129,130]. The interior regularity is studied in [72,126,227]. Proposition 2.12 provides a continuity property of solutions with respect to the reaction term in quasilinear elliptic problems, which enables us to pass to the limit in sequences of problems. Section 2.3 is devoted to the study of strong maximum principle for nonlinear elliptic problems. The main result is formulated as Theorem 2.19, which is taken from Motreanu–Tanaka [183]. It establishes a strong maximum principle applicable to the differential inequality problem (2.46), in particular to the Dirichlet problem  − p u − q u = f (x, u, ∇u) in , u=0 on ∂ and the Neumann problem  − p u − q u = f (x, u, ∇u) ∂u ∂ν

=0

in , on ∂

driven by (p, q)-Laplacian. Many other differential operators can be considered in the principal part of the elliptic equation. An essential feature of this result is that it allows having dependence on the solution u and its gradient ∇u on the right-hand side of the elliptic equation (often called convection term). Theorem 2.19 covers novel situations with respect to the strong maximum principles in Pucci–Serrin [206], Vázquez [229] and Felmer–Montenegro–Quaas [83]. This is demonstrated by the variety of examples mentioned in Examples 4.30 and 4.33. Usually, the strong maximum principle is utilized jointly with the nonlinear regularity theory. The scheme starts by showing that the solution, say u, belongs to L∞ (), which can be done through the Moser iteration process as developed in Proposition 2.10. Then, by the regularity up to the boundary in [129,130], it is seen that actually u ∈ C 1,α () for some 0 < α < 1. If the solution satisfies u ≥ 0 and u ≡ 0, as well as the growth condition (2.49) along it, one can invoke the strong maximum principle in Theorem 2.19 to obtain that u(x) > 0 for every x ∈  and ∂u(x)/∂ν < 0 for every x ∈ ∂, where ν denotes the outward unit normal to ∂.

Chapter 3

Nonlinear Elliptic Eigenvalue Problems 3.1 EIGENVALUE PROBLEM FOR p-LAPLACIAN UNDER DIFFERENT BOUNDARY CONDITIONS The aim of this section is to give an overview on the Fuˇcík spectrum, which is closely related to the ordinary spectrum, of the negative p-Laplacian −p with 1 < p < +∞ and different boundary conditions on a bounded domain  in RN . For any real-valued function u on , we pose u+ = max{u, 0} and u− = max{0, −u}. Let V be a closed subspace of the Sobolev space W 1,p () such that 1,p W0 () ⊂ V ⊂ W 1,p () and let V ∗ denote the dual space, with the duality pairing ·, · between V and V ∗ . It is well known that the operator −p : V → V ∗ is bounded, continuous, pseudomonotone, and has the (S)+ -property (i.e., from un  u in V and lim sup−p un , un − u ≤ 0 n→∞

it follows that un → u in V ) (see Proposition 1.56). We refer to [38,169] for various nonlinear boundary problems involving −p . In order to avoid tedious arguments, as well as certain repetitive ideas, in the approach, we outline the main elements of the theory of Fuˇcík spectrum of −p with Dirichlet, Neumann, and Steklov boundary conditions, and discuss in more details the Fuˇcík spectrum of −p with Robin boundary condition. Definition 3.1. The Fuˇcík spectrum of the negative p-Laplacian −p with homogeneous Dirichlet boundary condition is defined as the set p of those (a, b) ∈ R2 such that  −p u = a(u+ )p−1 − b(u− )p−1 in , (3.1) u=0 on ∂ 1,p

has a nontrivial (weak) solution u, which means that u ∈ W0 (), u ≡ 0, and it satisfies the equation   1,p p−2 |∇u| ∇u · ∇v dx = (a(u+ )p−1 − b(u− )p−1 )v dx, ∀v ∈ W0 (). 



Nonlinear Differential Problems with Smooth and Nonsmooth Constraints. DOI: 10.1016/B978-0-12-813386-6.00003-1 Copyright © 2018 Elsevier Inc. All rights reserved.

59

60 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Remark 3.2. We note that if a = b = λ, problem (3.1) reduces to  −p u = λ|u|p−2 u in , u=0 on ∂,

(3.2)

which is called the Dirichlet eigenvalue problem with respect to −p . According to Proposition 1.57, it is known that the first eigenvalue λ1 of (3.2) is positive, simple, and its corresponding eigenfunctions have constant sign (see also Anane [14] and Lindqvist [131]). In fact, the spectrum σ (−p ) of the negative p-Laplacian −p associated to (3.2) includes an unbounded sequence of eigenvalues {λk }, called the variational eigenvalues, which fulfills 0 < λ1 < λ2 ≤ · · · ≤ λk ≤ · · · → +∞. The variational eigenvalues satisfy minimax characterizations. Proposition 3.3. The Fuˇcík spectrum p of the negative p-Laplacian −p with homogeneous Dirichlet boundary condition contains the two lines {λ1 } × R and R × {λ1 }. Additionally, p contains the sequence of points (λk , λk ), k ∈ N. Proof. It can be easily seen from Definition 3.1 and Remark 3.2. Let us set for every s ≥ 0,   p |∇u| dx − s (u+ )p dx, Js (u) = 



1,p

u ∈ W0 ().

The function Js is of class C 1 on W0 (). Consider the restriction J˜s = Js |S , with S given by    1,p p |u| = 1 . S = u ∈ W0 () : 1,p



1,p Since S is a C 1 submanifold of W0 (), it follows that J˜s is of class C 1 on S in the sense of manifolds.

Theorem 3.4. The curve in R2 given by C := {(s + c(s), c(s)), (c(s), s + c(s)) : s ≥ 0},

(3.3)

where c(s) is described by the minimax values c(s) = inf

max

γ ∈ u∈γ ([−1,1])

J˜s (u),

with

= {γ ∈ C([−1, 1], S) : γ (−1) = −uˆ 1 and γ (1) = uˆ 1 },

(3.4)

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

61

is contained in p . In (3.4), uˆ 1 denotes the eigenfunction of (3.2) corresponding to λ1 satisfying uˆ 1 > 0 in  and uˆ 1 p = 1 (see Proposition 1.57). Moreover, the curve C in (3.3) is Lipschitz continuous and decreasing. The limit of c(s) as s → +∞ is equal to the first eigenvalue λ1 of (3.2). Proof. First, we can prove that the curve s ∈ (0, +∞) → (s + c(s), c(s)) ∈ R2 is contained in p (see [60, Theorem 2.10]). Taking into account that p is symmetric with respect to the diagonal of the plane, it turns out that the curve C in (3.3) is contained in p . The other properties of the first curve C in p that are mentioned in the statement of the proposition are proven in [60, Proposition 4.1] and [60, Proposition 4.4]. Remark 3.5. It is shown in [60, Theorem 3.1] that C given in (3.3) is the first nontrivial curve in p , which means that the first point in p belonging to the parallel to the diagonal drawn through a point of ([0, +∞) × {λ1 }) × ({λ1 } × R) must be on C. As a consequence, we infer that the curve C passes through (λ2 , λ2 ). In conjunction with the description of C in (3.3) and the minimax formula for c(s), this yields that λ2 has the variational characterization in Proposition 1.58, namely  max |∇u|p dx, (3.5) λ2 = inf γ ∈ u∈γ ([−1,1]) 

with introduced in (3.4). We illustrate the applicability of the Fuˇcík spectrum p by considering the following problem with homogeneous Dirichlet boundary condition and jumping nonlinearity  −p u = a(u+ )p−1 − b(u− )p−1 + g(x, u) in , (3.6) u=0 on ∂, where g :  × R → R is a Carathéodory function satisfying lim

t→0

g(x, t) =0 |t|p−1

uniformly for a.a. x ∈ .

In Carl–Perera [49] it is proven that problem (3.6) has at least three nontrivial solutions provided the point (a, b) ∈ R2 lies above the first nontrivial curve C in p constructed in (3.3). Moreover, a complete sign information for the three solutions is available: two solutions have opposite constant sign and the third one is sign changing (nodal solution). This information is obtained by means of the method of subsolution–supersolution whose application to problem (3.6) strongly relies on the hypothesis that the point (a, b) ∈ R2 is situated above the first nontrivial curve C in p . We pass to the Fuˇcík spectrum of the negative p-Laplacian −p with (homogeneous) Neumann boundary condition.

62 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Definition 3.6. The Fuˇcík spectrum of the negative p-Laplacian −p with Neumann boundary condition, denoted by p , consists of all pairs (a, b) ∈ R2 such that  −p u = a(u+ )p−1 − b(u− )p−1 in , (3.7) ∂u on ∂ ∂ν = 0 is solved nontrivially, meaning that u ∈ W 1,p (), u ≡ 0, and it verifies the equality   |∇u|p−2 ∇u · ∇v dx = (a(u+ )p−1 − b(u− )p−1 )v dx, ∀v ∈ W 1,p (). 



In (3.7), ∂u/∂ν denotes the normal derivative, with ν being the unit outward normal to ∂. Remark 3.7. In the case where a = b = λ, problem (3.7) becomes the Neumann eigenvalue problem of the negative p-Laplacian −p given by 

−p u = λ|u|p−2 u ∂u ∂ν

=0

in ,

(3.8)

on ∂.

The first eigenvalue λ1 = 0 of (3.8) is simple with the corresponding eigenspace R, so all eigenfunctions associated to λ1 do not change sign in , which does not happen for the higher order eigenvalues (see [127]). Proposition 3.8. The Fuˇcík spectrum p of the negative p-Laplacian −p with Neumann boundary condition contains (0, 0), (λ2 , λ2 ), where λ2 is the second eigenvalue of (3.8), and the two lines {0} × R and R × {0}. Therefore, ˜ p = p \ (({0} × R) ∪ (R × {0})) of p is contained in the nontrivial part  (0, +∞) × (0, +∞). Proof. It is a straightforward consequence of Definition 3.6, (3.7), and Remark 3.7. ˜ p can be done similarly to The construction of a first nontrivial curve in  1,p the Dirichlet Fuˇcík spectrum. Namely, let Js : W () → R be the functional given by   Js (u) = |∇u|p dx − s (u+ )p dx 



and let J˜s be its restriction to    |u|p = 1 . S = u ∈ W 1,p () : 

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

63

Notice that S is a C 1 submanifold of W 1,p (), so J˜s is of class C 1 on S in the sense of manifolds. This enables us to consider the notions of critical points and critical values for the functional J˜s . Theorem 3.9. The curve in R2 given by C := {(s + c(s), c(s)), (c(s), s + c(s)) : s ≥ 0}, where c(s) is expressed by the minimax value c(s) = inf

max

γ ∈ u∈γ ([−1,1])

J˜s (u),

with

= {γ ∈ C([−1, 1], S) : γ (−1) = −ϕ1 and γ (1) = ϕ1 },

(3.9)

is contained in p . In (3.9) one has ϕ1 = 1/||1/p , with || denoting the Lebesgue measure of . Moreover,  λ1 = 0 if p ≤ N, lim c(s) = (3.10) s→+∞ λˆ if p > N, where  ˆλ = inf |∇u|p dx : u ∈ W 1,p (), uLp () = 1,   u vanishes somewhere in  . Proof. The study for p is carried out along the same lines as in the case of the Dirichlet Fuˇcík spectrum p in Theorem 3.4. The asymptotic properties in (3.10) are shown in [16, Theorems 2.3 and 2.6]. Remark 3.10. In view of Theorem 1.20 (iii), the definition of λˆ in (3.10) is meaningful because for p > N the elements u ∈ W 1,p () are continuous functions on . Corollary 3.11. The first curve C passes through (λ2 , λ2 ), where λ2 denotes the second eigenvalue of (3.8). Consequently, λ2 admits the variational expression  max |∇u|p dx, λ2 = inf γ ∈ u∈γ ([−1,1]) 

with introduced in (3.9). Proof. This follows readily from Proposition 3.8 and Theorem 3.9.

64 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

The next objective of this section is to briefly discuss the Fuˇcík spectrum of −p with Steklov boundary condition. p Definition 3.12. The Steklov Fuˇcík spectrum of −p is defined as the set  2 of all pairs (a, b) ∈ R such that  −p u = −|u|p−2 u in , (3.11) ∂u + p−1 − p−1 = a(u ) − b(u ) on ∂ ∂ν has a weak solution u ≡ 0. Explicitly, u ∈ W 1,p () is a weak solution of (3.11) if it satisfies the equality   |∇u|p−2 ∇u · ∇v dx = − |u|p−2 uv dx   + (a(u+ )p−1 − b(u− )p−1 )v dσ ∂

for all v ∈ W 1,p (). Here the notation dσ stands for the (N − 1)-dimensional surface measure. Remark 3.13. If a = b = λ, (3.11) becomes the so-called Steklov eigenvalue problem, namely  −p u = −|u|p−2 u in , (3.12) ∂u p−2 u on ∂. ∂ν = λ|u| The first eigenvalue of the Steklov eigenvalue problem (3.12) is positive, simple, and every eigenfunction corresponding to the first eigenvalue does not change sign in . Actually, there is an eigenfunction associated to the first eigenvalue λ1 belonging to int(C 1 ()+ ), where the interior of the positive cone C 1 ()+ = {u ∈ C 1 () : u(x) ≥ 0, ∀x ∈ } in the Banach space C 1 () is nonempty and is given by   u ∈ C 1 () : u(x) > 0, ∀x ∈  (see [149]). It is established in [84] that there exists a sequence of eigenvalues λn of (3.12) such that λn → +∞ as n → ∞. In order to study variationally the Steklov Fuˇcík spectrum defined in Definition 3.12, for each s ≥ 0 one defines a C 1 functional Js : W 1,p () → R by    |∇u|p dx + |u|p dx − s (u+ )p dσ. Js (u) = 



∂

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

Restricting Js to

  S = u ∈ W 1,p () :

65

 |u|p dσ = 1 ,

∂

one obtains a C 1 functional J˜s on the C 1 submanifold S of W 1,p (). ˜ p is expressed as Theorem 3.14. The first nontrivial curve in  C = {(s + c(s), c(s)), (c(s), s + c(s)) : s ≥ 0}, where c(s) = inf

max

γ ∈ u∈γ ([−1,1])

J˜s (u),

= {γ ∈ C([−1, 1], S) : γ (−1) = −ϕ1 and γ (1) = ϕ1 } with ϕ1 ∈ int (C 1 ()+ ) being the first positive eigenfunction of (3.12) provided ϕ1 p =1. The first nontrivial curve C is Lipschitz continuous and decreasing. Moreover,  λ1 if p ≤ N, lim c(s) = s→+∞ ˜λ > λ1 if p > N, where p

λ˜ = inf max u∈L r∈R

rϕ1 + uW 1,p () p

rϕ1 + uLp (∂)

with L = {u ∈ W 1,p () : u vanishes somewhere on ∂}. The proof can be found in [150, Theorems 2.1 and 4.1]. As before, we can derive from Theorem 3.14 a variational characterization of the second eigenvalue λ2 of the Steklov eigenvalue problem (3.12). Corollary 3.15. The second eigenvalue λ2 of (3.12) admits the variational expression 

|∇u|p + |u|p dx. (3.13) max λ2 = inf γ ∈ u∈γ ([−1,1]) 

As an application, consider the following nonlinear elliptic equation subject to Steklov-type boundary condition with perturbation:  in , −p u = f (x, u) − |u|p−2 u (3.14) ∂u + p−1 − p−1 − b(u ) + g(x, u) on ∂, ∂ν = a(u )

66 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

for Carathéodory functions f :  × R → R and g : ∂ × R → R which are bounded on bounded sets and satisfy f (x, s) = 0 uniformly for a.a. x ∈ , s→0 |s|p−1 g(x, s) lim = 0 uniformly for a.a. x ∈ ∂, s→0 |s|p−1 f (x, s) = −∞ uniformly for a.a. x ∈ , lim |s|→+∞ |s|p−2 s g(x, s) lim = −∞ uniformly for a.a. x ∈ ∂. |s|→+∞ |s|p−2 s lim

˜ p , problem (3.14) If the point (a, b) is above the first nontrivial curve C in  possesses three nontrivial solutions: one solution with positive sign, one solution with negative sign, and the third one being sign changing (see [231]). Finally, we discuss the Fuˇcík spectrum of −p with a Robin boundary condition. Definition 3.16. The Fuˇcík spectrum of −p with Robin boundary condition p of all pairs (a, b) ∈ R2 for which a nontrivial weak is defined as the set  1,p solution u ∈ W () of the problem  −p u = a(u+ )p−1 − b(u− )p−1 in , (3.15) ∂u p−2 u on ∂ ∂ν = −β|u| exists, where β is a nonnegative constant. A weak solution of (3.15) means that   |∇u|p−2 ∇u · ∇v dx + β |u|p−2 uv dσ ∂  + p−1 − p−1 (a(u ) − b(u ) )v dx = 

for all v ∈ W 1,p (). p reduces to the Fuˇcík spectrum p of the negative Remark 3.17. If β = 0,  Neumann p-Laplacian. The special case a = b = λ leads to  −p u = λ|u|p−2 u in , (3.16) ∂u p−2 u on ∂, ∂ν = −β|u| which is the Robin eigenvalue problem of the negative p-Laplacian −p . The first eigenvalue in (3.16), denoted as usual by λ1 , is simple, isolated, and can be variationally characterized as     λ1 = inf |∇u|p dx + β |u|p dσ : |u|p dx = 1 . u∈W 1,p ()



∂



Nonlinear Elliptic Eigenvalue Problems Chapter | 3

67

This infimum is attained exactly on the eigenspace associated to λ1 . It is also known that the eigenfunctions corresponding to λ1 are of constant sign and belong to C 1,α () for some 0 < α < 1, and in fact λ1 is the only eigenvalue whose eigenfunctions do not change sign. For all these properties we refer to [127]. p in DefiniOur aim is to determine elements of the Fu˘cik spectrum  tion 3.16. They are found as critical points of a functional that is constructed by means of the Robin problem (3.15) following the approach in [186]. For a fixed s ∈ R, s ≥ 0, and corresponding to β ≥ 0 given in problem (3.15), we introduce the functional Js : W 1,p () → R by    p p |∇u| dx + β |u| dσ − s (u+ )p dx, Js (u) = 

∂



thus Js ∈ C 1 (W 1,p ()). Next we introduce the set    |u|p dx = 1 , S = u ∈ W 1,p () :

(3.17)



which is a C 1 submanifold of W 1,p (), and thus Js = Js |S is a C 1 function in the sense of manifolds. We note that u ∈ S is a critical point of Js (in the sense of manifolds) if and only if there exists t ∈ R such that    |∇u|p−2 ∇u · ∇v dx + β |u|p−2 uv dσ − s (u+ )p−1 v dx ∂   (3.18) p−2 |u| uv dx =t 

for all v ∈ W 1,p (); in fact, inserting v = u in (3.18) yields t = Js (u). Now we p . describe the relationship between the critical points of Js and the spectrum  Lemma 3.18. Given r, s ∈ R, s ≥ 0, one has that (r + s, r) belongs to the p if and only if there exists a critical point u ∈ S of Js such that spectrum  r = Js (u). p if and only Proof. The formula for the weak solution shows that (r + s, r) ∈  if there is u ∈ S that solves (3.18) with t = r. The lemma ensues. p through the critical points of Js . Lemma 3.18 enables us to find points in  In order to implement this, first we look for minimizers of Js . Proposition 3.19. There hold: (i) the first eigenfunction ϕ1 is a global minimizer of Js ; p . (ii) the point (λ1 , λ1 − s) ∈ R2 belongs to 

68 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Proof. (i) Since β, s ≥ 0, using the characterization of λ1 we have    |∇u|p dx + β |u|p dσ − s (u+ )p dx Js (u) =    ∂ p + p ≥ λ1 |u| dx − s (u ) dx ≥ λ1 − s = Js (ϕ1 ), ∀u ∈ S. 



(ii) On the basis of (i), we can apply Lemma 3.18. We produce a second critical point of Js as a local minimizer. Proposition 3.20. There hold: (i) the negative eigenfunction −ϕ1 is a strict local minimizer of Js ; p . (ii) the point (λ1 + s, λ1 ) ∈ R2 belongs to  Proof. (i) Arguing indirectly, let us suppose that there exists a sequence {un } ⊂ S with un = −ϕ1 , un → −ϕ1 in W 1,p () and Js (un ) ≤ λ1 = Js (−ϕ1 ). If un ≤ 0 for a.a. x ∈ , we obtain   Js (un ) = |∇un |p dx + β |un |p dσ > λ1 , 

∂

because un = −ϕ1 and un = ϕ1 , which contradicts the assumption Js (un ) ≤ λ1 . Consider now the complementary situation. Hence un changes sign whenever n is sufficiently large, thereby we can set wn =

u+ n + p un L ()

p

p

and rn = ∇wn Lp () + βwn Lp (∂) .

(3.19)

We claim that, along a relabeled subsequence, rn → +∞ as n → ∞. Suppose by contradiction that {rn } is bounded. This implies through (3.19) that {wn } is bounded in W 1,p (), so there exists a subsequence denoted again by {wn } such that wn → w in Lp (), for some w ∈ W 1,p (). Since wn Lp () = 1 and wn ≥ 0 a.e., we get wLp () = 1 and w ≥ 0. This contradicts that un → −ϕ1 in Lp (), thus proving the claim. On the other hand, from (3.19) and by using the variational characterization of λ1 , we infer that    + p − p p  |∇un | dx + β (u− Js (un ) = (rn − s) (un ) dx + n ) dσ   ∂   + p p ≥ (rn − s) (un ) dx + λ1 (u− n ) dx, 

whereas the choice of {un } gives Js (un ) ≤ λ1 = λ1



 

p (u+ n ) dx

 + λ1 

p (u− n ) dx.

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

69

Combining the above inequalities results in  p (λ1 − rn + s) (u+ n ) dx ≥ 0, 

therefore λ1 ≥ rn − s. This is against the unboundedness of {rn }, which completes the proof of (i). Part (ii) follows from Lemma 3.18 because Js (−ϕ1 ) = λ1 . Using the two local minima obtained in Propositions 3.19 and 3.20, we seek for a third critical point of Js via a version of the mountain pass theorem on C 1 manifolds. Taking into account (3.17), we define a norm of the derivative of the restriction Js of Js at the point u ∈ S by Js (u)∗ = min{t ∈ R : Js (u) − tT  (u)W 1,p ()∗ }, where T (u) =  |u|p dx. The definition of the Palais–Smale condition is here in the sense of functions on manifolds. Definition 3.21. The functional Js : S → R is said to satisfy the Palais–Smale condition on S if for any sequence {un } ⊂ S such that {Js (un )} is bounded and Js (un )∗ → 0 as n → ∞, there exists a strongly convergent subsequence in W 1,p (). We can check the Palais–Smale condition for Js on the manifold S. Lemma 3.22. The functional Js : S → R satisfies the Palais–Smale condition on S. Proof. Let {un } ⊂ S be a sequence such that {Js (un )} is bounded and Js (un )∗ → 0 as n → ∞, which means that there exists a sequence {tn } ⊂ R such that

  

p−2 p−2 p−1 |∇u | ∇u · ∇v dx + β |u | u v dσ − s (u+ v dx

n n n n n)

 ∂ 



p−2 − tn |un | un v dx ≤ n vW 1,p () (3.20)

 for all v ∈ W 1,p () and with n → 0+ . Note that Js (un ) ≥ ∇un Lp () − s. Since {un } ⊂ S and {Js (un )} is bounded, we derive that {un } is bounded in W 1,p (). Thus, along a relabeled subsequence we may suppose that un  u in W 1,p (), un → u in Lp () and un → u in Lp (∂). Taking v = un in (3.20) and using again the inclusion {un } ⊂ S shows that the sequence {tn } is bounded. Then, if we choose v = un − u, it follows that  |∇un |p−2 ∇un · ∇(un − u) dx → 0 as n → ∞. p



70 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Recalling that the negative p-Laplacian −p : W 1,p () → W 1,p ()∗ fulfills the (S)+ -property (see Proposition 1.56), we conclude that un → u in W 1,p (). The proof of the following version of the mountain pass theorem can be found in [98, Theorem 3.2]. Theorem 3.23. Let E be a Banach space and let g, f ∈ C 1 (E). Further, let M = {u ∈ E : g(u) = 0}, u0 , u1 ∈ M, and let  > 0 be such that u1 − u0 E >  and inf{f (u) : u ∈ M and u − u0 E = } > max{f (u0 ), f (u1 )}. Assume that f satisfies the Palais–Smale condition on M and that

= {γ ∈ C([−1, 1], M) : γ (−1) = u0 and γ (1) = u1 } is nonempty. Then c = inf

max

γ ∈ u∈γ ([−1,1])

f (u)

is a critical value of f |M . Now we obtain, in addition to ϕ1 and −ϕ1 , a third critical point of Js on S. Proposition 3.24. For each s ≥ 0, one has: (i) c(s) := inf

max

γ ∈ u∈γ ([−1,1])

Js (u),

(3.21)

where

= {γ ∈ C([−1, 1], S) : γ (−1) = −ϕ1 and γ (1) = ϕ1 }, is a critical value of Js satisfying c(s) > max{Js (−ϕ1 ), Js (ϕ1 )} = λ1 . In particular, there exists a critical point of Js that is different from −ϕ1 and ϕ1 . p . (ii) The point (s + c(s), c(s)) belongs to  Proof. (i) By Proposition 3.20 we know that −ϕ1 is a strict local minimizer of Js with Js (−ϕ1 ) = λ1 , while Proposition 3.19 ensures that ϕ1 is a global minimizer of Js with Js (ϕ1 ) = λ1 − s. Then we can show that inf{Js (u) : u ∈ S and u − (−ϕ1 )W 1,p () = } > max{Js (−ϕ1 ), Js (ϕ1 )} = λ1 , (3.22)

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

71

whenever  > 0 is sufficiently small. The proof that the inequality (3.22) is strict can be done as in [60, Lemma 2.9] (see also [150, Lemma 2.6]) on the basis of Ekeland’s variational principle. In order to fulfill the mountain-pass geometry, we choose  > 0 even smaller if necessary to have 2ϕ1 W 1,p () = ϕ1 − (−ϕ1 )W 1,p () > . Since Js : S → R satisfies the Palais–Smale condition on the manifold S as shown in Lemma 3.22, we may invoke the version of mountain pass theorem on manifolds in Theorem 3.23. This guarantees that c(s) introduced in (3.21) is a critical value of Js with c(s) > λ1 , providing a critical point different from −ϕ1 and ϕ1 . p . (ii) Thanks to Lemma 3.18 and part (i), we infer that (s + c(s), c(s)) ∈  p for s ≥ 0. As  p is Proposition 3.24 provides a curve (s + c(s), c(s)) ∈  symmetric with respect to the diagonal, we can complete it with its symmetric p , namely part obtaining a curve in  C := {(s + c(s), c(s)), (c(s), s + c(s)) : s ≥ 0}.

(3.23)

The proposition below establishes an important sign property related to the curve C. Proposition 3.25. Let (a0 , b0 ) ∈ C with C in (3.23) and let a, b ∈ L∞ () satisfy λ1 ≤ a(x) ≤ a0 , λ1 ≤ b(x) ≤ b0 for a.a. x ∈  such that λ1 < a(x) and λ1 < b(x) on subsets of positive measure. Then any nontrivial solution u of 

−p u = a(x)(u+ )p−1 − b(x)(u− )p−1

in ,

p−2 u |∇u|p−2 ∂u ∂ν = −β|u|

on ∂

(3.24)

changes sign in . Proof. Let u be a nontrivial solution of Eq. (3.24). Then −u is a nontrivial solution of  −p z = b(x)(z+ )p−1 − a(x)(z− )p−1 in , ∂z |∇z|p−2 ∂ν = −β|z|p−2 z

on ∂.

Hence we may suppose that the point (a0 , b0 ) ∈ C is such that a0 ≥ b0 . We argue by contradiction and assume that u does not change sign in . Without loss of generality, we may admit that u ≥ 0 a.e. in , so u is a solution of the Robin weighted eigenvalue problem with weight a(x): 

−p u = a(x)up−1

in ,

|∇u|p−2 ∂u ∂ν

on ∂.

= −βup−1

72 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

It means that u is an eigenfunction corresponding to the eigenvalue 1 for this problem. The first eigenvalue λ1 (a) of the above weighted problem is expressed as p p  |∇v| dx + β ∂ |v| dσ inf . λ1 (a) = p v∈W 1,p ()  a(x)|v| dx v ≡0

The fact that u ≥ 0 entails λ1 (a) = 1 because the only eigenfunction whose eigenfunctions do not change sign is λ1 (a) (see [127]). Then the hypothesis that λ1 < a(x) on a set of positive measure leads to the contradiction 1=

p  |∇ϕ1 | dx

+β λ1



p ∂ ϕ1 dσ

>

p  |∇ϕ1 | dx



+β



p ∂ ϕ1 dσ

p  a(x)ϕ1 dx

≥ λ1 (a)

= 1, which completes the proof. We use this sign information in the below proposition for establishing that p . the lines {λ1 } × R and R × {λ1 } are isolated in  p with an > λ1 and Proposition 3.26. There exists no sequence {(an , bn )} ⊂  bn > λ1 such that (an , bn ) → (a, b) with a = λ1 or b = λ1 . p Proof. Proceeding indirectly, assume that there exist sequences {(an , bn )} ⊂  1,p and {un } ⊂ W () with the properties: an → λ1 , bn → b, an > λ1 , bn > λ1 , un Lp () = 1, and 

p−1 − b (u− )p−1 −p un = an (u+ n n n)

in ,

n |∇un |p−2 ∂u ∂ν

on ∂.

= −β|un |p−2 un

(3.25)

If we act on (3.25) with the test function un , we get p



∇un Lp () = an



p (u+ n ) dx + bn

 

p (u− n ) dx − β

 |un |p dσ ≤ an + bn , ∂

which proves the boundedness of {un } in W 1,p (). Hence, along a subsequence, un  u in W 1,p () and un → u in Lp () and Lp (∂). Now, testing (3.25) with un − u, we infer that  lim

n→∞ 

|∇un |p−2 ∇un · ∇(un − u) dx = 0.

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

73

The (S)+ -property of −p on W 1,p () yields that un → u in W 1,p (). Thus, u is a solution of the equation  |∇u|p−2 ∇u · ∇v dx     (3.26) + p−1 − p−1 v dx − b (u ) v dx − β |u|p−2 uv dσ = λ1 (u ) 



∂

for all v ∈ W 1,p (). Inserting v = u+ in (3.26) leads to    |∇u+ |p dx = λ1 (u+ )p dx − β (u+ )p dσ. 



∂

This, in conjunction with the characterization of λ1 and since uLp () = 1, ensures that either u+ = 0 or u+ = ϕ1 . If u+ = 0, then u ≤ 0 and (3.26) implies that u is an eigenfunction. Recalling from Remark 3.17 that λ1 is the only eigenvalue whose eigenfunctions do not change sign, we deduce that u = −ϕ1 . This renders that {un } converges either to ϕ1 or to −ϕ1 in Lp (). Note that it forces to have either

|{x ∈  : un (x) < 0}| → 0

or

|{x ∈  : un (x) > 0}| → 0, (3.27)

respectively, where | · | denotes the Lebesgue measure. Indeed, assuming, for instance, un → ϕ1 in Lp (), since for any compact subset K ⊂  there holds   p |un − ϕ1 |p dx ≥ ϕ1 dx ≥ C|{un < 0} ∩ K|, {un 0}| q u+ n W 1,p () , 

with a constant C > 0. We infer that |{x ∈  : un (x) > 0}|

1− pq

≥

1 , (an + 1)C

74 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

and in the same way |{x ∈  : un (x) < 0}|

1− pq

≥

1 . (bn + 1)C

Hence we reach a contradiction with (3.27), which completes the proof. The following auxiliary fact is helpful to link with the preceding critical point results. Lemma 3.27. For every r > inf Js = λ1 − s, each connected component of S

{u ∈ S : Js (u) < r} contains a critical point, in fact a local minimizer of Js . Proof. Let C be a connected component of {u ∈ S : Js (u) < r} and denote d = inf{Js (u) : u ∈ C}. We claim that there exists u0 ∈ C such that Js (u0 ) = d. To this end, let {un } ⊂ C be a sequence such that d ≤ Js (un ) ≤ d +

1 . n2

Applying Ekeland’s variational principle to Js on C provides a sequence {vn } ⊂ C such that Js (vn ) ≤ Js (un ), 1 un − vn W 1,p () ≤ , n 1 Js (vn ) ≤ Js (v) + v − vn W 1,p () , n

(3.28) (3.29) ∀v ∈ C.

(3.30)

If n is sufficiently large, by (3.28) we obtain 1 Js (vn ) ≤ Js (un ) ≤ d + 2 < r. n Moreover, owing to (3.30), it can be shown that {vn } is a Palais–Smale sequence for Js . Then Lemma 3.22 and (3.29) ensure that, up to a relabeled subsequence, un → u0 in W 1,p () with u0 ∈ C and Js (u0 ) = d. We note that u0 ∈ C because otherwise the maximality of C as a connected component would be contradicted, so u0 is a local minimizer of Js and we are done. The next result points out that C (see (3.23)) is the first nontrivial curve p . in 

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

75

Theorem 3.28. Let s ≥ 0. Then (s + c(s), c(s)) ∈ C is the first point in the p and the ray {(s, 0) + t (1, 1) : t > λ1 }. intersection between  p with Proof. Assume, by contradiction, the existence of a point (s + μ, μ) ∈  λ1 < μ < c(s). p is closed enable us to suppose that μ is Proposition 3.26 and the fact that  the minimum number with the required property. By virtue of Lemma 3.18, μ is a critical value of the functional Js and there is no critical value of Js in the interval (λ1 , μ). We complete the proof by reaching a contradiction to the definition of c(s) in (3.21). To this end, it suffices to construct a path in along which there holds Js ≤ μ. Let u ∈ S be a critical point of Js with Js (u) = μ. Then u fulfills    |∇u|p−2 ∇u · ∇v dx = (s + μ) (u+ )p−1 v dx − μ (u− )p−1 v dx     p−2 |u| uv dσ −β ∂

for all v ∈ W 1,p (). Setting v = u+ and v = −u− yields    |∇u+ |p dx = (s + μ) (u+ )p dx − β (u+ )p dσ 

and





− p



|∇u | dx = μ 

(3.31)

∂

− p



(u ) dx − β 

(u− )p dσ,

(3.32)

∂

respectively. Since u changes sign (see Proposition 3.25), the following paths are well defined on S: (1 − t)u + tu+ (1 − t)u+ + tu− , u (t) = , 2 (1 − t)u + tu+ Lp () (1 − t)u+ + tu− Lp () −tu− + (1 − t)u u3 (t) =  − tu− + (1 − t)uLp ()

u1 (t) =

for all t ∈ [0, 1]. By means of direct calculations based on (3.31) and (3.32) we infer that Js (u1 (t)) = Js (u3 (t)) = μ

for all t ∈ [0, 1]

and Js (u2 (t)) = μ −

st p u− Lp () ≤μ (1 − t)u+ + tu− Lp ()

for all t ∈ [0, 1].

76 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Due to the minimality property of μ, the only critical points of Js in the set {w ∈ S : Js (w) < μ − s} are ϕ1 and possibly −ϕ1 provided μ − s > λ1 . We note that, because u− /u− Lp () does not change sign, it is not a critical point of Js . Therefore, there exists a C 1 path α : [−, ] → S with α(0) = u− /u− Lp () and d  dt Js (α(t))|t=0 = 0. Using this path and observing from (3.32) that Js (u− /u− Lp () ) = μ − s, we can move from u− /u− Lp () to a point v with Js (v) < μ − s. Applying Lemma 3.27, we find that the connected component of {w ∈ S : Js (w) < μ − s} containing v crosses {ϕ1 , −ϕ1 }. Let us say that it passes through ϕ1 , otherwise the reasoning is the same employing −ϕ1 . Consequently, there is a path u4 (t) from u− /u− Lp () to ϕ1 within the set {w ∈ S : Js (w) < μ − s}. Then the path −u4 (t) joins −u− /u− Lp () and −ϕ1 and, since u4 (t) ∈ S, we have Js (−u4 (t)) ≤ Js (u4 (t)) + s < μ − s + s = μ for all t. Connecting u1 (t), u2 (t) and u4 (t), we construct a path joining u and ϕ1 , and joining u3 (t) and −u4 (t) we get a path which connects u and −ϕ1 . These yield a path γ (t) on S joining ϕ1 and −ϕ1 . Furthermore, in view of the above discussion, it turns out that Js (γ (t)) ≤ μ for all t . This proves the theorem. Corollary 3.29. The second eigenvalue λ2 of (3.16) has the variational characterization    p p λ2 = inf max |∇u| dx + β |u| dσ , (3.33) γ ∈ u∈γ ([−1,1])



∂

with and S in Proposition 3.24. Proof. Theorem 3.28 for s = 0 ensures that c(0) = λ2 . Formula (3.33) follows by applying Proposition 3.24 (i) with s = 0. Proposition 3.30. The curve s → (s + c(s), c(s)) is Lipschitz continuous and decreasing. Proof. If s1 < s2 , it follows Js1 (u) ≥ Js2 (u) for all u ∈ S, which ensures that c(s1 ) ≥ c(s2 ). For every  > 0 there exists γ ∈ such that max

u∈γ ([−1,1])

Js2 (u) ≤ c(s2 ) + ,

hence 0 ≤ c(s1 ) − c(s2 ) ≤

max

u∈γ ([−1,1])

Js1 (u) −

max

u∈γ ([−1,1])

Js2 (u) + .

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

77

Taking u0 ∈ γ ([−1, 1]) such that max

u∈γ ([−1,1])

Js1 (u) = Js1 (u0 )

yields 0 ≤ c(s1 ) − c(s2 ) ≤ Js1 (u0 ) − Js2 (u0 ) +  = s1 − s2 + . As  > 0 was arbitrary, this ensures that s → (s + c(s), c(s)) is Lipschitz continuous. In order to prove that the curve is decreasing, it suffices to argue for s > 0. p , TheLet 0 < s1 < s2 . Then, since (s1 + c(s1 ), c(s1 )), (s2 + c(s2 ), c(s2 )) ∈  orem 3.28 implies that s1 + c(s1 ) < s2 + c(s2 ). On the other hand, as already remarked, c(s1 ) ≥ c(s2 ), which completes the proof. Next we investigate the asymptotic behavior of the curve C. Theorem 3.31. Let p ≤ N . Then lim c(s) = λ1 . s→+∞

Proof. Let us proceed by contradiction and suppose that c(s) does not converge to λ1 as s → +∞. Then there exists δ > 0 such that max

u∈γ ([−1,1])

Js (u) ≥ λ1 + δ

for all γ ∈ and all s ≥ 0.

Since p ≤ N , we can choose a function ψ ∈ W 1,p () which is unbounded above. Then we define γ ∈ by γ (t) =

tϕ1 + (1 − |t|)ψ , tϕ1 + (1 − |t|)ψLp ()

t ∈ [−1, 1].

For every s > 0, let ts ∈ [−1, 1] satisfy max Js (γ (t)) = Js (γ (ts )).

t∈[−1,1]

Denoting vs = ts ϕ1 + (1 − |ts |)ψ, we infer that     |∇vs |p dx + β |vs |p dσ − s (vs+ )p dx ≥ (λ1 + δ) |vs |p dx. 

∂





(3.34) Letting s → +∞, we can assume along a subsequence that ts →  t ∈ [−1, 1]. The family {vs : s > 0} being bounded in W 1,p (), from (3.34) one sees that  (vs+ )p dx → 0 as s → +∞, 

which forces  t|)ψ ≤ 0. tϕ1 + (1 − |

78 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Due to the choice of ψ , this is impossible unless  t = −1. Passing to the limit in (3.34) as s → +∞ and using  t = −1, we arrive at the contradiction δ ≤ 0, so the proof is complete. It remains to study the asymptotic properties of the curve C when p > N . For β = 0, problem (3.15) becomes a Neumann problem with homogeneous boundary condition that was studied in Theorem 3.9 yielding  λ1 = 0 if p ≤ N, lim c(s) = s→+∞ λˆ if p > N, where λˆ = inf

 

|∇u|p dx : u ∈ W 1,p (), uLp () = 1 and  u vanishes somewhere in  .

Therefore, we only have to treat the case β > 0. In this respect, the key idea is to work with an adequate equivalent norm on the space W 1,p (). So, for β > 0 we introduce the norm  1/p p p uβ = ∇uLp () + βuLp (∂) , (3.35) which is an equivalent norm on W 1,p (). Then we have the following result. Theorem 3.32. Let β > 0 and p > N . Then p

lim c(s) = inf max

s→+∞

u∈L r∈R

rϕ1 + uβ p

rϕ1 + uLp ()

=: λ,

where L = {u ∈ W 1,p () : u vanishes somewhere in , u ≡

0}. Moreover, λ > λ1 . Proof. First, we are going to prove the strict inequality λ > λ1 . The characterization of λ1 in Remark 3.17, (3.35), and the definition of λ yield p

λ1 =

inf

u∈W 1,p ()\{0}

uβ p

uLp ()

p

≤ inf

u∈L

uβ p

uLp ()

≤ λ.

(3.36)

Let us check that the first inequality in (3.36) is strict. Assuming on the contrary, we would find a sequence {un } ⊂ L satisfying

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

79

p

un β p

un Lp () Set vn =

un un β ,

→ λ1 as n → ∞.

hence vn β = 1 and 1 → λ1 as n → ∞. p vn Lp ()

Due to the compact embedding W 1,p () ⊂ C(), there is a subsequence of {vn }, still denoted by {vn }, such that vn  v in W 1,p () and vn → v uniformly on . These convergences imply vβ ≤ 1, v ∈ L, and finally p

vβ

1 = λ1 , n→∞ vn p p L ()

≤ lim

p vLp ()

which ensures that v is an eigenfunction in (3.16) corresponding to the first eigenvalue λ1 (see Remark 3.17). This is a contradiction because every eigenfunction associated to λ1 is strictly positive or negative on , whereas v ∈ L. Hence, recalling (3.36), we get λ > λ1 . Now we prove the first part in the theorem. We start by claiming that there exist u ∈ L such that p

max r∈R

rϕ1 + uβ p

rϕ1 + uLp ()

= λ.

(3.37)

By the definition of λ, we can find a sequence {un } ⊂ L such that p

max r∈R

rϕ1 + un β p

rϕ1 + un Lp ()

→ λ as n → ∞.

(3.38)

Without loss of generality, we can assume that un W 1,p () = 1. Hence we may suppose that un  u in W 1,p (), as well as un → u uniformly in , with some u ∈ L. The weak lower semicontinuity of  · β and (3.38) yield p

max r∈R

rϕ1 + uβ p

rϕ1 + uLp ()

p

≤ lim max n→∞ r∈R

rϕ1 + un β p

rϕ1 + un Lp ()

= λ.

Combining this with the definition of λ, we see that (3.37) holds true. To prove that c(s) → λ as s → +∞, we argue by contradiction admitting that there exists δ > 0 such that max Js (γ (t)) ≥ λ + δ for all γ ∈ and all s ≥ 0.

t∈[−1,1]

80 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Here the decreasing monotonicity of c(s) has been used (see Proposition 3.30). Consider the path γ ∈ defined by γ (t) =

tϕ1 + (1 − |t|)u , tϕ1 + (1 − |t|)uLp ()

t ∈ [−1, 1],

with u given in (3.37). Proceeding as in the proof of Theorem 3.31, for every s > 0 we fix ts ∈ [−1, 1] to satisfy max Js (γ (t)) = Js (γ (ts ))

t∈[−1,1]

and denote vs = ts ϕ1 + (1 − |ts |)u. We have     |∇vs |p dx + β |vs |p dσ − s (vs+ )p dx ≥ (λ + δ) |vs |p dx. (3.39) 

∂





From (3.39) and since {vs } is uniformly bounded, we obtain  (vs+ )p dx → 0 

t ∈ [−1, 1] as s → +∞, which yields  tϕ1 ≤ −(1 − | t|)u. As ϕ1 > 0 and ts →  and u vanishes somewhere in , we deduce that  t ≤ 0. In addition, passing to the limit in (3.39) leads to   p   |∇(tϕ1 + (1 − |t|)u)| dx + β | tϕ1 + (1 − | t|)u|p dσ  ∂  (3.40) tϕ1 + (1 − | t|)u|p dx. ≥ (λ + δ) | 

If  t = −1, (3.40) can be expressed as  t ϕ + uβ  1+  t 1

p

 t  1+ ϕ + uLp ()  t 1 p

≥ λ + δ.

Comparing with (3.37) reveals that a contradiction is reached. If  t = −1, in view of (3.40) and λ > λ1 , we also arrive at a contradiction, which establishes the result.

3.2 EIGENVALUE PROBLEMS FOR (p, q)-LAPLACIAN WITH INDEFINITE WEIGHTS The problem treated in this section is the following quasilinear elliptic problem involving the (p, q)-Laplacian  −p u − μq u = λ(mp (x)|u|p−2 u + μmq (x)|u|q−2 u) in , (3.41) u=0 on ∂,

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

81

on a bounded domain  ⊂ RN with C 2 boundary ∂, with 1 < q < p < +∞. Here we have two parameters λ ∈ R and μ ≥ 0, and two weights mp , mq ∈ L∞ () such that the Lebesgue measure of {x ∈  : mr (x) > 0} is positive whenever r ∈ {p, q}. A solution of problem (3.41) is understood in the weak sense, that is, any 1,p u ∈ W0 () for which it holds   |∇u|p−2 ∇u · ∇ϕ dx + μ |∇u|q−2 ∇u · ∇ϕ dx    = λ (mp |u|p−2 u + μmq |u|q−2 u)ϕ dx  1,p

whenever ϕ ∈ W0 (). In the sequel, ·r denotes the usual Lr -norm (1 ≤ r ≤ +∞). For any u ∈ R, we set u± := max{±u, 0}. The Lebesgue measure of  is denoted by ||. We recall from [14] (see also Proposition 1.57 in the case of nonnegative weights) that the first positive eigenvalue λ1 (r, mr ) of −r with weight function mr is obtained by minimizing the Rayleigh quotient:    r 1,r r  |∇u| dx : u ∈ W0 (), mr |u| dx > 0 . (3.42) λ1 (r, mr ) = inf r   mr |u| dx The first positive eigenvalue is simple and isolated. Since there exist no nonnegative eigenvalues corresponding to nonpositive weights, we set λ1 (r, mr ) = +∞ if

mr ≤ 0.

(3.43)

We also recall that λ1 (r, mr ) has a positive eigenfunction ϕ1 (r, mr ) ∈ C01,αr () with some αr ∈ (0, 1). It is worth mentioning that λ1 (r, mr ) and −λ1 (r, −mr ) are the only eigenvalues of −r with weight mr for which there exist constantsign eigenfunctions. We start by pointing out that finding a solution for (3.41) is equivalent to finding a solution in the case μ = 1, that is to solving the problem  −p u − q u = λ(mp (x)|u|p−2 u + mq (x)|u|q−2 u) in , (3.44) u=0 on ∂. Indeed, if u is a solution of (3.44), then multiplying Eq. (3.44) by s p−1 for s > 0 we see that v = su satisfies −p v − s p−q q v = λ(mp |v|p−2 v + s p−q mq |v|q−2 v) in . Choosing s p−q = μ, we obtain that v = su is a solution of (3.41). Conversely, if w is a solution of (3.41), then v = tw, with t = μ1/(q−p) , is a solution of (3.44).

82 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints 1,p

We introduce the functionals  and  on W0 () by 1 1 p q ∇up + ∇uq , p q   1 1 p (u) := mp |u| dx + mq |u|q dx p  q  (u) :=

(3.45) (3.46)

1,p

for all u ∈ W0 (). Proposition 3.33. Let  λ := inf

 (u) 1,p : u ∈ W0 (), (u) > 0 . (u)

(3.47)

Then λ = min{λ1 (p, mp ), λ1 (q, mq )}. corresponding to λ1 (p, mp ) Proof. Let ϕ1 and ψ1 be the positive eigenfunctions p q and λ1 (q, mq ), respectively, such that  mp ϕ1 dx = 1 and  mq ψ1 dx = 1. For sufficiently large t > 0, it turns out that    q (tϕ1 ) = t q t p−q + mq ϕ1 dx > 0 

because p > q. By (3.45), (3.46), (3.47), and since q − p < 0, we find (tϕ1 ) λ1 (p, mp ) + pt q−p  |∇ϕ1 |q dx/q λ≤ → λ1 (p, mp ) = q (tϕ1 ) 1 + pt q−p  mq ϕ1 dx/q as t → +∞, so λ ≤ λ1 (p, mp ). On the other hand, because (tψ1 ) λ1 (q, mq ) + qt p−q  |∇ψ1 |p dx/p → λ1 (q, mq ) = λ≤ p (tψ1 ) 1 + qt p−q  mp ψ1 dx/p as t → 0+ , we obtain that λ ≤ λ1 (q, mq ), which implies λ ≤ min{λ1 (p, mp ), λ1 (q, mq )}. In order to prove the converse inequality λ ≥ min{λ1 (p, mp ), λ1 (q, mq )}, suppose that λ < min{λ1 (p, mp ), λ1 (q, mq )}. Then, in view of (3.47), there ex1,p ists an element u ∈ W0 () satisfying (u) > 0 and (u) < min{λ1 (p, mp ), λ1 (q, mq )}. (u) We consider three cases:

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

 (i) (ii)

83

 mp |u| dx > 0 and p

 

mp |u|p dx ≤ 0 and

 

mp |u|p dx > 0 and

(iii) 

mq |u|q dx ≤ 0; mq |u|q dx > 0; mq |u|q dx > 0.



p Case (i). There hold p(u) ≤  mp |u|p dx and p(u) ≥ ∇up . The definition of λ1 (p, mp ) gives rise to the contradiction p

min{λ1 (p, mp ), λ1 (q, mq )} > Case (ii). Since q(u) ≤ arises:



q  mq |u|

∇up (u) ≥ λ1 (p, mp ). ≥ p (u) m  p |u| dx q

dx and q(u) ≥ ∇uq , the contradiction q

min{λ1 (p, mp ), λ1 (q, mq )} >

∇uq (u) ≥ ≥ λ1 (q, mq ). q (u) m  q |u| dx

Case (iii). It follows from (3.42) that  ∇urr

≥ λ1 (r, mr )

mr |u|r dx 

whenever r ∈ {q, p}. Hence we get   λ1 (q, mq ) λ1 (p, mp ) p mp |u| dx + mq |u|q dx (u) ≥ p q   ≥ min{λ1 (p, mp ), λ1 (q, mq )}(u) contradicting the assumption λ < min{λ1 (p, mp ), λ1 (q, mq )}. The next result is a key step in our approach. Proposition 3.34. Assume that λ1 (p, mp ) = λ1 (q, mq ) or ϕ1 = tψ1 for all t > 0, where ϕ1 and ψ1 are positive eigenfunctions corresponding to λ1 (p, mp ) and λ1 (q, mq ), respectively. Then the infimum in (3.47) is not attained. 1,p

Proof. Arguing by contradiction, suppose there exists u ∈ W0 () such that (u) > 0 and (u) = λ = min{λ1 (p, mp ), λ1 (q, mq )}. (u)

(3.48)

Here Proposition 3.33 gives the second equality. We argue by considering the three cases in the proof of Proposition 3.33.

84 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Case (i). By means of (3.48) and the second assumption made in Case (i) of Proposition 3.33, we may write p

λ=

q

p

∇up (u) ∇up + p∇uq /q ≥ ≥ λ1 (p, mp ) ≥ λ. ≥ p dx p (u) m |u| m  p  p |u| dx

This implies p



∇up = λ1 (p, mp )

mp |u|p dx

and

∇uq = 0,



which is against u = 0. Case (ii). On the basis of (3.48) and the second assumption in Case (ii) of Proposition 3.33, we get  q ∇uq = λ1 (q, mq ) mq |u|q dx and ∇up = 0, 

which contradicts u = 0. Case (iii). Reasoning as in Case (iii) of Proposition 3.33, we show that q |∇u|p dx   |∇u| dx (p, m ) = λ (q, m ) = = λ . 1 p 1 q p q  mp |u| dx  mq |u| dx The simplicity of the eigenvalue λ1 (r, mr ) (for r ∈ {p, q}) guarantees that u = tψ1 = sϕ1 for some t = 0 and s = 0. The hypothesis made in the statement of the proposition is contradicted. We are in a position to state our first main theorem, which is a nonexistence result for (3.44), hence also for (3.41) in view of the preceding comments. Theorem 3.35. If − min{λ1 (p, −mp ), λ1 (q, −mq )} < λ < min{λ1 (p, mp ), λ1 (q, mq )}, then (3.44) has no nontrivial solutions. Moreover, if one of the following conditions holds: (i) λ1 (p, mp ) = λ1 (q, mq ) (resp. λ1 (p, −mp ) = λ1 (q, −mq )); (ii) ϕ1 (p, mp ) = tψ1 (q, mq ) for all t > 0 (resp. ϕ1 (p, −mp ) = tψ1 (q, −mq ) (whenever max{λ1 (p, −mp ), λ1 (q, −mq )} < +∞) for all t > 0), where ϕ1 (p, ±mp ) and ψ1 (q, ±mq ) are positive eigenfunctions which correspond to λ1 (p, ±mp ) and λ1 (q, ±mq ), respectively, then problem (3.44) with λ = min{λ1 (p, mp ), λ1 (q, nq )} (resp. λ = − min{λ1 (p, −mp ), λ1 (q, −mq )}) has no nontrivial solutions.

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

85

Proof. Arguing by contradiction, let u be a nontrivial solution of (3.44). Then, for every s > 0, we have that v = su is a nontrivial solution of the Dirichlet problem   −p v − s p−q q v = λ mp |v|p−2 v + s p−q mq |v|q−2 v in , v = 0 on ∂. Choosing s p−q = p/q and then acting with su as test function on the above equation, we obtain 0 < p(su) = pλ(su) (see (3.45) and (3.46)). First, we examine the case of λ > 0. From the above estimate and according to Proposition 3.33, we obtain the contradiction λ=

(su) ≥ λ = min{λ1 (p, mp ), λ1 (q, mq )}, (su)

which yields the first assertion of the theorem. The second part of the theorem follows readily by applying Proposition 3.34. It remains to examine the case of λ < 0. We split the proof into three parts. Case 1. (mp )− ≡ 0 and (mq )− ≡ 0. The desired conclusion follows from the above argument by replacing λ, mp , mq with −λ, −mp , −mq , respectively. Case 2. (mp )− ≡ 0 and mq ≥ 0. On account of (3.43), we note that min{λ1 (p, −mp ), λ1 (q, −mq )} = λ1 (p, −mp ). For any s > 0, we obtain the estimate    p p (−mp )|su|p dx + (−mq )|su|q dx 0 < ∇(su)p < p(su) = −λ q    ≤ −λ (−mp )|su|p dx, whence have



  (−mp )|su|

> 0. Therefore, by the definition of λ1 (p, −mp ), we

p dx

p

∇(su)p ≥ λ1 (p, −mp ), p (−m p )|su| dx 

−λ >

contradicting the assumption of the theorem. Case 3. (mq )− ≡ 0 and mp ≥ 0. As before, this situation entails min{λ1 (p, −mp ), λ1 (q, −mq )} = λ1 (q, −mq ).

86 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Since 0


∇(su)q ≥ λ1 (q, −mq ), q (−m q )|su| dx 

which contradicts the assumption of the theorem, completing the proof. Remark 3.36. If λ1 (p, mp ) = λ1 (q, mq ) and ϕ1 (p, mp ) = tψ1 (q, mq ) for some t > 0, then ψ1 := ψ1 (q, mq ) and ϕ1 := ϕ1 (p, mp ) are both positive solutions of (3.44) with λ = λ1 (p, mp ) = λ1 (q, mq ). Indeed, the definitions of ϕ1 and ψ1 yield p−1

−p ϕ1 = λmp ϕ1

q−1

and − q ψ1 = λmq ψ1

,

while the (p − 1)-homogeneity of these equations combined with the equality ϕ1 = tψ1 implies that p−1

−p ψ1 = λmp ψ1

q−1

and − q ϕ1 = λmq ϕ1

.

For a later use, we also quote the following auxiliary result. Lemma 3.37 ([224, Lemma 13]). Set   1,p p p X(d) := u ∈ W0 () : ∇up ≤ dup

(3.49)

for d > 0. Then there exists C = C(d) > 0 such that ∇up ≤ Cuq

for all u ∈ X(d).

An essential part in our approach for problem (3.44) (thus, (3.41)) is played by the underlying variational structure. We define the functional Iλ = I(λ,mp ,mq ) 1,p

on W0 () by (3.50) Iλ (u) = I(λ,mp ,mq )   1 1 λ λ p q := ∇up + ∇uq − mp (u+ )p dx − mq (u+ )q dx p q p  q  1,p

for all u ∈ W0 ().

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

87

Remark 3.38. If u is a nontrivial critical point of Iλ , then u is a positive solution of (3.44). Indeed, using −u− as a test function leads to 0 = Iλ (u), −u−  = ∇u− p + ∇u− q , p

q

whence u− = 0. This amounts to saying that u is a nonnegative solution of (3.44). The nonlinear regularity theory up to the boundary in [129, Theo1,β rem 1] and [130, p. 320] (see also Proposition 2.12) ensures that u ∈ C0 () with some β ∈ (0, 1). Moreover, the strong maximum principle in Theorem 2.19 guarantees that u > 0 in  and ∂u/∂ν < 0 on ∂, because u ≡ 0 and u ≥ 0. Our main existence result for problem (3.44) (or (3.41)) in the nonresonant case is as follows. Theorem 3.39. Assume that the condition λ1 (p, mp ) = λ1 (q, mq )

(resp. λ1 (p, −mp ) = λ1 (q, −mq ))

holds. If min{λ1 (p, mp ), λ1 (q, mq )} < λ < max{λ1 (p, mp ), λ1 (q, mq )} − max{λ1 (p, −mp ), λ1 (q, −mq )} < λ < − min{λ1 (p, −mp ), λ1 (q, −mq )}),

(resp.

then (3.44) has at least one positive solution. Proof. On the basis of Remark 3.38, it suffices to provide a nontrivial critical point of Iλ . According to the relevant intervals for λ, the proof will be separately developed in four cases: (i) (ii) (iii) (iv)

λ1 (q, mq ) < λ < λ1 (p, mp ); λ1 (q, −mq ) < −λ < λ1 (p, −mp ); λ1 (p, mp ) < λ < λ1 (q, mq ); λ1 (p, −mp ) < −λ < λ1 (q, −mq ).

(i) Iλ is sequentially weakly lower semicontinuous since mp , mq ∈ L∞ () 1,p and the embeddings of W0 () into Lp () and Lq () are compact. Fix ε > 0 such that (1 − ε)λ1 (p, mp ) > λ,

(3.51) 1,p

which is possible due to the assumption in case (i). For every u ∈ W0 () with + )p dx ≤ 0, through Hölder’s inequality we obtain m (u  p 1 p ∇up + p 1 p ≥ ∇up − p

Iλ (u) ≥

1 λ q q ∇uq − mq ∞ u+ q q q λ q mq ∞ up ||1−q/p q

88 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

≥ 1,p

For u ∈ W0 () with

λmq ∞ ||1−q/p 1 p q ∇up . ∇up − p qλ1 (p, 1)q/p

+ p  mp (u ) dx

> 0, by (3.42) we have  + p ∇u p ≥ λ1 (p, mp ) mp (u+ )p dx. 

Then, taking into account (3.51), we derive ε p ∇up + p ε p ≥ ∇up − p ε p ≥ ∇up − p

Iλ (u) ≥

 (1 − ε)λ1 (p, mp ) − λ λ q mp (u+ )p dx − mq ∞ u+ q p q  λ q mq ∞ up ||1−q/p q λmq ∞ ||1−q/p q ∇up . (3.52) qλ1 (p, 1)q/p

Since q < p, it follows from (3.51) and (3.52) that Iλ is coercive and bounded below. Consequently, by a standard result (see, e.g., [152, Theorem 1.1]), there exists a global minimizer u0 of Iλ . In order to have u0 = 0 it suffices to show that (Iλ (u0 ) =) min Iλ < 0. Let 1,p

W0 ()

be the eigenfunction corresponding to λ1 (q, mq ) that satisfies q m ψ1 dx = 1. Because λ > λ1 (q, mq ), for sufficiently small t > 0, q 

ψ1

 Iλ (tψ1 ) = t q

λt p−q t p−q p ∇ψ1 p − p p

 

p

mp ψ1 dx +

λ1 (q, mq ) − λ q

 < 0,

which completes the proof of case (i). (ii) Since λ1 (q, −mq ) < +∞, by (3.43) we must have (−mq )+ ≡ 0. We distinguish two cases. Case Where mp Changes Sign. Applying the result in case (i) with −λ, −mp , −mq in place of λ, mp , mq , respectively, permits achieving the conclusion. Case Where mp ≥ 0. In this case, λ1 (p, −mp ) = +∞. We note that −λ > 0, whence  λ − mp (u+ )p dx ≥ 0 p  1,p

for all u ∈ W0 (). Hence we have Iλ (u) ≥

λmq ∞ ||1−q/p 1 p q ∇up ∇up − p qλ1 (p, 1)q/p

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

89

1,p

for all u ∈ W0 (), whence the functional Iλ is coercive and bounded below. As in the proof of case (i), replacing ψ1 by the normalized positive eigenfunction ψ1 (q, −mq ) corresponding to λ1 (q, −mq ), we can show that Iλ has a global minimizer u0 with Iλ (u0 ) < 0, which completes the proof of case (ii). (iii) We claim that there exist δ > 0 and ρ > 0 such that Iλ (u) ≥ δ

whenever uq = ρ.

(3.53)

Choose d > max{1, λmp ∞ , λmp ∞ /λ1 (p, 1)}.

(3.54)

Thanks to Lemma 3.37, there exists a constant C > 0 such that ∇up ≤ Cuq

for all u ∈ X(d),

with X(d) as in (3.49). For any u ∈ X(d) satisfying by (3.54) and (3.55) we have

(3.55)

+ q  mq (u )

dx ≤ 0,

λmp ∞ 1−d λ1 (q, 1) d p q p p ∇up + uq + ∇up − up p q p p   λmp ∞ 1−d λ1 (q, 1) 1 p q p ≥ ∇up + uq + ∇up d − p q p λ1 (p, 1) (q, 1) λ (1 − d)C p 1 p q (3.56) uq + uq . ≥ p q For any u ∈ X(d) satisfying  mq (u+ )q dx ≤ 0, owing to (3.49) and (3.54) we find Iλ (u) ≥

Iλ (u) ≥

d − λmp ∞ λ1 (q, 1) λ1 (q, 1) p q q up + uq ≥ uq . p q q

1,p

If u ∈ W0 () fulfills

(3.57)



+ q  mq (u )

dx > 0, through (3.42) we get  q q ∇uq ≥ ∇u+ q ≥ λ1 (q, mq ) mq (u+ )q dx.

(3.58)



Our assumption on λ enables us to fix 0 < ε < 1 with (1 − ε)λ1 (q, mq ) > λ.

(3.59)

If in addition u ∈ X(d), then due to (3.54), (3.58), and (3.59) we have the estimate Iλ (u) ≥

d − λmp ∞ ε p q up + ∇uq p q

90 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints



1 + mq (u+ )q dx (1 − ε)λ1 (q, mq ) − λ q  ελ1 (q, 1) q ≥ (3.60) uq . q Finally, if u ∈ X(d) and  mq (u+ )q dx > 0, then (3.54), (3.55), (3.58), and (3.59) imply 

1−d ε 1 p q mq (u+ )q dx ∇up + ∇uq + (1 − ε)λ1 (q, mq ) − λ Iλ (u) ≥ p q q  λmp ∞ d p p up + ∇up − p p dλ1 (p, 1) − λmp ∞ (1 − d)C p ελ1 (q, 1) p q p ≥ uq + uq + up p q p (1 − d)C p ελ1 (q, 1) p q ≥ (3.61) uq + uq . p q Using that q < p, the claim in (3.53) follows from (3.56), (3.57), (3.60), and (3.61). Now we claim that there is R > 0 such that Rϕ1 q > ρ

and

Iλ (Rϕ1 ) < 0,

(3.62)

where ρ > 0 is the constant in (3.53) and ϕp1 is the positive eigenfunction corresponding to λ1 (p, mp ) satisfying  mp ϕ1 dx = 1. The claim in (3.62) is true because for a sufficiently large R > 0 we have    Iλ (Rϕ1 ) λ1 (p, mp ) − λ 1 q q =  − m ϕ dx λ1 (p, mp ) and p > q. Since mp , mq ∈ L∞ () and 0 ≤ λ = λ1 (p, mp ), the functional Iλ satisfies the Palais–Smale condition. The proof can be done along the lines of [224, Lemma 12]. Therefore the properties stated in (3.53) and (3.62) allow us to apply the mountain pass theorem (see Theorem 1.44), which guarantees the existence of a critical value c ≥ δ of Iλ , with δ > 0 in (3.53), namely c = inf max Iλ (γ (t)), 

γ ∈ t∈[0,1]

 1,p  := γ ∈ C([0, 1], W0 ()) : γ (0) = 0, γ (1) = Rϕ1 . This completes the proof of case (iii). (iv) Since λ1 (p, −mp ) < +∞, we must have (−mp )+ ≡ 0. We distinguish two cases.

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

91

Case Where mq Changes Sign. The conclusion can be achieved through the proof in case (iii) with −λ, −mp , −mq in place of λ, mp , mq , respectively. Case Where mq ≥ 0. In this case, λ1 (q, −mq ) = +∞. Under the assumption of case (iv), we notice that −λ > 0, which results in   + q λ mq (u ) dx = (−λ) (−mq )(u+ )q dx ≤ 0 



1,p

for all u ∈ W0 (). Arguing as in (3.56) and (3.57) with −λ, −mq instead of λ, mq , respectively, enables us to find ⎧ λ1 (q, 1) (1 − d)C p ⎪ p q ⎪ ⎨ uq + uq if u ∈ X(d), p q Iλ (u) ≥ ⎪ λ (q, 1) q ⎪ ⎩ 1 if u ∈ X(d), uq q where d > 0 is a constant satisfying d > max{1, −λmp ∞ , −λmp ∞ /λ1 (p, 1)} and C = C(d) is determined by Lemma 3.37. Thus there exist δ > 0 and ρ > 0 such that (3.53) holds true. Moreover, proceeding as in case (iii), this time with the normalized positive eigenfunction  ϕ1 := ϕ1 (p, −mp ) corresponding to λ1 (p, −mp ) in place of ϕ1 provides a constant R > 0 such that R ϕ1  q > ρ

and

Iλ (R ϕ1 ) < 0.

Because Iλ = I(λ,mp ,mq ) = I(−λ,−mp ,−mq ) fulfills the Palais–Smale condition as can be seen on the pattern of part (iii) (note that −λ > λ1 (p, −mp )), the same argument as in case (iii) ensures the existence of a nontrivial critical point of Iλ , which in view of Remark 3.38 completes the proof. Next we present our results for problem (3.44) in the resonant cases. Theorem 3.40. Assume that λ = λ1 (p, mp ) > λ1 (q, mq ) and



resp. λ = −λ1 (p, −mp ) < −λ1 (q, −mq ) (3.63) 

 |∇ϕ1 (p, mp )| dx − λ q

mq ϕ1 (p, mp )q dx > 0    |∇ϕ1 (p, −mp )|q dx − λ mq ϕ1 (p, −mp )q dx > 0 . 

 resp.







Then (3.44) has at least one positive solution.

(3.64)

92 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Proof. We suppose that λ = λ1 (p, mp ) > λ1 (q, mq ). The situation λ = −λ1 (p, −mp ) < −λ1 (q, −mq ) can be analogously handled taking −λ, −mp , −mq instead of λ, mp , mq , respectively. Since the Lebesgue measure of {x ∈  : mp (x) > 0} is positive, there exists an n0 ∈ N such that 1,p (mp − 1/n0 )+ ≡ 0. For n ≥ n0 , we define the functional In on W0 () by In (u) := I(λ,mp ,mq ) (u) +

λ + p u p = I(λ,mp −1/n,mq ) (u) pn

1,p

for all u ∈ W0 () (see (3.50)). Using that λ1 (p, mp − 1/n) > λ1 (p, mp ) = λ (where the first inequality follows from (3.42)), we are able to apply Theorem 3.39 obtaining a positive solution un of the Dirichlet problem 

−p v − q v = λ (mp − 1/n)|v|p−2 v + mq |v|q−2 v in , v=0 on ∂. Through the proof of Theorem 3.39 (see the case (i) therein), we may assume that un is a global minimizer of In and In (un ) < 0. In addition, observing that In ≤ In0 provided n ≥ n0 , we infer that 0 > In0 (un0 ) ≥ In (un0 ) ≥ min In = In (un ) 1,p

W0 ()

for all n ≥ n0 . At this point, we note that if un  := ∇un p is bounded, then {un } is a bounded Palais–Smale sequence of Iλ because In (un ) = 0 and Iλ (un )W −1,p () = Iλ (un ) − In (un )

−1,p  W0 ()

≤

λλ1 (p, 1)−1/p p−1 (un )+ p . n

On the other hand, by a standard argument based on the (S)+ -property of −p (see Proposition 1.56), it can be readily shown that {un } has a subsequence converging to some critical point u0 of Iλ . We note that u0 = 0 because Iλ (u0 ) = lim In (un ) ≤ In0 (un0 ) < 0. n→∞

Therefore u0 is a positive solution of (3.44) (see Remark 3.38). This ensures that the conclusion of the theorem is achieved once the sequence {un } is bounded. In order to prove the boundedness of {un }, suppose that un  → +∞ along a relabeled subsequence. Setting wn := un /un , we may admit that {wn }

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

93

1,p

converges to w0 weakly in W0 () and strongly in Lp () for some w0 ∈ 1,p W0 (). We observe that wn ≥ 0 and so by taking (wn − w0 )/un p−1 as test function, we obtain w n − w0 0 = In (un ),  un p−1  |∇wn |p−2 ∇wn · ∇(wn − w0 ) dx =   1 |∇wn |q−2 ∇wn · ∇(wn − w0 ) dx + un p−q   1 p−1 − λ (mp − )wn (wn − w0 ) dx n   λ q−1 mq wn (wn − w0 ) dx − un p−q   = |∇wn |p−2 ∇wn · ∇(wn − w0 ) dx + o(1), 

where o(1) → 0 as n → ∞. Due to the (S)+ -property of −p given in Propo1,p sition 1.56, this implies that wn → w0 strongly in W0 (). Then, for any 1,p ϕ ∈ W0 (), by taking ϕ/un p−1 as test function we have  1 |∇wn |p−2 ∇wn · ∇ϕ dx + |∇wn |q−2 ∇wn · ∇ϕ dx un p−q     1 p−1 λ q−1 − λ (mp − )wn ϕ dx − mq wn ϕ dx. p−q n u  n   

0=

Letting n → ∞, it follows that p−1

−p w0 = λmp (x)w0

in ,

w0 = 0 on ∂,

with w0 ≥ 0 and ∇w0 p = 1. According to the strong maximum principle (see Theorem 2.19), we see that w0 > 0 in . This entails w0 = ϕ1 /∇ϕ1 p because λ = λ1 (p, mp ) is a simple eigenvalue, where ϕ1 = ϕ1 (p, mp ) is a positive eigenfunction corresponding to λ1 (p, mp ). The facts that In (un ) < 0 for all n ≥ n0 and un is a critical point of In result in     1 1 In (un ) q q = − ∇wn q − λ mq wn dx . 0≥ un q q p  q

Letting n → ∞ leads to ∇ϕ1 q − λ The proof is thus complete.



q  mq ϕ1

dx ≤ 0, which contradicts (3.64).

94 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Theorem 3.41. Assume that λ = λ1 (q, mq ) > λ1 (p, mp )



resp. λ = −λ1 (q, −mq ) < −λ1 (p, −mp ) (3.65)

and 

 |∇ψ1 (q, mq )|p dx − λ

 resp.

mq ψ1 (q, mq )p dx > 0     p |∇ψ1 (q, −mq )| dx − λ mq ψ1 (q, −mq )p dx > 0 . 

(3.66)



Then (3.44) has at least one positive solution. Proof. As in the proof of Theorem 3.40, we can consider according to (3.65) that λ = λ1 (q, mq ) > λ1 (p, mp ) and choose n0 ∈ N satisfying (mq − 1/n0 )+ ≡ 0. 1,p For n ≥ n0 , we define the functional Jn on W0 () by Jn (u) := I(λ,mp ,mq ) (u) +

λ + q 1,p u q = I(λ,mp ,mq −1/n) (u) for all u ∈ W0 (). qn

Using that λ1 (q, mq − 1/n) > λ1 (q, mq ) = λ for any n ≥ n0 , we are in a position to apply Theorem 3.39 obtaining a positive solution un of the Dirichlet problem 

−p v − q v = λ mp |v|p−2 v + (mq − 1/n)|v|q−2 v in , v=0 on ∂. Following the pattern of the proof of Theorem 3.40, this time proceeding as in the case (iii) in the proof of Theorem 3.39, we deduce that Jn (un ) > 0 for 1,p all n ≥ n0 . Furthermore, the sequence {un } is bounded in W0 (). Namely, if un  := ∇un p → +∞ along a subsequence, the same argument as in the proof of Theorem 3.40 yields that wn := un /un  has a subsequence converging to a positive eigenfunction w0 of −p with weight mp , corresponding to λ. This yields that λ = λ1 (p, mp ) and w0 = ϕ1 /ϕ1 , where ϕ1 = ϕ1 (p, mp ) is a positive eigenfunction corresponding to λ1 (p, mp ); indeed, any positive eigenvalue other than λ1 (p, mp ) has no positive eigenfunctions and the eigenvalue λ1 (p, mp ) is simple. However, this is a contradiction because we are considering the case where λ = λ1 (q, mq ) > λ1 (p, mp ). The bounded sequence {un } is a Palais–Smale sequence for the functional Iλ as can be seen from the estimate Iλ (un )W −1,p () = Iλ (un ) − Jn (un )W −1,p () ≤

q−1

Cun q n

,

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

95

with a positive constant C independent of n. Proceeding as in the proof of Theorem 3.40, {un } possesses a subsequence converging to a critical point u0 of Iλ . In order to complete the proof, due to Remark 3.38, it suffices to justify that u0 = 0. By way of contradiction we assume that up to a subsequence un → 0 1,p strongly in W0 (). Set vn := un /un . Then, up to a subsequence, we may 1,p suppose that {vn } converges to v0 weakly in W0 () and strongly in Lp () 1,p for some v0 ∈ W0 (). It is clear that v0 ≥ 0. Using (vn − v0 )/un q−1 as test function, we obtain vn − v0  un q−1  |∇vn |p−2 ∇vn · ∇(vn − v0 ) dx = un p−q   + |∇vn |q−2 ∇vn · ∇(vn − v0 ) dx    1 q−1 p−1 mp vn (vn − v0 ) dx − λ (mq − )vn (vn − v0 ) dx − λun p−q n    |∇vn |q−2 ∇vn · ∇(vn − v0 ) dx + o(1), =

0 = Jn (un ),

 1,q

where o(1) → 0 as n → ∞. Invoking the (S)+ -property of −q on W0 () 1,q (see Proposition 1.56), we infer that vn → v0 strongly in W0 (). We claim that v0 = 0. By taking un as test function, we may write 0 = Jn (un ), un 

   1 p p q q un dx mq − = ∇un p − λ mp un dx + ∇un q − λ n   p p ≥ ∇un p − λ mp un dx. 



(3.67)

q Here we have utilized the inequality ∇uq − λ  mq uq dx ≥ 0 for all u ∈ 1,q W0 (), which follows from the definition of λ1 (q, mq ) and since λ = p p λ1 (q, mq ). This renders ∇un p ≤ λmp ∞ un p , whence un ∈ X(d) with d = λmp ∞ (see (3.49) for the definition of X(d)). Therefore, Lemma 3.37 guarantees the existence of a constant C > 0 such that ∇un p ≤ Cun q ≤ Cλ1 (q, 1)−1/q ∇un q for all n ≥ n0 . Consequently, recalling that vn → v0 1,q strongly in W0 (), we derive ∇un q 1 ≥ > 0, n→∞ ∇un p Cλ1 (q, 1)−1/q

∇v0 q = lim ∇vn q = lim n→∞

thus proving our claim.

96 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints 1,p

For any ϕ ∈ W0 (), by taking ϕ/un q−1 as test function we get ϕ  un q−1   |∇vn |p−2 ∇vn · ∇ϕ dx + |∇vn |q−2 ∇vn · ∇ϕ dx = un p−q       1 q−1 p−1 p−q − λun  mp vn ϕ dx − λ vn ϕ dx mq − n     q−1 |∇vn |q−2 ∇vn · ∇ϕ dx − λ mq vn ϕ dx + o(1), =

0 = Jn (un ),





where o(1) → 0 as n → ∞. Letting n → ∞ shows that −q v0 = λmq |v0 |q−2 v0 in  and v0 = 0 on ∂. The simplicity of the eigenvalue λ1 (q, mq ) ensures that v0 = ψ1 /ψ1 , where ψ1 = ψ1 (q, mq ) stands for a positive eigenfunction corresponding to λ1 (q, mq ). Using that Jn (un ) > 0 for all n ≥ n0 in conjunction with (3.67), we obtain     1 1 Jn (un ) p p =  − λ m v dx . − ∇v 0≤ n p p n un p p q  Hence, on account of 1/p − 1/q < 0, it turns out that  p p ∇vn p − λ mp vn dx ≤ 0.  1,p

Since {vn } converges to v0 = ψ1 /ψ1  weakly in W0 () and strongly in p p Lp (), by passing to the limit inferior we have ∇ψ1 p − λ  mp ψ1 dx ≤ 0. This contradicts assumption (3.66). Consequently, our claim u0 = 0 holds true. The proof of the theorem is complete. We produce some examples where the hypotheses of the formulated results are fulfilled. Example 3.42. Let us take N = 1,  = (0, π), 1 < q = 2 < p, mq ≡ 1 and mp,n (x) = 1 − h(x)/n with 0 ≤ h ∈ L∞ () and h ≡ 0. Then it is clear that λ1 (2, m2 ) = 1 and ψ1 = ψ1 (2, m2 ) = sin x. By easy computation, we have  π   π 1 π p |ψ1 |p dx − mp,n ψ1 dx = h(x) sinp x dx > 0 n 0 0 0 for every n ∈ N, so (3.66) holds true. On the other hand, we see that π  p |ψ | dx =1 λ1 (p, 1) < 0 π 1p 0 ψ1 dx

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

97

by the definition of λ1 (p, 1) and the facts that ψ1 = tϕ1 (p, 1) for any t > 0 and λ1 (p, 1) is simple. By the continuity of λ1 (p, mp ) with respect to mp , it follows that for a sufficiently large n we have λ1 (p, mp,n ) < 1, so (3.65) is valid, too. Similarly, we can construct examples satisfying (3.63) and (3.64).

3.3 EIGENVALUE PROBLEMS FOR ASYMPTOTICALLY HOMOGENEOUS DIFFERENTIAL ELLIPTIC OPERATORS The object of this section is to study the eigenvalue problem consisting in determining the existence of λ ∈ R called eigenvalue provided  −div A(x, ∇u) = λ|u|p−2 u in , (3.68) u=0 on ∂ has a nontrivial solution. Here  ⊂ RN is a bounded domain with C 2 boundary ∂, 1 < p < +∞, and A :  × RN → RN is a map which is strictly monotone in the second variable and satisfies certain regularity conditions. We do not suppose that A is (p − 1)-homogeneous in the second variable. We assume that the map A satisfies the following conditions: (HA ) A(x, y) = a(x, |y|)y, where a(x, t) > 0 for all (x, t) ∈  × (0, +∞), and (i) A ∈ C( × RN , RN ) ∩ C 1 ( × (RN \ {0}), RN ); (ii) there exists C1 > 0 such that Dy A(x, y) ≤ C1 |y|p−2

for every x ∈ , and y ∈ RN \ {0};

(iii) there exists C0 > 0 such that Dy A(x, y)ξ · ξ ≥ C0 |y|p−2 |ξ |2 for every x ∈ , y ∈ RN \ {0} and ξ ∈ RN ; (iv) there exists C2 > 0 such that Dx A(x, y) ≤ C2 (1 + |y|p−1 )

for every x ∈ , y ∈ RN \ {0};

(v) there exist C3 > 0 and 0 < t0 ≤ 1 such that Dx A(x, y) ≤ C3 |y|p−1 (− log |y| ) for every x ∈ , y ∈ RN with 0 < |y| < t0 . Without any loss of generality we also suppose that C0 ≤ p − 1 ≤ C1 . Notice that the case C0 = p − 1 = C1 characterizes the p-Laplacian. 1,p The Banach space W0 () is endowed with the norm u := ∇up , where  · q denotes the usual Lq -norm for 1 ≤ q ≤ +∞ (see Remark 1.18). We also denote by || the Lebesgue measure of .

98 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Set 

|y|

G(x, y) :=

(3.69)

a(x, t)t dt. 0

We can easily see that ∇y G(x, y) = A(x, y)

and

G(x, 0) = 0

for every x ∈ . Remark 3.43. The following assertions hold under condition (HA ): (i) for all x ∈ , A(x, y) is maximal monotone and strictly monotone in y; C1 (ii) |A(x, y)| ≤ |y|p−1 for every (x, y) ∈  × RN ; p−1 C0 (iii) A(x, y) · y ≥ |y|p for every (x, y) ∈  × RN ; p−1 (iv) G(x, y) defined in (3.69) is strictly convex in y for all x and satisfies the following inequalities: A(x, y) · y ≥ G(x, y) ≥

C0 |y|p p(p − 1)

and G(x, y) ≤

C1 |y|p p(p − 1) (3.70)

for every (x, y) ∈  × RN , where C0 and C1 are the positive constants in (HA ). For later use we mention the following useful property. 

Proposition 3.44 ([168, Proposition 1]). Let V : W0 () → W −1,p () be the map defined by  V (u), v = A(x, ∇u) · ∇v dx 1,p

 1,p

for u, v ∈ W0 (). Then, V has the (S)+ -property, that is, any sequence {un } weakly convergent to u is strongly convergent to u whenever lim supV (un ), un − u ≤ 0. n→∞

1,p

Let us introduce the function J : W0 () → R by  G(x, ∇u) dx

J (u) = 

1,p

for all u ∈ W0 (),

(3.71)

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

which is clearly of class C 1 . We note that   1,p rS := u ∈ W0 () : up = r

for r > 0

99

(3.72)

1,p

is a C 1 Finsler manifold, actually, a C 1 submanifold of W0 () (see [71, Sections 27.4, 27.5]) because r is a regular value of the function u → up on 1,p W0 () \ {0}. Hence, the norm of the derivative at u ∈ rS of the restriction J˜ of J to rS is defined as   J (u)∗ := min J  (u) − t (u)W −1,p () : t ∈ R   = sup J  (u), v : v ∈ Tu (rS), v = 1 , where (u) :=

1 p up p

and Tu (rS) designates the tangent space of rS at u, that is,  1,p |u|p−2 uv dx = 0 }. Tu (rS) = {v ∈ W0 () : 

The restriction J˜ = J |rS is a C 1 function on rS in the sense of manifolds (see (3.71), (3.72)). 1,p Recall that λ1 is the first eigenvalue of −p on W0 () (see Proposition 1.57). Proposition 3.45. For r > 0, the infimum  G(x, ∇u) dx μ1 (A, r) = inf u∈rS 

(3.73)

is attained at points ±uˆ r ∈ rS with uˆ r ∈ C 1,α () and uˆ r > 0 in . Moreover, ±uˆ r are solutions of (3.68) with λ = λ1 (A, uˆ r )/r p , where  C0 λ1 (A, uˆ r ) = A(x, ∇ uˆ r ) · ∇ uˆ r dx ≥ (3.74) λ1 r p . p−1  Proof. Let {un } ⊂ rS be a minimizing sequence for (3.73). Using (3.70), it fol1,p lows that {un } is bounded in W0 (), so along a relabeled subsequence there 1,p 1,p hold un  uˆ r in W0 () and un → uˆ r in Lp () for some uˆ r ∈ W0 (), thus uˆ r ∈ rS. Since G(x, ·) is convex and continuous for all x ∈ , J is weakly 1,p lower semicontinuous on W0 () (see, e.g., [152, Theorem 1.2]). Therefore, we derive that   μ1 (A, r) ≤ G(x, ∇ uˆ r ) dx ≤ lim inf G(x, ∇un ) dx, 

n→∞



100 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

which yields  μ1 (A, r) =

G(x, ∇ uˆ r ) dx. 

The fact that the functional J is even implies that |uˆ r | is also a global minimizer of J˜. Consequently, we may assume that uˆ r ≥ 0. On the other hand, the Lagrange multiplier rule in Theorem 1.37 leads to the existence of t ∈ R such that   p−1 1,p A(x, ∇ uˆ r ) · ∇v dx = t uˆ r v dx for all v ∈ W0 (). (3.75) 



It follows that uˆ r is a solution of (3.68) with λ = t . By the nonlinear regularity theory, we obtain that uˆ r ∈ C 1,α () (0 < α < 1) and uˆ r > 0 in . Moreover, inserting v = uˆ r in (3.75) entails  C0 C0 λ1 C0 λ 1 p p p p tr = A(x, ∇ uˆ r ) · ∇ uˆ r dx ≥ ∇ uˆ r p ≥ uˆ r p = r . p − 1 p − 1 p −1  (3.76) Therefore, we have t=

C0 λ1 λ1 (A, uˆ r ) ≥ . p r p−1

Since J is even, we derive that J (−uˆ r ) = J (uˆ r ) = μ1 (A, r) and −uˆ r is a negative solution of (3.68) with λ = λ1 (A, uˆ r )/r p . From (3.76), we see that (3.74) ensues. Set   K1 (A, r) := uˆ r ∈ rS : J (u) = μ1 (A, r) , which is nonempty for each r > 0 as shown in Proposition 3.45. We also introduce λ1 (A, r) :=

inf

u∈K1 (A, r)

where λ1 (A, u) =

λ1 (A, u)

and λ1 (A, r) :=

sup

λ1 (A, u),

u∈K1 (A, r)



 A(x, ∇u) · ∇u dx.

Lemma 3.46. For every r > 0, λ1 (A, r) and λ1 (A, r) are attained.

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

101

Proof. We only prove for λ1 (A, r) because the corresponding formula for λ1 (A, r) can be shown similarly. Fix any r > 0. Let un ∈ K1 (A, r) satisfy λ1 (A, un ) → λ1 (A, r) as n → ∞. We see that ∇un p is bounded because   C0 p G(x, ∇un ) dx = μ1 (A, r) ≤ G(x, ∇w) dx ∇un p ≤ p(p − 1)   for all w ∈ rS, where we use (3.70). Recall that each un is a solution of (3.68) with λ = λ1 (A, un )/r p . Moreover, we have C0 C1 p λ1 r p ≤ λ1 (A, un ) ≤ ∇un p p−1 p−1 by Remark 3.43 (ii) (see (3.74) for the first inequality), so λ1 (A, un )/r p is bounded. As a result (see Proposition 2.12), we may assume that there exists 1,p u0 ∈ W0 () such that un → u0 in C01 () by choosing a subsequence if nec1,p essary. Since J in (3.71) and λ1 (A, ·) are continuous on W0 (), it is seen that J (u0 ) = lim J (un ) = μ1 (A, r), u0 ∈ K1 (A, r) n→∞

and λ1 (A, u0 ) = lim λ1 (A, un ) = λ1 (A, r). n→∞

Thus the conclusion follows. Set



A(x, ∇u) · ∇u dx and p up   G(x, ∇u) μ1 (A) := inf dx. p 1,p up u∈W0 ()\{0} 

λ1 (A) :=

inf

1,p u∈W0 ()\{0}

1,p

Recall that λ1 stands for the first eigenvalue of −p on W0 (). Lemma 3.47.

  λ1 (A, r) C1 C0 , λ1 ≤ λ1 (A) ≤ min inf λ1 r>0 p−1 rp p−1 μ1 (A, r) μ1 (A) = inf . r>0 rp

and

1,p

(3.77)

Proof. First, we establish the result for λ1 (A). For every u ∈ W0 (), u = 0, by (ii) and (iii) in Remark 3.43 we have  p p C0 ∇up A(x, ∇u) · ∇u C1 ∇up ≤ dx ≤ , p p − 1 upp p − 1 upp up 

102 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

thus, C0 C1 λ1 ≤ λ1 (A) ≤ λ1 p−1 p−1 by passing to the infimum with respect to u. Now fix any ε > 0. Then there exists rε > 0 such that λ1 (A, r) + ε. r>0 rp

λ1 (A, rε )/rεp ≤ inf

By Lemma 3.46, we can choose uε ∈ rε S such that  A(x, ∇uε ) · ∇uε dx = λ1 (A, uε ) = λ1 (A, rε ). 

By the definition of λ1 (A), we obtain  A(x, ∇uε ) · ∇uε λ (A, r) λ (A, rε ) λ1 (A) ≤ dx = 1 p ≤ inf 1 p + ε. p r>0 r uε p rε  Since ε > 0 is arbitrary, we infer that λ1 (A) ≤ inf

r>0

λ1 (A, r) . rp

Next, we prove the formula for μ1 (A) in (3.77). Fix ε > 0. There exists an rε > 0 such that μ1 (A, rε ) μ1 (A, r) ≤ inf + ε. p r>0 rp rε On the other hand, because μ1 (A, rε ) is attained at some uε ∈ rε S (by Proposition 3.45),   G(x, ∇u) G(x, ∇uε ) μ1 (A, rε ) inf dx ≤ dx = p p p 1,p u u  rε  ε p p u∈W0 ()\{0}  ≤ inf

r>0

μ1 (A, r) + ε. rp

Since ε > 0 is arbitrary, this yields μ1 (A) ≤ infr>0 μ1 (A,r) rp . For any ε > 0, we take vε = 0 such that  G(x, ∇vε ) dx ≤ μ1 (A) + ε. p vε p 

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

103

Then we obtain rε := vε p > 0 and  μ1 (A, rε ) G(x, ∇vε ) ≤ dx ≤ μ1 (A) + ε. p p rε vε p  This leads to μ1 (A) ≥ inf

r>0

μ1 (A,r) rp .

The following nonexistence result is valid. Proposition 3.48. If λ < λ1 (A), then problem (3.68) has no nontrivial solutions. Proof. If u is a nontrivial solution of (3.68), by the definition of λ1 (A) we have λ1 (A) ≤  A(x,∇u)·∇u dx = λ. p u p

Set Ap :=

C1 p−1



C1 C0

p−1 ≥ 1,

(3.78)

which is equal to 1 exactly in the case of A(x, y) = |y|p−2 y (i.e., the special case of the p-Laplacian). Lemma 3.49 ([223, Lemma 16]). Given ε > 0, for every u, ϕ ∈ W 1,p () ∩ C 1 () ∩ L∞ () with u ≥ 0 and ϕ ≥ 0 in  we have    ϕp p A(x, ∇u) · ∇ dx ≤ Ap ∇ϕp . (u + ε)p−1  We deduce the following qualitative property. Hereafter, λ1 < λ2 are the 1,p first two eigenvalues of −p on W0 () and uˆ 1 stands for the positive eigenfunction of −p corresponding to λ1 such that uˆ 1 p = 1 (whose existence is ensured by Proposition 1.57). Proposition 3.50. If u is a nontrivial solution of (3.68) with λ > Ap λ1 , then u changes sign. Proof. On the contrary, we may assume that u ≥ 0 because A is odd. Due to the strong maximum principle in Theorem 2.19, we infer that u ∈ C01 () and u > 0 in . According to Lemma 3.49, we obtain    p uˆ 1 p A(x, ∇u) · ∇ Ap λ1 = Ap ∇ uˆ 1 p ≥ dx (u + ε)p−1  p−1   u p uˆ 1 dx =λ  u+ε whenever ε > 0. Letting ε → 0+ , we have λ ≤ Ap λ1 . This is a contradiction.

104 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

C0 C0 λ2 . If Ap λ1 < λ < λ2 , then p−1 p−1 problem (3.68) has no nontrivial solutions.

Proposition 3.51. Assume that Ap λ1
0 there exists r0 > 0 such that problem (3.68) has no nontrivial solutions in 1,p

Bp (r0 ) := {v ∈ W0 () : vp < r0 } provided λ < λ1, a0 − ε. Proof. We argue by contradiction, thus we assume that there exist ε0 > 0, {λn } and {un } such that λn < λ1, a0 − ε0 , un ∈ Bp (1/n) and un is a nontrivial solution of problem (3.68) with λ = λn . By taking un as test function, we have  λ1, a0 − ε0 C0 p p A(x, ∇un ) · ∇un dx = λn un p < →0 ∇un p ≤ p−1 np  (3.81) 1,p

as n → ∞. Therefore, un → 0 in W0 (). In addition, since 0 ≤ λn < λ1, a0 − ε0 and un is a nontrivial solution of (3.68) with λ = λn , Proposition 2.12 yields that {un } converges to 0 in C 1 (). Set vn := un /un p . It follows from (3.81) and the boundedness of {λn } 1,p that {vn } is bounded in W0 (). Hence, we may assume by choosing a sub1,p sequence that {vn } converges to some v0 weakly in W0 () and strongly in p p L (). Again by taking un /un p as test function in (3.68) with λ = λn , we obtain   a0 (x)|∇un |p a˜ 0 (x, |∇un |)|∇un |2 λ1, a0 − ε0 > λn = dx + dx p p un p un p     a˜ 0 (x, |∇un |)|∇un |2 = a0 (x)|∇vn |p dx + p un p    a˜ 0 (x, |∇un |)|∇un |2 ≥ λ1, a0 + =: λ1, a0 + I p un p  because of the characterization of λ1, a0 . Hypothesis H(A)0 guarantees that for every δ > 0 there exists ρ0 > 0 such that |a˜ 0 (x, t)| ≤ δ|t|p−2 if |t| ≤ ρ0 . Since un C 1 () → 0 and using (3.81), we get  |I | ≤ δ

|∇vn |p dx ≤ 

δ(p − 1) δ(p − 1) λn ≤ (λ1, a0 − ε0 ) C0 C0

for sufficiently large n. As a result, by taking a sufficiently small δ > 0, we have a contradiction for sufficiently large n.

106 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Theorem 3.53. Assume H(A)0 . Then for every ε > 0 there exists r1 > 0 such that problem (3.68) has no constant-sign solutions in Bp (r1 ) \ {0} provided λ > λ1, a0 + ε. Proof. Arguing by contradiction, we assume that there exist ε0 > 0, {μn } and {un } such that μn > λ1, a0 + ε0 , un ∈ Bp (1/n) and un is a nontrivial, constantsign solution of (3.68) with λ = μn . Because A is odd, we may suppose that un ≥ 0 by considering −un if necessary. Thus, by nonlinear regularity theory and strong maximum principle, un ∈ C 1 () and un > 0 in . Here we note that μn ≤ Ap λ1 holds by Proposition 3.50, where λ1 denotes the first eigenvalue of −p (see (3.78) for the definition of Ap ), and so the sequence {μn } is bounded. Therefore, we may assume that {μn } converges to some μ0 along a relabeled subsequence. In addition, by the same argument as in Theorem 3.52, we can show that un → 0 in C 1 (). p−1 Set An (x, y) := A(x, un p y)/un p and fn (x, t) := μn |t|p−2 t . Then, An satisfies conditions (HA ) (i)–(iv) with the same constants C0 , C1 and C2 . Moreover, |fn (x, t)| ≤ μn |t|p−1 ≤ Ap λ1 |t|p−1 for every t ∈ R, a.e. x ∈ . Note also that we have the boundedness of {un } due to the inequality  p p C0 ∇un p /(p − 1) ≤ A(x, ∇un ) · ∇un dx = μn un p . 

Since vn := un /un p is a positive solution of −div An (x, ∇u) = fn (x, u)

in ,

u = 0 on ∂,

Proposition 2.12 guarantees that {vn } has a convergent subsequence in C 1 (). By choosing a subsequence, we may suppose that vn → v0 = 0 in C 1 () (note 1,p that v0 p = 1). Observe that for every w ∈ W0 (), we obtain 

a˜ 0 (x, |∇un |)∇un 

p−1 un p

 · ∇w dx = 

a˜ 0 (x, |∇un |)∇un · ∇w|∇vn |p−1 dx → 0 |∇un |p−1

as n → ∞ by H(A)0 , un → 0 and vn → v0 in C 1 (). As a result, letting n → ∞ in the equality   a˜ 0 (x, |∇un |)∇un a0 (x)|∇vn |p−2 ∇vn · ∇w dx + · ∇w dx p−1   un p  = μn |vn |p−2 vn w dx 

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

107

1,p

for each w ∈ W0 (), we see that v0 is a positive solution of (3.80) with λ = μ0 . This confirms that μ0 = λ1, a0 because otherwise (3.80) has no positive solutions. Therefore, we have a contradiction because μ0 = lim μn ≥ n→∞ λ1, a0 + ε0 . Proposition 3.54. Suppose H(A)0 . Then for every ε > 0 there exists r0 > 0 such that λ1 (A, r) ≥ λ1, a0 − ε rp

for every 0 < r < r0 .

Proof. Assume that there exist ε > 0 and rn > 0 such that rn → 0 as n → ∞ and p

λ1 (A, rn )/rn < λ1, a0 − ε for every n ∈ N. By Proposition 3.45 and Lemma 3.46 (note that A is odd in the second variable), we can choose a positive function un ∈ (rn S) ∩ C 1 () satisfying  A(x, ∇un ) · ∇un dx = λ1 (A, rn ),    min G(x, ∇v) dx = G(x, ∇un ) dx. v∈rn S 



Observe that C0 p ∇un p ≤ p−1



p

A(x, ∇un ) · ∇un dx = λ1 (A, rn ) < (λ1, a0 − ε)rn → 0. (3.82) 1,p

By (3.82), we obtain un → 0 in W0 (). Because un is a solution of (3.68) p with λ = λ1 (A, rn )/rn (see Proposition 3.45), by combining the inequality λ1, a0 − ε >

λ1 (A, rn ) = p rn



 a0 (x)|∇vn |p dx +





a˜ 0 (x, |∇un |)|∇un |2 dx, p un p p

and an argument as in Theorem 3.52 with λn = λ1 (A, rn )/rn , we arrive at a contradiction. Proposition 3.55. Suppose H(A)0 . Then for every ε > 0 there exists r1 > 0 such that λ1 (A, r) ≤ λ1, a0 + ε rp

for every 0 < r < r1 .

108 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Proof. Assume that there exist ε0 > 0 and rn > 0 such that rn → 0 as n → ∞ p and λ1 (A, rn )/rn > λ1, a0 + ε0 for every n ∈ N. According to Lemma 3.46 and Proposition 3.45, we can find a positive function un ∈ (rn S) ∩ C 1 () satisfying  A(x, ∇un ) · ∇un dx = λ1 (A, rn ),    G(x, ∇v) dx = G(x, ∇un ) dx. min v∈rn S 



Recalling that ϕa0 denotes the Lp -normalized positive eigenfunction corresponding to λ1, a0 , it turns out that   C0 p G(x, ∇un ) dx ≤ G(x, rn ∇ϕa0 ) dx ∇un p ≤ p(p − 1)   p C1 rn p ∇ϕa0 p . ≤ p(p − 1) We see that un → 0 in C 1 () due to Proposition 2.12 because un is a positive p solution of (3.68) with λ = λ1 (A, rn )/rn and λ1, a0 + ε0
0, from Propositions 3.54 and 3.55 the following result is readily obtained. Corollary 3.56. Under H(A)0 , lim

r→0+

λ1 (A, r) λ (A, r) = lim 1 p = λ1, a0 . p + r r r→0

Next we get the following. Proposition 3.57. Under H(A)0 , lim

r→0+

μ1 (A, r) λ1, a0 = . rp p

Proof. Due to Proposition 3.45, for every r > 0 there exists a positive solution ur ∈ (rS) ∩ C 1 () of (3.68) with λ = λ1 (A, ur )/r p and μ1 (A, r) = J (ur ). Then we can prove that ur → 0 in C 1 () as r → 0+ and ur /ur p is bounded 1,p in W0 () as r → 0+ by a similar reasoning to the one in Proposition 3.55 on the basis of Corollary 3.56.

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

109

Set ˜ 0 (x, y) := G



|y|

a˜ 0 (x, t)t dt

0

for y ∈ RN . We point out that λ1 (A, r) ≤ λ1 (A, ur ) ≤ λ1 (A, r) and

  1 ˜ 0 (x, ∇ur ) dx G(x, ∇ur ) dx = a0 (x)|∇ur |p dx + G p      λ1 (A, ur ) 1 ˜ 0 (x, ∇ur ) dx. G = a˜ 0 (x, |∇ur |)|∇ur |2 dx + − p p   

μ1 (A, r) =

Thus, by Corollary 3.56 and r = ur p , it suffices to prove that  lim

r→0+



a˜ 0 (x, |∇ur |)|∇ur |2 dx = 0 and p ur p

 lim

r→0+



˜ 0 (x, ∇ur ) G dx = 0. p ur p

Now we fix any ε > 0. Then, by H(A)0 , there exists δ > 0 such that |a˜ 0 (x, t)| ≤ εt p−2

˜ 0 (x, y)| ≤ ε and |G

|y|p p

for every 0 < t ≤ δ, |y| ≤ δ.

Because ur → 0 in C 1 () as r → 0+ , we may assume that ur C 1 () ≤ δ for sufficiently small r > 0. Therefore, we obtain p

 a˜ (x, |∇u |)|∇u |2 ∇ur p 0 r r

dx ≤ ε

p p , ur p ur p  p

 G ˜ 0 (x, ∇ur )

∇ur p

dx ≤ ε

p p . ur p pur p  Since ∇ur p /ur p is bounded as r → 0+ and ε > 0 is arbitrary, the desired conclusion follows. Our next objective is to examine the asymptotic homogeneity near +∞. We consider the case where A is asymptotically (p − 1)-homogeneous near +∞ in the following sense: H(A)∞ there exist a positive function a∞ ∈ C 1 () and a function a˜ ∈ C( × R) such that A(x, y) = a∞ (x)|y|p−2 y + a(x, ˜ |y|)y for every x ∈ , y ∈ RN , a(x, ˜ t) = 0 uniformly in x ∈ . lim t→+∞ t p−2

110 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

For the weight function a∞ , we define   p λ1, a∞ := inf a∞ (x)|∇u| dx : up = 1 . 

Because 0 < min a∞ (x) ≤ max a∞ (x) < +∞, x∈

x∈

as for the first eigenvalue of −p we can prove: (i) λ1, a∞ is the first eigenvalue of   −div a∞ (x)|∇u|p−2 ∇u = λ|u|p−2 u

in ,

u = 0 on ∂; (3.83)

(ii) λ1, a∞ has a positive eigenfunction ϕa∞ ∈ C 1 () with ϕa∞ p = 1 and it is simple; (iii) problem (3.83) has no nontrivial constant-sign solutions provided λ = λ1, a∞ . Example 3.58. An example covering the desired asymptotical (p − 1)-homogeneity near 0 and near +∞ can be considered as follows: A(x, y) = (a0 (x)|y|p−2 + a∞ (x)|y|q−2 )(1 + |y|q )

p−q q

y

for 1 < p ≤ q < +∞, a0 , a∞ ∈ C 1 () with min a0 > 0 and min a∞ > 0. This 



map A satisfies A(x, y) − a0 (x)|y|p−2 y = o(|y|p−1 ) A(x, y) − a∞ (x)|y|

p−2

y = o(|y|

p−1

)

as |y| → 0, as |y| → +∞.

An auxiliary result is needed. 1,p

Lemma 3.59. Assume H(A)∞ and let {un } ⊂ W0 () be a sequence satisfying 1,p un p → +∞ as n → ∞. If vn := un /un p is bounded in W0 (), then the following assertions hold:  a(x, ˜ |∇un |)|∇un |2 (i) lim dx = 0; p n→∞  un p  a(x, ˜ |∇un |)∇un · ∇w 1,p (ii) lim dx = 0 for every w ∈ W0 (); p−1 n→∞  un p  ˜ |y| G(x, ∇un ) ˜ (iii) lim dx = 0, where G(x, y) := 0 a(x, ˜ t)t dt for y ∈ RN . p n→∞  un p

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

111

Proof. (i) Fix ε > 0. By the property of a, ˜ there exist constants R > 0 and C > 0 such that |a(x, ˜ t)| ≤ ε|t|p−2 if t ≥ R

|a(x, ˜ t)| ≤ C if 0 ≤ t ≤ R.

and

(3.84)

Therefore, by (3.84) we deduce  

 a(x, ˜ |∇un |)|∇un |2

C|∇un |2

p dx ≤ ε|∇v | dx +

n p p dx un p  {|∇un |>R} {|∇un |≤R} un p p

≤ ε∇vn p +

CR 2 || CR 2 || p p ≤ εD + p , un p un p

where D := sup ∇vn p . Letting n → ∞, we have n

 a(x, ˜ |∇un |)|∇un |2

lim sup dx ≤ εD p p un p n→∞ 

because un p → +∞ as n → ∞. Since ε > 0 is arbitrary, the conclusion is achieved. 1,p (ii) For any ε > 0 and w ∈ W0 (), by Hölder’s inequality and (3.84) we have

 a(x, ˜ |∇un |)∇un · ∇w

dx

p−1  un p   C|∇un | |∇w| ε|∇vn |p−1 |∇w| dx + dx ≤ p−1 {|∇un |>R} {|∇un |≤R} un p p−1

≤ ε∇vn p

∇wp +

CR∇wp ||(p−1)/p p−1

un p

.

Combining this inequality and the argument in (i) leads to the conclusion. (iii) It is straightforward to show that for every ε > 0 there exists a constant C > 0 such that ˜ |G(x, y)| ≤ ε|y|p + C Consequently,

for every y ∈ RN .



p ˜ ∇un ) dx ≤ ε∇un p + C||

G(x, 

holds. This estimate implies the conclusion. Now we are able to set forth our existence and nonexistence results under 1,p condition H(A)∞ . Recall that Bp (R) = {u ∈ W0 () : up < R}. Theorem 3.60. Assume H(A)∞ . Then for every ε > 0 there exists R0 > 0 1,p such that problem (3.68) has no solutions in W0 () \ Bp (R0 ) provided λ < λ1, a∞ − ε.

112 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Proof. By way of contradiction, we assume that there exist ε0 > 0, {λn } and {un } such that λn < λ1, a∞ −ε0 , lim un p = +∞ and un is a solution of (3.68) n→∞

with λ = λn . Inserting un as a test function gives  C0 p p p A(x, ∇un ) · ∇un dx = λn un p ≤ (λ1, a∞ − ε0 )un p ∇un p ≤ p−1  (refer to (iii) in Remark 3.43). Therefore, vn := un /un p is bounded in 1,p W0 (). p Again by taking un /un p as test function in (3.68) with λ = λn , we obtain 

 a∞ (x)|∇un |p a(x, ˜ |∇un |)|∇un |2 dx + dx p p un p un p     a(x, ˜ |∇un |)|∇un |2 p = a∞ (x)|∇vn | dx + p un p 

λ1, a∞ − ε0 > λn =

≥ λ1, a∞ + o(1) due to the definition of λ1, a∞ and Lemma 3.59 (i), which yields a contradiction. Theorem 3.61. Assume condition H(A)∞ . Then for every ε > 0 there exists 1,p R1 > 0 such that problem (3.68) has no constant-sign solutions in W0 () \ Bp (R1 ) provided λ > λ1, a∞ + ε. Proof. Arguing indirectly, assume that there exist ε0 > 0, {μn } and {un } such that μn > λ1, a∞ + ε0 , lim un p = +∞ and un is a constant-sign solution n→∞

of (3.68) with λ = μn . Because A is odd, we may suppose that un ≥ 0 by considering −un if necessary. Thus, by Proposition 2.12 and Theorem 2.19, we know that un ∈ C 1 () and un > 0 in . Here, we note that μn ≤ Ap λ1 holds by Proposition 3.50, where λ1 denotes the first eigenvalue of −p (see (3.78) for the definition of Ap ), and so {μn } is bounded. Hence, we may assume that {μn } converges to some μ0 ∈ [λ1, a∞ + ε0 , Ap λ1 ] passing to a relabeled subsequence. In addition, taking un as test function in (3.68) with λ = μn , we obtain  C0 p p A(x, ∇un ) · ∇un dx = μn un p , ∇un p ≤ p−1  1,p

hence, letting vn := un /un p , the sequence {vn } is bounded in W0 (). Thus, along a relabeled subsequence, we may suppose that {vn } converges to some v 1,p weakly in W0 () and strongly in Lp (). We claim that v is a positive solution of   −div a∞ (x)|∇v|p−2 ∇v = μ0 |v|p−2 v in , v = 0 on ∂. (3.85)

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

113

If the claim holds, then necessarily μ0 = λ1, a∞ is true because (3.83) has no positive solutions in the case of λ = λ1, a∞ . Hence, this contradicts λ1, a∞ + ε0 ≤ lim μn = μ0 . n→∞

Now we prove the claim. First, we show that {vn } converges to v strongly 1,p p−1 in W0 (). Indeed, taking (vn − v)/un p as test function in (3.68) with λ = μn results in   p−1 a∞ (x)|∇vn |p−2 ∇vn · ∇(vn − v) dx μn vn (vn − v) dx =    a(x, ˜ |∇un |)∇un · ∇(vn − v) dx + p−1  un p  a∞ (x)|∇vn |p−2 ∇vn · ∇(vn − v) dx + o(1) = 

as n → ∞ due to (i) and (ii) in Lemma 3.59. Since vn → v in Lp (), this implies that  a∞ (x)|∇vn |p−2 ∇vn · ∇(vn − v) dx → 0 

as n → ∞. Noting that    o(1) = a∞ (x) |∇vn |p−2 ∇vn − |∇v|p−2 ∇v · ∇(vn − v) dx      |∇vn |p−2 ∇vn − |∇v|p−2 ∇v · ∇(vn − v) dx ≥ min a∞     p−1 p−1 ≥ min a∞ (∇vn p − ∇vp )(∇vn p − ∇vp ) ≥ 0 

and 0 < min a∞ ≤ max a∞ < +∞, 



1,p

we have vn → v in W0 (). As a result, v is a solution of (3.85) by letting n → ∞ in the equality   a(x, ˜ |∇un |)∇un · ∇w a∞ (x)|∇vn |p−2 ∇vn · ∇w dx + dx p−1   un p  p−1 = μn vn w dx  1,p

for every w ∈ W0 () (note that the second term converges to 0 by Lemma 3.59 (ii)). Since vn = un /un p > 0 in , v is nonnegative, and so vp = 1

114 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

and v is positive by Theorem 2.19. The claim is proven, which completes the proof. Proposition 3.62. Suppose H(A)∞ . Then for every ε > 0 there exists R0 > 0 satisfying λ1 (A, r) ≥ λ1, a∞ − ε rp

for every r > R0 .

Proof. Assume that there exist ε0 > 0 and rn > 0 such that rn → +∞ as n → ∞ and λ1 (A, rn ) < λ1, a∞ − ε0 p rn for every n. Using Proposition 3.45 and Lemma 3.46, we can choose a positive function un ∈ (rn S) ∩ C 1 () satisfying  A(x, ∇un ) · ∇un dx = λ1 (A, rn ),    G(x, ∇v) dx = G(x, ∇un ) dx. min v∈rn S 

Notice that C0 p ∇un p ≤ p−1



 

p

A(x, ∇un ) · ∇un dx = λ1 (A, rn ) < (λ1, a∞ − ε0 )rn , 1,p

so {un /rn } is bounded in W0 (). Since un is a solution of (3.68) with λ = p λ1 (A, rn )/rn (see Proposition 3.45), the same argument as in Theorem 3.60 p with μn = λ1 (A, rn )/rn yields a contradiction. Proposition 3.63. Suppose H(A)∞ . Then for every ε > 0 there exists R1 > 0 such that λ1 (A, r) ≤ λ1, a∞ + ε rp

for every r > R1 .

Proof. Assume that there exist ε0 > 0 and rn > 0 satisfying rn → +∞ as p n → ∞ and λ1 (A, rn )/rn > λ1, a∞ + ε0 for every n. According to Proposition 3.45 and Lemma 3.46, there is a positive function un ∈ (rn S) ∩ C 1 () with the property  A(x, ∇un ) · ∇un dx = λ1 (A, rn ),    min G(x, ∇v) dx = G(x, ∇un ) dx. v∈rn S 



Nonlinear Elliptic Eigenvalue Problems Chapter | 3

115

Using ϕa∞ introduced below (3.83), we note that   C0 p G(x, ∇un ) dx ≤ G(x, rn ∇ϕa∞ ) dx ∇un p ≤ p(p − 1)   p C1 rn p ≤ ∇ϕa∞ p . p(p − 1) 1,p

Consequently, we obtain that {un /rn } is bounded in W0 (). Since un is a p positive solution of (3.68) with λ = λ1 (A, rn )/rn , the same argument as in Thep orem 3.61 with μn = λ1 (A, rn )/rn leads to a contradiction. Propositions 3.62 and 3.63 imply the following result. Corollary 3.64. Under H(A)∞ , lim

r→+∞

λ1 (A, r) λ (A, r) = lim 1 p = λ1, a∞ . r→+∞ rp r

Proposition 3.65. Under H(A)∞ , we have lim

r→+∞

μ1 (A, r) λ1, a∞ = . rp p

Proof. Due to Proposition 3.45, for every r > 0 there exists a positive solution ur ∈ (rS) ∩ C 1 () of (3.68) with λ = λ1 (A, ur )/r p and μ1 (A, r) = 1,p  G(x, ∇ur ) dx. Then, {ur /r} is bounded in W0 () as seen from   C0 p G(x, ∇ur ) dx ≤ G(x, r∇w) dx ∇ur p ≤ p(p − 1)   r p C1 p ≤ ∇wp p(p − 1) 1,p

for any w ∈ W0 () with wp = 1. Set  ˜ G(x, y) :=

|y|

a(x, ˜ t)t dx

0

for all y ∈ RN . Notice that λ1 (A, r) ≤ λ1 (A, ur ) ≤ λ1 (A, r)  and μ1 (A, r) = G(x, ∇ur ) dx    1 ˜ a∞ (x)|∇ur |p dx + = G(x, ∇ur ) dx p     λ1 (A, ur ) 1 2 ˜ = a(x, ˜ |∇ur |)|∇ur | dx + − G(x, ∇ur ) dx. p p  

116 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

According to Corollary 3.64 and Lemma 3.59 (i) and (iii) (note that ur p = r → +∞), our conclusion is achieved. The rest of this section is devoted to existence results of positive solutions to the equation  −div A(x, ∇u) = f (x, u) in , (3.86) u=0 on ∂. The nonlinear term f is supposed to satisfy: H(f ) f is a Carathéodory function on  × R with f (x, 0) = 0 for a.e. x ∈ , f is bounded on bounded sets and f (x, s) = α0 uniformly for a.e. x ∈ , s p−1 f (x, s) = α uniformly for a.e. x ∈ , lim s→+∞ s p−1 lim

s→0+

(3.87) (3.88)

for some constants α0 and α. Theorem 3.66. Assume H(A)0 , H(A)∞ , and H(f ). If one of the following conditions: (i) α0 > λ1, a0 and α < λ1, a∞ ; (ii) α0 < λ1, a0 and α > λ1, a∞ , where λ1, a0 and λ1, a∞ are the first eigenvalues obtained by (3.80) and (3.83), respectively, is true, then problem (3.86) has at least one positive solution. In particular, this ensures the existence of a whole interval of eigenvalues for our eigenvalue problem (3.68) because we can apply Theorem 3.66 to the function f (x, s) = λ|s|p−2 s. Corollary 3.67. Assume H(A)0 , H(A)∞ , and λ1, a0 = λ1, a∞ . Then for every λ between λ1, a0 and λ1, a∞ , (3.68) has a nontrivial (positive) solution, therefore, λ is an eigenvalue. In order to prove Theorem 3.66, we develop a variational approach relying 1,p on the C 1 functional I on W0 () defined by   1,p G(x, ∇u) dx − F+ (x, u) dx for u ∈ W0 (), I (u) := 



where  F+ (x, t) :=

0

t

f+ (x, s) ds

and

f+ (x, t) = f (x, t) if t ≥ 0, f+ (x, t) = 0 if t ≤ 0.

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

117

1,p

Remark 3.68. If u ∈ W0 () is a nontrivial critical point of I , then u is a positive solution of problem (3.86). Indeed, by acting with −u− as a test function, we obtain   0 = I  (u), −u−  = A(x, ∇u) · (−∇u− ) dx − f+ (x, u)(−u− ) dx    C0 p = A(x, ∇u) · (−∇u− ) dx ≥ ∇u− p . p − 1  Thus, u ≥ 0 holds. By Theorem 2.19, since u ≡ 0 and f+ (x, u) = f (x, u), we see that u is a positive solution of (3.86). From now on in this section, we assume that hypothesis H(f ) is satisfied. Lemma 3.69. If α = λ1, a∞ , then the functional I satisfies the Palais–Smale condition. Proof. Let {un } be a Palais–Smale sequence of I , that is, I (un ) → c

and

I  (un )W −1,p () → 0

as n → ∞ 1,p

for some c ∈ R. It is sufficient to prove the boundedness of {un } in W0 () 1,p because the operator V (u) = −div A(x, ∇u) on W0 () has the (S)+ -property 1,p (see Proposition 3.44) and the embedding of W0 () into Lp () is compact. Then, thanks to the inequality   C0 p G(x, ∇un ) dx = I (un ) + F+ (x, un ) dx ∇un p ≤ p(p − 1)   p (3.89) ≤ I (un ) + Cun p , it is sufficient to prove the boundedness of {un } in Lp (). Seeking a contradiction, assume that un p → +∞ as n → ∞ by choosing a subsequence if necessary. Set vn := un /un p . Inequality (3.89) provides that {vn } is bounded 1,p in W0 (). Hence, we may suppose, by choosing a subsequence, that vn  v0 1,p in W0 () and vn → v0 in Lp () for some v0 . We see that v0 ≥ 0 a.e. in . Indeed, acting with −(un )− as a test function shows that  −  − o(1)∇(un ) p = I (un ), −(un )  = A(x, ∇un ) · (−∇(un )− ) dx 

C0 p ≥ ∇(un )− p . p−1 Since p > 1, we get ∇(un )− p → 0 as n → ∞. Thus, (vn )− → 0 in W0 (), and hence (v0 )− = 0 a.e. x ∈ . 1,p

118 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Let us prove that lim

f+ (·, un ) − α((un )− )p−1 p p−1

un p

n→∞

= 0,

(3.90)

where p  = p/(p − 1). Let an arbitrary ε > 0. It follows from (3.88) that there exists a constant Cε > 0 such that |f (x, s) − αs p−1 | ≤ ε|s|p−1 + Cε

for every s ≥ 0, a.e. x ∈ .

Then we obtain      p |f+ (x, un ) − α((un )+ )p−1 |p dx ≤ 2p −1 (ε p −1 (un )+ p + Cεp −1 ||). 

Since un p → +∞ as n → ∞, this shows that (3.90) holds true because ε > 0 is arbitrary. We know from Lemma 3.59 that  a(x, ˜ |∇un |)∇un · ∇(vn − v0 ) dx lim p−1 n→∞  un p  a(x, ˜ |∇un |)∇un · ∇ϕ dx = 0 (3.91) = lim p−1 n→∞  un p 1,p

for every ϕ ∈ W0 (). Thus, by considering o(1) =

I  (un ), vn − v0  p−1

un p

 = 

a∞ (x)|∇vn |p−2 ∇vn · ∇(vn − v0 ) dx + o(1), 1,p

we see that {vn } strongly converges to v0 in W0 () (refer to the (S)+ -property of u → −div (a∞ (x)|∇u|p−2 ∇u)). Therefore, passing to the limit in o(1) = p−1 1,p I  (un ), ϕ/un p for any ϕ ∈ W0 () and taking into account (3.90) and (3.91), we infer that v0 is a nontrivial solution (note that v0 p = 1) of −div (a∞ (x)|∇u|p−2 ∇u) = α|u|p−2 u

in ,

u = 0 on ∂.

Since v0 ≥ 0 a.e. x ∈ , v0 is a positive solution of (3.83) with λ = α. This implies that α = λ1, a∞ because (3.83) has no positive solutions if λ = λ1, a∞ , which contradicts that α = λ1, a∞ . Hence, {un } is bounded in Lp (), which completes the proof. Lemma 3.70. Assume H(A)∞ and that α < λ1, a∞ . Then the functional I is 1,p coercive, bounded below, and weakly lower semicontinuous on W0 ().

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

119

Proof. Because of α < λ1, a∞ , we can find sufficiently small constants ε > 0 and 0 < δ < 1 satisfying (1 − δ)(λ1, a∞ − ε) > α + ε.

(3.92)

By (3.88), there exists C > 0 such that |F+ (x, s)| ≤ (α + ε)s p /p + C for every s ≥ 0, a.e. x ∈ . Due to Proposition 3.65 and the definition of μ1 (A, r), there 1,p exists R > 0 such that for every u ∈ W0 () with up ≥ R it holds  λ1, a∞ − ε p G(x, ∇u) dx ≥ μ1 (A, up ) ≥ (3.93) up . p  1,p

Hence, for every u ∈ W0 () with up ≥ R, by (3.70), (3.92), and (3.93) we obtain δC0 α+ε + p (1 − δ)(λ1, a∞ − ε) p p up + ∇up − u p − C|| p p(p − 1) p δC0 p ≥ ∇up − C||. p(p − 1)

I (u) ≥

This shows that I is coercive. Moreover, because I is bounded below on Bp (R), 1,p we see that I is bounded below on W0 (). Since J is weakly lower semicon1,p tinuous (see the proof of Proposition 3.45) and W0 () → Lp () is compact, 1,p I is weakly lower semicontinuous on W0 (). Lemma 3.71. Assume H(A)0 and α0 < λ1, a0 with α0 in (3.87). Let p < q ≤ p ∗ , where p ∗ = Np/(N −p) if N > p, p ∗ = +∞ if N ≤ p. Then there exists ρ0 > 0 such that   inf I (u) : uq = ρ > 0 for every 0 < ρ < ρ0 . Proof. In view of the hypothesis α0 < λ1, a0 , we can take sufficiently small constants ε > 0 and 0 < δ < 1 satisfying (1 − δ)(λ1, a0 − ε) > α0 + ε.

(3.94)

According to Proposition 3.57, there exists r0 > 0 such that μ1 (A, r) λ1, a0 − ε ≥ rp p

for every 0 < r < r0 .

(3.95)

In addition, hypothesis H(f ) guarantees the existence of Dq > 0 satisfying F+ (x, s) ≤

α0 + ε p s + Dq s q p

for every s ≥ 0, a.e. x ∈ .

(3.96)

120 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints 1,p

Since the embedding W0 () → Lq () is continuous, there is a constant 1,p Cq > 0 such that uq ≤ Cq ∇up for every u ∈ W0 (). We choose a constant ρ > 0 satisfying ⎧  1/(q−p) ⎫ ⎨ ⎬ δC0 ρ < min r0 ||1/q−1/p , (3.97) =: ρ0 . p ⎩ ⎭ 2p(p − 1)Dq Cq Note that up < r0 if uq = ρ by Hölder’s inequality and (3.97). Therefore, whenever uq = ρ, we have    F+ (x, u) dx I (u) = (1 − δ) G(x, ∇u) dx + δ G(x, ∇u) dx − 



μ1 (A, up ) δC0 p p ≥ (1 − δ) up + ∇up p p(p − 1) up α0 + ε + p q − u p − Dq u+ q p

1 p ≥ (1 − δ)(λ1, a0 − ε) − α0 − ε up p   δC0 q−p p + uq p − Dq uq p(p − 1)Cq ≥



δC0 p pρ 2p(p − 1)Cq

by the definition of μ1 (A, r), (3.70), (3.96), (3.95), (3.94), and (3.97). The conclusion is ensued. Proof of Theorem 3.66 (i). Lemma 3.70 guarantees the existence of a global minimizer of the functional I . Thus it suffices to prove that min I < 0 1,p

W0 ()

in order to show the existence of a nontrivial critical point of I . Choose a positive constant ε > 0 such that α0 > λ1, a0 + 2ε. Let ϕa0 ∈ C 1 () be a positive eigenfunction corresponding to λ1, a0 with ϕa0 p = 1 (refer to the text below (3.80)). It is seen that  ˜ 0 (x, r∇ϕa0 ) dx/r p → 0 G 

0+

as r → (refer to the proof of Proposition 3.57). Hence there exists r0 > 0 such that    ˜ rp G0 (x, r∇ϕa0 ) p p G(x, r∇ϕa0 ) dx = a0 (x)|∇ϕa0 | dx + r dx p rp   

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

≤

λ1, a0 + ε p λ1, a0 + ε p r = rϕa0 p p p

121

(3.98)

for every 0 < r < r0 . On the other hand, it follows from (3.87) that there exists δ > 0 such that F+ (x, s) ≥

α0 − ε p s p

for every s ∈ [0, δ], a.e. x ∈ .

(3.99)

Therefore, by (3.98) and (3.99), for every 0 < r < min{r0 , δ/ϕa0 ∞ } we have I (ru0 ) ≤

rp p (λ1, a0 + 2ε − α0 )ϕa0 p < 0 p

(recall that λ1, a0 + 2ε − α0 < 0), whence min I < 0. 1,p

W0 ()

Proof of Theorem 3.66 (ii). Let p < q ≤ p∗ . By Lemma 3.71, we find ρ > 0 for which one has δ0 := inf{I (u) : uq = ρ } > 0. 1,p

We claim the existence of w ∈ W0 () such that wq > ρ

and

(3.100)

I (w) < δ0 .

Admitting the validity of this claim, we introduce   1,p

:= γ ∈ C([0, 1], W0 ()) : γ (0) = 0, γ (1) = w

and

c := inf max I (γ (t)). γ ∈ t∈[0,1]

1,p

Obviously, = ∅ and γ ([0, 1]) ∩ {u ∈ W0 () : uq = ρ } = ∅ for every 1,p γ ∈ since W0 () → Lq () is continuous. Thus, the mountain pass theorem (see Theorem 1.44) guarantees that c(≥ δ0 ) is a nontrivial critical value of I because I satisfies the Palais–Smale condition by Lemma 3.69. Finally, we prove the existence of w satisfying (3.100). Since α > λ1, a∞ , we can choose ε0 > 0 such that α > λ1, a∞ + 2ε0 .

(3.101)

Let ϕa∞ ∈ C 1 () be a positive eigenfunction corresponding to λ1, a∞ with ϕa∞ p = 1 (refer to the text below (3.83)). It follows from Lemma 3.59 (iii) that  ˜ G(x, r∇ϕa∞ ) dx/r p → 0 

122 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

as r → +∞ (note that rϕa∞ p = r). Hence there exists R0 > 0 such that   ˜ rp G0 (x, r∇ϕa∞ ) a∞ (x)|∇ϕa∞ |p dx + r p dx p  rp  λ1, a∞ + ε0 p λ1, a∞ + ε0 p ≤ (3.102) r = rϕa∞ p p p

 

G(x, r∇ϕa∞ ) dx =

for every r ≥ R0 . In addition, it follows from (3.88) that there exists D > 0 such that F+ (x, u) ≥

α − ε0 p u −D p

for every u ≥ 0, a.e. x ∈ .

(3.103)

Consequently, from (3.101), (3.102), and (3.103), we obtain I (rϕa0 ) ≤

rp p (λ1, a∞ + 2ε0 − α)ϕa0 p + D|| → −∞ p

as r → +∞. This implies the existence of w satisfying (3.100). In order to consider resonant cases, we introduce the following hypotheses on ˜ G(x, y) =



|y|

a(x, ˜ t)t dt

0

and ˜ 0 (x, y) = G



|y|

a˜ 0 (x, t)t dt,

0

where a˜ and a˜ 0 are the functions in hypotheses H(A)∞ and H(A)0 , respectively. H (R)+ There exist constants 1 ≤ q < p and H0 > 0 such that ˜ p G(x, y) − a(x, ˜ |y|)|y|2 = +∞ for a.e. x ∈ , |y|→+∞ |y|q ˜ p G(x, y) − a(x, ˜ |y|)|y|2 ≥ −H0 (1 + |y|q ) for a.e. x ∈ , every y ∈ RN , and f (x, t)t − pF (x, t) ≥ −H0 (1 + t q ) for a.e. x ∈ , every t ≥ 0. lim

H (R)− There exist constants 1 ≤ q < p and H0 > 0 such that ˜ p G(x, y) − a(x, ˜ |y|)|y|2 = −∞ for a.e. x ∈ , |y|→+∞ |y|q ˜ p G(x, y) − a(x, ˜ |y|)|y|2 ≤ H0 (1 + |y|q ) for a.e. x ∈ , every y ∈ RN , and f (x, t)t − pF (x, t) ≤ H0 (t q + 1) for a.e. x ∈ , every t ≥ 0. lim

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

123

(R)+ There exist constants 1 ≤ q < p and H0 > 0 such that H ˜ p G(x, y) − a(x, ˜ |y|)|y|2 ≥ −H0 (1 + |y|q )

for a.e. x ∈ , every y ∈ RN ,

f (x, t)t − pF (x, t) ≥ −H0 (1 + t q ) for a.e. x ∈ , every t ≥ 0, f (x, t)t − pF (x, t) and lim = +∞ for a.e. x ∈ . t→+∞ tq (R)− There exist constants 1 ≤ q < p and H0 > 0 such that H ˜ p G(x, y) − a(x, ˜ |y|)|y|2 ≤ H0 (1 + |y|q )

for a.e. x ∈ , every y ∈ RN ,

f (x, t)t − pF (x, t) ≤ H0 (1 + t q ) for a.e. x ∈ , every t ≥ 0, f (x, t)t − pF (x, t) and lim = −∞ for a.e. x ∈ . t→+∞ tq ∗ H (R)+ 0 There exist constants p ≤ r < p and H0 > 0 such that

˜ 0 (x, y) − a˜ 0 (x, |y|)|y|2 pG = +∞ for a.e. x ∈ , |y|→0 |y|r ˜ 0 (x, y) − a˜ 0 (x, |y|)|y|2 ≥ −H0 |y|r for a.e. x ∈ , every |y| ≤ 1, pG lim

and f (x, t)t − pF (x, t) ≥ −H0 t r

for a.e. x ∈ , every t ∈ [0, 1].

∗ H (R)− 0 There exist constants p ≤ r < p and H0 > 0 such that

˜ 0 (x, y) − a˜ 0 (x, |y|)|y|2 pG = −∞ for a.e. x ∈ , |y|→0 |y|r ˜ 0 (x, y) − a˜ 0 (x, |y|)|y|2 ≤ H0 |y|r for a.e. x ∈ , every |y| ≤ 1, pG lim

and f (x, t)t − pF (x, t) ≤ H0 t r

for a.e. x ∈ , every t ∈ [0, 1].

(R)+ There exist constants p ≤ r < p ∗ and H0 > 0 such that H 0 ˜ 0 (x, y) − a˜ 0 (x, |y|)|y|2 ≥ −H0 |y|r pG

for a.e. x ∈ , every |y| ≤ 1,

f (x, t)t − pF (x, t) ≥ −H0 t for a.e. x ∈ , every t ∈ [0, 1], f (x, t)t − pF (x, t) and lim = +∞ for a.e. x ∈ . tr t→0+ r

(R)− There exist constants p ≤ r < p ∗ and H0 > 0 such that H 0 ˜ 0 (x, y) − a˜ 0 (x, |y|)|y|2 ≤ H0 |y|r pG

for a.e. x ∈ , every |y| ≤ 1,

f (x, t)t − pF (x, t) ≤ H0 t for a.e. x ∈ , every t ∈ [0, 1], f (x, t)t − pF (x, t) and lim = −∞ for a.e. x ∈ . tr t→0+ r

124 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

We discuss preliminary results needed in the sequel. Set f±n (x, t) := f (x, t) ±

p p−2 t, |t| n

1,p

and define approximate functionals on W0 () by   1 p G(x, ∇u) dx − (F±n )+ (x, u) dx = I (u) ∓ u+ p . I±n (u) := n   From now on, we assume that f satisfies H(f ) (in particular (3.87)–(3.88) hold). (R)+ (resp. either H (R)− or H (R)− ) Lemma 3.72. If either H (R)+ or H holds and {un } satisfies sup I±n (un ) < +∞



and

n∈N

resp. inf I±n (un ) > −∞ and n∈N

 lim I±n (un )W −1,p () = 0,

n→∞

lim I  (un )W −1,p () n→∞ ±n

 =0 ,

1,p

then {un } is bounded in W0 (). Proof. We note that the boundedness of {un p } guarantees that {un } is bounded because  o(1)un  = I±n (un ), un  ≥

C0 1 p p un p − C(1 + un p ) ∓ (un )+ p p−1 n

for some C > 0 independent of n. Reasoning by contradiction, assume that un p → +∞ as n → ∞. Let vn := un /un p . Then by the same argument as in Lemma 3.69, we see that {vn } has a subsequence strongly convergent to a positive solution v0 of −div (a∞ (x)|∇u|p−2 ∇u) = α|u|p−2 u

in ,

u = 0 on ∂.

(3.104)

Problem (3.104) has positive solutions only if α = λ1, a∞ . So, we have α = λ1, a∞ and v0 = ϕa∞ (note that v0 p = 1). To simplify the notation, we denote the above subsequence again by {vn }. Thus, un (x) → +∞ as n → ∞ for a.e. x ∈  (note that v0 = ϕa∞ > 0 in ). (R)− . Then it turns out that (R)+ or H Assume H  f+ (x, un )un − pF+ (x, un ) dx → ±∞ K1 := q un p  (3.105) (R)± holds, respectively, if H (R)± . Indeed, it follows from H (R)+ that where q is the constant in H q (f+ (x, t)t − pF+ (x, t))/t is bounded below on  × [1, +∞). Since un (x) →

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

125

(R)+ holds, by applying Fatou’s lemma +∞ a.e. x ∈ , we have (3.105) if H to the inequality  f+ (x, un )un − pF+ (x, un ) q 2H0 K1 ≥ vn dx − q p ||, u u {un (x)≥1} n p n (R)+ . In the case of H (R)− , by considering where H0 > 0 is the constant in H −f instead of f , we can check the claim in (3.105), too. (R)− , through Hölder’s inequality we get Furthermore, in the case of H 

˜ p G(x, ∇un ) − a(x, ˜ |∇un |)|∇un |2 dx q un p   1 q (p−q)/p + o(1) ≤ H0 (|∇vn |q + q ) dx ≤ H0 ∇vn p || un p 

K2 :=

q

≤ H0 ∇v0 p ||(p−q)/p + o(1)

(3.106)

because vn → v0 in W 1,p (), where q ∈ [1, p) and H0 > 0 are the constants in (R)− . Similarly, in the case of H (R)+ , we obtain H q

K2 ≥ −H0 ∇v0 p ||(p−q)/p + o(1).

(3.107)

Hence, due to (3.105), (3.106) or (3.107), we reach a contradiction passing to the limit inferior or superior in the equality  (u ), u  pI±n (un ) − I±n n n q

un p

= K 2 + K1 .

Now assume H (R)+ or H (R)− . Since v0 is a positive solution of (3.104), we have |∇un (x)|→ + ∞ as n→∞ for a.e. x ∈ 0 := {x  ∈  : |∇v0 (x  )| = 0 }. Because |0 | > 0, we can show that  

˜ p G(x, ∇un ) − a(x, ˜ |∇un |)|∇un |2 dx → ±∞ q un p

if H (R)± holds,

respectively,

by a similar argument to the one for f as above. In addition, we can easily obtain the inequality  f+ (x, un )un − pF+ (x, un ) q q ± dx ≥ −H0 vn q + o(1) = −H0 v0 q + o(1) q u   n p in the case of H (R)± , respectively. Thereby, a contradiction arises by consider (u ), u )/u q . ing the limit of (pI±n (un ) − I±n n n n p The following preliminary result is related to the case of α0 = λ1, a0 .

126 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints − + +   Lemma 3.73. Assume H (R)− 0 or H (R)0 (resp. H (R)0 or H (R)0 ). Let 1,p  un ∈ W0 () \ {0} satisfy I±n (un ) = 0 for every n and inf I±n (un ) ≥ 0 (resp. n

sup I±n (un ) ≤ 0). Then lim inf un p > 0. n→∞

n

Proof. For otherwise, assume that lim un p = 0 along a relabeled subn→∞ sequence. Notice that the boundedness of {un p } yields that {un } and {un /un p } are bounded owing to  o(1)un  = I±n (un ), un  ≥

C0 p p p un p − C(1 + (un )+ p ) ∓ (un )+ p p−1 n (3.108)

for some C > 0 independent of n. Then, since un is a positive solution of −div A(x, ∇u) = f±n (x, un ) in  (refer to Remark 3.68), it follows from Proposition 2.12 that un → 0 in C 1 () (note that |(f±n )+ (x, t)| ≤ C(t + )p−1 , and use H(f ) and the fact that un → 0 in Lp ()). Therefore, we may assume that un C 1 () ≤ 1 by considering a sufficiently large n. Since

f (x, u  t) n p

±n

≤ Ct p p−1 un p for a.e. x ∈ , every t ≥ 0 (C > 0 is independent of n and use H(f ) and (3.108)), by a similar argument to Theorem 3.53, letting vn := un /un p , we see that {vn } has a subsequence convergent to a positive solution v0 in C 1 () of the problem −div (a0 (x)|∇u|p−2 ∇u) = α0 |u|p−2 u in ,

u = 0 on ∂.

(3.109)

If α0 = λ1, a0 , then we arrive at a contradiction because (3.80) does not have a positive solution unless λ = λ1, a0 . So, we may assume that α0 = λ1, a0 and v0 = ϕa0 (recalling that v0 p = 1). To simplify the notation, we denote the above subsequence by {vn }. − Assume H (R)+ 0 or H (R)0 . Then we can prove that  L1 := 

if

˜ 0 (x, ∇un ) − a˜ 0 (x, |∇un |)|∇un |2 pG dx → ±∞ un rp

H (R)± 0

(3.110)

holds, respectively,

− where r ∈ [p, p ∗ ) is the constant in H (R)+ 0 or H (R)0 . Indeed, since ∇v0 p > 0, we can choose a constant ε0 > 0 such that |{x ∈  : |∇v0 | >

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

127

2ε0 }| > 0. Under assumption H (R)+ 0 , we may write  L1 ≥

{|∇vn |>ε0 }



−

H0 |∇vn |r dx

{|∇vn |≤ε0 }

 ≥

˜ 0 (x, ∇un ) − a˜ 0 (x, |∇un |)|∇un |2 pG |∇vn |r dx |∇un |r

{|∇vn |>ε0 }

˜ 0 (x, ∇un ) − a˜ 0 (x, |∇un |)|∇un |2 pG |∇vn |r dx − ε0r H0 ||, |∇un |r (3.111)

where H0 is the positive constant in H (R)+ 0 . Applying Fatou’s lemma, the claim is proven because the Lebesgue measure of {x ∈  : |∇v0 | > 2ε0 } is ˜ 0 (x, ∇un ), we can positive. Similarly, by considering a˜ 0 (x, |∇un |)|∇un |2 − p G . prove (3.110) under H (R)− 0 On the other hand, letting  L2 := 

f+ (x, un )un − pF+ (x, un ) , un rp

− by using H (R)+ 0 or H (R)0 , we are able to find that

 ±L2 ≥ −H0 

((vn )+ )r dx ≥ −H0 vn rr = −H0 v0 rr + o(1)

(3.112)

(note that un C 1 () ≤ 1 and vn → v0 in C 1 ()). If sup(±I±n (un )) ≤ 0, obn

serving that ±(L1 + L2 ) = ±

 (u ), u  pI±n (un ) − I±n pI±n (un ) n n =± ≤ 0, un rp un rp

(3.113)

respectively, we get a contradiction by taking the limit superior or inferior in (3.113) because of (3.111) and (3.112). (R)− . By an argument similar to the one used for L1 (R)+ or H Assume H 0 0 ± under H (R)0 , we can show that 

f+ (x, un )un − pF+ (x, un ) dx un rp   f+ (x, un )un − pF+ (x, un ) ((vn )+ )r dx → ±∞ = ((un )+ )r {vn >0} (R)± holds, if H 0

128 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

(R)± , it is easily seen that respectively. Moreover, under H 0  ± 

˜ 0 (x, ∇un ) − a˜ 0 (x, |∇un |)|∇un |2 pG dx ≥ ∓H0 ∇vn rr un rp = ∓H0 ∇v0 rr + o(1),

respectively (note that un C 1 () ≤ 1 and vn → v0 in C 1 ()). The conclusion follows proceeding as above. Next we prepare the investigation of resonant cases with other preliminary results. Choose smooth nonnegative functions ϕ and ψ on [0, +∞) satisfying ϕ(t) = 1 if 0 ≤ t ≤ 2, ϕ(t) = 0 if t ≥ 4, ψ(t) = 0 if t ≤ 5 and ψ(t) = 1 if 1,p t ≥ 10. We define approximate functionals on W0 () by 1 1 p p p p I˜±n (u) := I (u) ∓ ψ(up )u+ p ± ϕ(up )u+ p . n n Lemma 3.74. If α ± p/n = λ1, a∞ , then I˜±n satisfies the Palais–Smale condition, respectively. Proof. Let {um } be a Palais–Smale sequence of I˜+n or I˜−n . If um p → +∞, then the equality I˜±n (um ) = I±n (um ) holds for sufficiently large m. So, by applying Lemma 3.69 to f±n (due to the fact that α ± p/n = λ1, a∞ ), we get a contradiction if um p → +∞. Consequently, we see that {um p } is bounded. Then, the same proof as in Lemma 3.69 provides that {um } has a convergent 1,p subsequence in W0 (). − +  Lemma 3.75. Assume conditions H (R)− 0 or H (R)0 (respectively, H (R)0 1,p  (u ) = 0 for every n and (R)+ ). Let un ∈ W () \ {0}, satisfy I˜±n or H n 0 0 inf I˜±n (un ) ≥ 0 (respectively, sup I˜±n (un ) ≤ 0). Then lim inf un p > 0. n

n→∞

n

Proof. Notice that I˜±n (u) = I∓n (u) provided up ≤ 2. Then the result can be established by the same argument as in Lemma 3.73. (R)+ (resp. either H (R)− or H (R)− ) Lemma 3.76. If either H (R)+ or H holds and {un } satisfies sup I˜±n (un ) < +∞



and

n∈N

resp. inf I˜±n (un ) > −∞ and n∈N

1,p

then {un } is bounded in W0 ().

 lim I˜±n (un )W −1,p () = 0,

n→∞

  ˜ lim I±n (un )W −1,p () = 0 ,

n→∞

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

129

Proof. Because I˜±n (u) = I±n (u) provided up ≥ 10, the result can be proven along the same argument as in Lemma 3.72. Our results regarding the resonant cases are as follows. Theorem 3.77. Assume H(f ), H(A)0 , and H(A)∞ . If one of the following conditions holds: (R)+ or H (R)+ ; (i) α0 > λ1, a0 , α = λ1, a∞ and H (R)− or H (R)− ; (ii) α0 < λ1, a0 , α = λ1, a∞ and H (R)+ or H (R)+ ; (iii) α0 = λ1, a0 , α < λ1, a∞ and H 0 0 (R)− or H (R)− ; (iv) α0 = λ1, a0 , α > λ1, a∞ and H 0 0 (R)+ or H (R)+ ) and (H (R)+ or H (R)+ ); (v) α0 = λ1, a0 , α = λ1, a∞ , (H 0 0 − − (R) or H (R) ) and (H (R)− or H (R)− ), (vi) α0 = λ1, a0 , α = λ1, a∞ , (H 0 0 then problem (3.86) has at least one positive solution. Proof. (i) Since α0 > λ1, a0 , there exists n0 ∈ N such that α0 − p/n0 > λ1, a0 . Note that p f−n (x, t) → α0 − > λ1, a0 as t → 0+ n t p−1 whenever n ≥ n0 , and f−n (x, t) p p → α − = λ1, a∞ − < λ1, a∞ p−1 n n t

as t → +∞.

Hence, by using the proof of Theorem 3.66 (i) for f−n , we can find a global minimizer un of I−n with I−n (un ) < 0 for each n ≥ n0 . Here, we remark that sup I−n (un ) < 0. In fact, for every n ≥ n0 , we have n≥n0

1 1 p p I−n (un ) ≤ I−n (un0 ) = I (un0 ) + un0 p ≤ I (un0 ) + un0 p n n0 = I−n0 (un0 ) < 0, where in the first inequality we use the fact that un is a global minimizer of I−n . 1,p Now, due to Lemma 3.72, we see that {un } is bounded in W0 (). Therefore, p  I  (un )W −1,p () = I  (un ) − I−n (un )W −1,p () ≤ p un p−1 → 0 nλ1 1,p

as n → ∞, where λ1 denotes the first eigenvalue of −p on W0 (). Since I is bounded on bounded sets, we may assume that {un } is a bounded Palais– Smale sequence of I . Then, by Proposition 3.44, it follows that {un } has a convergent subsequence to some v0 . It is clear that I (v0 ) ≤ sup I−n (un ) = I−n0 (un0 ) < 0, n≥n0

and so v0 is a nontrivial critical point of I .

130 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

(ii) By Lemma 3.71 and α0 < λ1, a0 , we can choose q0 ∈ (p, p∗ ] and ρ > 0 such that inf{I (u) : uq0 = ρ} > 0. Since p

I+n (u) ≥ I (u) −

uq0 ||1−p/q0 n

1,p

for every u ∈ W0 (),

we can take n0 ∈ N such that α0 +

p < λ1, a0 n0

and

δ0 := inf{I+n0 (u) : uq0 = ρ } > 0.

Hence inf{I+n (u) : uq0 = ρ } ≥ δ0 1,p

for every n ≥ n0 because I+n (u) ≥ I+n0 (u) for all n ≥ n0 and u ∈ W0 (). By means of p f+n (x, t) → α + > α = λ1, a∞ p−1 n t

as t → +∞

and applying Lemma 3.69 to f+n in place of f , I+n satisfies the Palais–Smale condition. Therefore, the proof of Theorem 3.66 (ii) implies that for every 1,p n ≥ n0 there exists a critical point un ∈ W0 () of I+n such that I+n (un ) ≥ δ0 . 1,p According to Lemma 3.72, the sequence {un } is bounded in W0 (). Thus, because we have a bounded Palais–Smale sequence of I , a similar proof as in the case of (i) yields a nontrivial critical point of I taking into account that inf I (un ) ≥ inf I+n (un ) ≥ δ0 > 0.

n≥n0

n≥n0

(iii) Fix n0 ∈ N so that α + p/n0 < λ1, a∞ . The proof of Theorem 3.66 (i) guarantees that, for every n ≥ n0 , I+n has a global minimizer un satisfying I+n (un ) < 0 because f+n (x, t) p → α0 + > α0 = λ1, a0 p−1 n t

as t → 0+

and p f+n (x, t) → α + < λ1, a∞ p−1 n t

as t → +∞ whenever n ≥ n0 . 1,p

Noting that I+n (u) ≥ I+n0 (u) for every u ∈ W0 () and n ≥ n0 , {un } is 1,p 1,p bounded in W0 () since I+n0 is coercive on W0 () by Lemma 3.70. Thus

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131

{un } is a bounded Palais–Smale sequence of I by the same argument as in case (i). Therefore, as before, {un } has a convergent subsequence to some u0 1,p in W0 (). On the other hand, Lemma 3.73 and the fact that sup I+n (un ) ≤ 0 n≥n0

guarantee that u0 = 0. Consequently, u0 is a nontrivial critical point of I . (iv) Let n0 ∈ N be so that α − p/n0 > λ1, a∞ . By applying Lemma 3.71 to f−n for n ≥ n0 in view of α0 − p/n < λ1, a0 , we can choose q0 ∈ (p, p ∗ ] and ρn > 0 such that δn := inf{I−n (u) : uq0 = ρn } > 0. By noting that p f−n (x, t) → α − > λ1, a∞ p−1 n t

as t → +∞

and applying, for every n ≥ n0 , Lemma 3.69 to f−n in place of f , I−n satisfies the Palais–Smale condition. Therefore, the proof of Theorem 3.66 (ii) implies 1,p that for every n ≥ n0 there exists a critical point un ∈ W0 () of I−n such that I−n (un ) ≥ δn > 0. According to Lemma 3.72, the sequence {un } is bounded 1,p in W0 (). Thus, as in case (i), {un } has a convergent subsequence to some 1,p u0 in W0 (). On the other hand, Lemma 3.73 ensures that u0 = 0 (note that inf I−n (un ) ≥ 0). Thereby, u0 is a nontrivial critical point of I . n≥n0

(v) Note that I˜−n (u) = I−n (u) provided up ≥ 10 and I˜−n (u) = I+n (u) if up ≤ 2. So, by an argument similar to (i), I˜−n has a global minimizer un . Moreover, as in (iii), observing that p f+n (x, t) → α0 + > λ1, a0 as t → 0+ and n t p−1 p f−n (x, t) → α − < λ1, a∞ as t → +∞, p−1 n t

we have I˜−n (un ) < 0, whence un = 0. Lemma 3.76 implies the boundedness of {un }, through the same justification as in (i). We conclude that {un } is a bounded Palais–Smale sequence of I . Therefore, we may assume that {un } 1,p converges to some u0 in W0 () by choosing a relabeled subsequence. On the other hand, Lemma 3.73 yields lim infn→∞ un p > 0. Hence u0 = 0, which means that u0 is a nontrivial critical point of I . (vi) We point out that I˜+n (u) = I+n (u) provided up ≥ 10 and I˜+n (u) = I−n (u) if up ≤ 2. So, because p f−n (x, t) → α0 − < λ1, a0 as t → 0+ and p−1 n t p f+n (x, t) → α + > λ1, a∞ as t → +∞, p−1 n t

132 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

carrying out a similar argument to (ii) and (iv) for each n, this time on the basis of Lemmas 3.74, 3.75, and 3.76, we obtain a nontrivial critical point un of I˜+n with I˜+n (un ) > 0. Then, as in (v), we can find a nontrivial critical point of I .

3.4 NOTES Section 3.1 is based on the references [186,187]. The general idea of the Fuˇcík spectrum originates in the following context. Given a bounded domain  ⊂ RN , let T be a self-adjoint linear operator on L2 () with compact resolvent whose eigenvalues satisfy 0 < λ0 < λ 1 < · · · < λ k < · · · . The Fuˇcík spectrum  of T is defined as the set of all pairs (a, b) ∈ R2 such that the equation T u = au+ − bu−

(3.114)

has a nontrivial solution. In (3.114) we denoted u+ = max{u, 0} (the positive part of u) and u− = min{−u, 0} (the negative part of u). Fuˇcík [88] and Dancer [65] recognized that the set  plays an important part in the study of semilinear equations of type T u = g(x, u), where g :  × R → R is a Carathéodory function with jumping nonlinearities g(x, s) →a s

as s → +∞,

g(x, s) →b s

as s → −∞.

A systematic study of this spectrum was developed by Fuˇcík [89] in the case of the negative Laplacian in one-dimension, i.e., for N = 1, with periodic boundary condition proving that this spectrum is composed of two families of curves in R2 emanating from the points (λk , λk ) determined by the eigenvalues λk of the negative periodic Laplacian in one-dimension. Afterwards, many authors studied the Fuˇcík spectrum 2 for the negative Laplacian − with Dirichlet boundary condition on a bounded domain  ⊂ RN (see, e.g., [15,34,66,69,147, 216]). In this respect, Dancer [66] proved that the lines R × {λ1 } and {λ1 } × R are isolated in 2 , whereas de Figueiredo–Gossez [69] constructed a first nontrivial curve in 2 passing through (λ2 , λ2 ). Here λ1 and λ2 denote the first and second eigenvalue of − with Dirichlet boundary condition, respectively. The next step in this direction was to investigate the Fuˇcík spectrum p of the negative p-Laplacian −p u = −div(|∇u|p−2 ∇u),

1 < p < +∞,

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

133

which is a nonlinear operator if p = 2, aiming to extend the results known for −. Drábek [74] has shown that, for p = 2, in the case of a one-dimensional domain, p has similar properties as in the linear case, i.e., for p = 2. In Section 3.1 we discuss the Fuˇcík spectrum of −p under different boundary conditions. It is shown that there exists a close relationship between these Fuˇcík spectra and the ordinary spectra of −p subject to different boundary conditions. A fundamental fact is that every such Fuˇcík spectrum contains a first nontrivial curve C which is Lipschitz continuous and decreasing. However, the asymptotic behavior of these curves is different relatively to the imposed boundary condition. In the applications of these spectra to certain nonlinear elliptic problems with jumping nonlinearities, subtle phenomena can occur due to the interaction of the involved nonlinearities with the spectra, in particular resonance to spectral elements. Problems of this type can be considered beyond the setting of quasilinear elliptic equations. As already mentioned, the Fuˇcík spectrum of −p depends strongly on the choice of the boundary condition related to . Specifically, we provide essential results related to the Fuˇcík spectrum of −p with Dirichlet, Neumann, Steklov, and Robin boundary conditions. The work of Cuesta–de Figueiredo–Gossez [60] was the first that gave a complete study of the beginning of the Fuˇcík spectrum of −p in the general case of 1 < p < +∞ with homogeneous Dirichlet boundary condition. The variational approach through the mountain pass theorem built in [60] was the starting point for investigating the Fuˇcík spectrum under other boundary conditions (Neumann, Steklov, Robin). An important consequence is the variational 1,p characterization of the second eigenvalue λ2 of −p on W0 () as stated in (3.5). Generally, the complete description of the Fuˇcík spectrum is an open problem. Multiple solutions results concerning problems of type (3.6) and using the representation of the first nontrivial curve C, in particular the characterization 1,p of the second eigenvalue λ2 of −p on W0 () in (3.5), can be found in numerous publications; see, for example, [40,44,178]. We also refer to versions of such results in the case of nonsmooth potential associated to (3.6) (see, e.g., [42,45,46]). The basic paper dealing with the Fuˇcík spectrum of the negative Neumann p-Laplacian is due to Arias–Campos–Gossez [16]. A striking difference between the Dirichlet Fuˇcík spectrum p and the Neumann Fuˇcík spectrum p consists in the asymptotic behavior of the first nontrivial curve C. In the Neumann case, to describe the asymptotic properties of the curve C, it is required to consider the situations p ≤ N and p > N separately. An extension to the Fuˇcík spectrum of the negative Neumann p-Laplacian with weights has been achieved by Arias–Campos–Cuesta–Gossez [17]. For the weights given by the measurable functions m(x) and n(x) on , the authors consider the set  of all pairs (a, b) ∈ R2 such that

134 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints



−p u = am(x)(u+ )p−1 − bn(x)(u− )p−1 ∂u ∂ν

=0

in , on ∂

has a nontrivial solution. Under suitable assumptions on the data it is shown that  contains a first nontrivial curve. In Motreanu–Tanaka [179] the Fuˇcík spectrum of the negative Neumann p-Laplacian is used to study quasilinear elliptic equations of the form  −div A(x, ∇u) = f (x, u) in , ∂u ∂ν

=0

on ∂,

where in the principal part of the equation one has an operator A ∈ C( × RN , RN ) ∩ C 1 ( × (RN \ {0}), RN ) of the form A(x, y) = a(x, |y|)y, with a(x, t) > 0 for all (x, t) ∈  × (0, +∞), which is strictly monotone with respect to the second variable and fulfills some further regularity assumptions, while f :  × R → R is a Carathéodory function having a representation similar to (3.6). The obtained results apply in particular to the case of the Neumann p-Laplacian, i.e., when divA(x, ∇u) = p u. The Steklov eigenvalue problem (3.12) was first studied by Martínez– Rossi [149] (see also Lê [127]). The Steklov Fuˇcík spectrum defined in (3.11) has been studied by Martínez–Rossi [150]. The fundamental difference of it with respect to the Dirichlet and Neumann Fuˇcík spectra is that in the Steklov case a boundary integral is involved, a fact that substantially modifies the analysis regarding the relevant values a and b. Similarly to the Neumann Fuˇcík spectrum, in order to derive the asymptotic properties of the first curve in the Fuˇcík spectrum of −p with Steklov boundary condition, it is needed to take into account two cases, p ≤ N and p > N . In particular, one obtains the variational characterization of the second eigenvalue λ2 in (3.13). Applications to quasilinear elliptic equations are given in Winkert [231,232]. The Fuˇcík spectrum −p on W 1,p () with Robin boundary condition as introduced in (3.15) was investigated in [187]. The corresponding eigenvalue problem (3.16) was studied before by Lê [127] proving that similar results hold as for the eigenvalue problems for the negative p-Laplacian with Dirichlet, Neumann, and Steklov boundary conditions. In comparison to the corresponding functionals related to the Fuˇcík spectra for the Dirichlet and Steklov problems, the functional associated to the Fuˇcík spectrum of −p with Robin boundary condition exhibits an essential difference because its expression does not incorporate the norm of the space W 1,p (), and it is also different from the functional used to treat the Neumann problem because it has the additional boundary term involving β. It is shown that the Robin Fuˇcík spectrum of −p on W 1,p () contains a first nontrivial curve expressed as C = {(s + c(s), c(s)), (c(s), s + c(s)) : s ≥ 0},

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where c(s) has an explicit minimax description in terms of the positive normalized eigenfunction of (3.16) associated to the first eigenvalue λ1 . The asymptotic behavior of C requires, as in the Neumann and Steklov cases, some more involved considerations distinguishing the cases p ≤ N and p > N . One can suppose that β > 0 since the case β = 0 is included in (3.10). The key idea is to argue with an adequate equivalent norm on the space W 1,p (), namely uβ = ∇uLp () + βuLp (∂) (see also Deng [70, Theorem 2.1]). The required asymptotic properties are achieved in Theorems 3.31 and 3.32. Section 3.2 is cast according to Motreanu–Tanaka [182] focusing on the eigenvalue problem (3.41) which involves the (p, q)-Laplacian operator. Letting μ → 0+ , problem (3.41) becomes the following (p − 1)-homogeneous equation known as the usual weighted eigenvalue problem for the p-Laplacian:  −p u = λmp (x)|u|p−2 u in , (3.115) u=0 on ∂. Moreover, after multiplying our equation by 1/μ and then letting μ → +∞, (3.41) becomes the equation  −q u = λmq (x)|u|q−2 u in , (3.116) u=0 on ∂. Thus, we can understand that there is a close relationship between (3.41) and the usual weighted eigenvalue problems for the p-Laplacian and q-Laplacian operators, i.e., problems (3.115) and (3.116). We briefly discuss the motivation for the given formulation of the eigenvalue problem (3.41). Let us observe that setting Ar (t) := |t|r−2 , where r ∈ {p, q}, the basic eigenvalue problems (3.115), (3.116) can be written as −div (Ar (|∇u|)∇u) = λmr (x)Ar (u)u

in ,

u = 0 on ∂.

Now in line with this, if m := mp ≡ mq and μ = 1, then setting Ap,q (t) := Ap (t) + Aq (t) there corresponds the problem −div (Ap,q (|∇u|)∇u) = λm(x)Ap,q (u)u

in ,

u = 0 on ∂.

We have thus arrived at the statement of problem (3.41) encompassing the natural formulation for the eigenvalue problem for the (p, q)-Laplacian operator. Nonlinear eigenvalue problems for elliptic equations have been thoroughly studied (see [210] for a comprehensive survey of different developments). Among the papers that studied (p, q)-Laplacian equations we cite [81,145,

136 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

218,233–235]. There are few results on the eigenvalue problems for the (p, q)-Laplacian, that is, the operator p + q . Benouhiba–Belyacine [25] showed the existence of the principal eigenvalue and of a continuous family of eigenvalues for the equation −p u − q u = λg(x)|u|p−2 u in RN . In [54, Theorem 4.2], Cingolani and Degiovanni proved the existence of a nontrivial solution for the problem −p u − μu = λ|u|p−2 u + g(u) in ,

u=0

on ∂

in the case where p > 2 = q, g ∈ C 1 (R) and λ ∈ σ (−p ) (the spectrum of 1,p −p on W0 ()). Under the Neumann boundary condition, Mih˘ailescu [157] determined the set of eigenvalues for the equation −p u − u = λu in , where p > 2 = q. Tanaka [224] has completely described the eigenvalues λ for which the problem  −r u − μρ u = λmr (x)|u|r−2 u in , u=0 on ∂ has a positive solution, where 1 < r = ρ < +∞ and μ > 0. The above eigenvalue problems are incorporated in (3.41) by posing mp ≡ 0 or mq ≡ 0. We emphasize that problem (3.41) (or, equivalently, (3.44) in the case of μ = 1) exhibits striking differences with respect to the classical eigenvalue problem (3.115). This is mainly caused by the fact that the elliptic problem (3.44), as well as (3.41), is driven by a nonhomogeneous operator −p − q , whereas (3.115) is driven by a homogeneous, precisely (p − 1)-homogeneous, operator. As an example of such a difference, we mention that for the homogeneous problem (3.115) the infimum in (3.42) is attained by a positive eigenfunction, whereas in our nonhomogeneous problem (3.41) (or, equivalently, (3.44)) there exist no such functions except in the special case where λ1 (p, mp ) = λ1 (q, mq ) and the first eigenvalues λ1 (p, mp ) and λ1 (q, mq ) have the same eigenspace (see Proposition 3.34 and Theorem 3.35). We refer to [75, 180,211,225] for related results on generalized eigenvalue problems involving nonhomogeneous operators. Existence of positive solutions in such cases can be found in [90] and [203]. An essential point in our approach is that problem (3.41) is equivalent to another eigenvalue problem (3.44) where we have only one parameter λ and just μ = 1, which evidently simplifies the setting. This equivalence allows us to make an efficient use of the Rayleigh quotient. In this respect, we recall that a standard argument relying on the Rayleigh quotient permits to show that if −λ1 (p, −mp ) < λ < λ1 (p, mp ), then (3.115) has no nontrivial solutions. In line with this classical result, we obtain through the Rayleigh quotient for

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

137

our problem a nonexistence result formulated as Theorem 3.35, which is inspired from [224, Theorem 1]. Our main result that guarantees the existence of positive solutions for problem (3.41) is stated as Theorem 3.39. It covers the range − min{λ1 (p, −mp ), λ1 (q, −mq )} < λ < min{λ1 (p, mp ), λ1 (q, mq )} for the parameter λ in problem (3.44). Since we have both λ1 (p, mp ) and λ1 (q, mq ) well-defined (although λ1 (r, 0) = +∞, see (3.43)), here the resonant cases read as λ = λ1 (p, mp ) > λ1 (q, mq ) and λ = λ1 (q, mq ) > λ1 (p, mp ). Our existence results in the resonant cases λ = λ1 (p, mp ) > λ1 (q, mq ) and λ = λ1 (q, mq ) > λ1 (p, mp ) related to problem (3.44) are presented in Theorems 3.40 and 3.41, respectively. Section 3.3 presents the results established by Motreanu–Tanaka [180]. Hypotheses closely related to assumption (HA ) are already considered in the study of quasilinear elliptic problems (see [176, Example 2.2], [64,156,168,223]). We also refer to [92,120,121,211] for generalized p-Laplacian operators. In particular, for A(x, y) = |y|p−2 y, that is, div A(x, ∇u) stands for the usual p-Laplacian p u, we can take C0 = C1 = p − 1 in (HA ). Conversely, in the case where C0 = C1 = p − 1 holds in (HA ), by the inequalities in Remark 3.43 (ii) and (iii), we see that a(x, t) = |t|p−2 whence A(x, y) = |y|p−2 y. Hence problem (3.68) contains the corresponding p-Laplacian eigenvalue problem as a special case. In the p-Laplacian case, the first eigenvalue λ1 is obtained by the Rayleigh quotient, namely p  |∇u| dx inf λ1 = p 1,p up u∈W0 ()\{0} as stated in Proposition 1.57. In the case of our operator A(x, ·), possibly nonhomogeneous, in general inf{λ ∈ R : λ is eigenvalue of A } is not obtained by such Rayleigh quotient corresponding to A. However, the Rayleigh quotient continues to play a significant role. We examine its behavior as up → 0 or up → +∞ under an additional condition describing certain asymptotic (p − 1)-homogeneity. Precisely, under conditions H(A)0 and H(A)∞ we prove that    |∇u(x)| a(x, t)t λ1, a0 λ1, a∞ dt dx : up = r → or min p r p p  0 as r → 0+ or r → +∞ respectively, where   p λ1, a0 = min a0 (x)|∇u| dx : up = 1 ,   p a∞ (x)|∇u| dx : up = 1 . λ1, a∞ = min 

138 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Concerning eigenvalue problems with a nonhomogeneous operator under the Neumann boundary condition, we can refer to [211,225]. A new insight of eigenvalue problem (3.68) is earned by incorporating it in the problem of seeking a positive solution for the quasilinear elliptic problem  −div A(x, ∇u) = f (x, u) in , (3.117) u=0 on ∂, where f satisfies the assumption H(f ), which demands that f (x, ·) be asymptotically (p − 1)-linear near 0+ and +∞ meaning f (x, s) = α0 uniformly for a.e. x ∈ , s p−1 f (x, s) = α uniformly for a.e. x ∈ , lim s→+∞ s p−1 lim

s→0+

with some constants α0 and α. Early results regarding the existence of a positive solution for problem (3.117) with nonhomogeneous operators are available, e.g., in [90] and [203], as well as [92] for the existence of a positive radial solution. However, we cannot apply these results to our nonlinear term which is only asymptotically (p − 1)-linear near 0+ and +∞, and furthermore with possibly different weights. The starting point of our analysis is that for the p-Laplacian equation, if (α − λ1 )(α0 − λ1 ) < 0 (where λ1 denotes the first eigenvalue of −p under Dirichlet boundary condition), then −p u = f (x, u)

in ,

u = 0 on ∂,

has a positive solution (see [67]). In Theorem 3.66 we extend this existence result from the p-Laplacian equation to the corresponding problem involving our nonhomogeneous operator A. We also mention that in the special case of A(x, y) = A(y), the result in [125] provides the existence of a positive solution if there hold α < λ1 C0 /(p − 1) and λ1 C1 /(p − 1) < α0 (note that we can apply this result only to the case where α < α0 ). We emphasize that for our general operator the case λ1, a0 = λ1, a∞ can occur. Note that in such a situation, contrary to the case of the p-Laplacian, we can still apply our theorem when α0 = α provided this number is between λ1, a0 and λ1, a∞ . The known result for the case of the p-Laplacian is obtained from our theorem simply by setting a0 ≡ 1 and a∞ ≡ 1. Theorem 3.66 implies that if λ1, a0 = λ1, a∞ is true, then every λ within the interval determined by the numbers λ1, a0 and λ1, a∞ is an eigenvalue of A. This is pointed out in Corollary 3.67. Besides there is a positive eigenfunction. This 1,p shows that generally the spectrum of the operator −div A(x, ∇·) on W0 () is

Nonlinear Elliptic Eigenvalue Problems Chapter | 3

139

not discrete. We cannot extend this conclusion to the case of the p-Laplacian because there we have the equality λ1, a0 = λ1, a∞ . The final part of Section 3.3 deals with the study of problem (3.117) at resonance. In this respect we treat the one-sided resonant and doubly resonant cases under additional conditions on the term f (x, s). For the p-Laplacian equation under resonance and doubly resonance, we refer to [222]. Theorem 3.77 provides the existence of a positive solution in all cases of resonance for problem (3.117) with a nonhomogeneous operator in the principal part.

Chapter 4

Nonlinear Elliptic Equations with General Dependence on the Gradient of the Solution 4.1 GRADIENT DEPENDENT PROBLEMS WITH p-LAPLACIAN The problem considered in this section is stated as follows:  −p u = f (x, u, ∇u) in , u=0 on ∂.

(4.1)

In the problem statement  is a bounded domain in RN of class C 2 , 1 < p < N , p is the p-Laplacian operator and f :  × R × RN → R is a Carathéodory function, that is, f (·, s, ξ ) is measurable for every s ∈ R and ξ ∈ RN , f (x, ·, ·) is continuous for almost every x ∈ . We note that the right-hand side of the equation depends on the point x in the domain , on the solution u(x) and on the gradient ∇u(x) of the solution. Here our main concern is the dependence on the gradient ∇u(x), which prevents the application of variational methods. Our main goal is to obtain positive solutions to problem (4.1). As before, we denote by || the Lebesgue measure of , whereas p  stands 1,p for the conjugate of p, i.e., p1 + p1 = 1. We endow the Sobolev space W0 () with the standard norm 1  p |∇u|p dx u = 

(see Remark 1.18). As usual, we set u+

= max{u, 0} and u− = max{0, −u}. It is

1,p 1,p well known that if u ∈ W0 (), then u+ , u− ∈ W0 () (see Corollary 1.16). In what follows, λ1 denotes the first eigenvalue of the negative p-Laplacian 1,p operator on W0 () with the associated eigenfunction uˆ 1 ∈ int (C01 ()+ ) (see (1.39)) satisfying uˆ 1 p = 1 as provided in Proposition 1.57. Our first as-

sumptions on the reaction term f (x, u, ∇u) are:

H(f )1 for every M > 0, there exist constants kM > 0 and 0 < θM < λ1 such that |f (x, s, ξ )| ≤ kM + θM |s|p−1 for a.e. x ∈ , s ∈ R and ξ ∈ RN with |ξ | ≤ M; Nonlinear Differential Problems with Smooth and Nonsmooth Constraints. DOI: 10.1016/B978-0-12-813386-6.00004-3 141 Copyright © 2018 Elsevier Inc. All rights reserved.

142 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

H(f )+ 2 for every M > 0 there exists a constant ηM > λ1 such that lim inf s→0+

f (x, s, ξ ) ≥ ηM > λ1 s p−1

uniformly for a.e. x ∈  and all ξ ∈ RN with |ξ | ≤ M; for every M > 0 there exists a constant ζM > 0 such that

H(f )+ 3

lim sup s→0+

f (x, s, ξ ) ≤ ζM s p−1

uniformly for a.e. x ∈  and all ξ ∈ RN with |ξ | ≤ M. + Remark 4.1. Assumptions H(f )+ 2 and H(f )3 imply that f (x, 0, ξ ) = 0 for N a.e. x ∈  and every ξ ∈ R . So, in particular, u = 0 is a solution of (4.1). Since we seek positive solutions for problem (4.1), without loss of generality we may assume that f (x, s, ξ ) = 0 for a.e. x ∈ , s ≤ 0, ξ ∈ RN .

An indication that our growth conditions are justified in the context of looking for positive solutions is offered by the below remark. Remark 4.2. Notice that if u is a positive solution of (4.1), then f (x, u(x), ∇u(x)) ≤ λ1 up−1 (x) on a set of positive measure in . Indeed, otherwise u would solve the problem  −p u = m(x)up−1 in , u=0 on ∂, for a.e. x ∈ , where m(x) =

f (x, u(x), ∇u(x)) > λ1 . up−1 (x)

Then u has to change sign (see Proposition 1.57), which contradicts that u is positive. We are going to consider an auxiliary problem by freezing the gradient in (4.1). Namely, for every w ∈ C01 (), let us formulate the Dirichlet problem  −p u = f (x, u, ∇w) in , (4.2) u=0 on ∂. 1,p

By a solution of problem (4.2) we mean a weak solution, i.e., any u ∈ W0 () such that   p−2 |∇u| ∇u · ∇v dx = f (x, u, ∇w)v dx 



Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 143 1,p

for all v ∈ W0 (). More generally, a function uw ∈ W 1,p (), with uw ≥ 0 on ∂ (in the sense of trace), is a supersolution for problem (4.2) if   |∇uw |p−2 ∇uw · ∇v dx ≥ f (x, uw , ∇w)v dx 



1,p

for all v ∈ W0 (), v ≥ 0 a.e. in . Similarly, uw ∈ W 1,p (), with uw ≤ 0 on ∂ (in the sense of trace), is a subsolution for problem (4.2) if   p−2 |∇uw | ∇uw ∇v dx ≤ f (x, uw , ∇w)v dx 



1,p

for all v ∈ W0 (), v ≥ 0 a.e. in . Lemma 4.3. Assuming H(f )1 , for every w ∈ C01 () there exists uw ∈ int (C01 ()+ ), which is a supersolution of (4.2). Proof. Fix w ∈ C01 () and set M = wC 1 () . From assumption H(f )1 , we have that |f (x, s, ∇w(x))| ≤ kM + θM |s|p−1

(4.3)

for a.e. x ∈  and all s ∈ R, where kM > 0 and 0 < θM < λ1 . Consider the following Dirichlet problem:  −p u = kM + θM |u|p−1 in , u=0 on ∂. 1,p

Since the embedding of W0 () into Lp () is compact (see Theorem 1.20),  1,p the superposition operator KM : W0 ()→W −1,p () defined by KM (u(·)) = kM + θM |u(·)|p−1 is completely continuous. Taking into account from Proposition 1.56 that the p-Laplacian operator is strictly monotone, continuous, and bounded, we have that −p − KM is pseudomonotone and bounded. Thanks to the estimate     1 1 p p |∇u| dx − kM |u| + θM |u| dx p  p    1 −1 1 θM ≥ 1− up − kM || p λ1 p u, p λ1 and using θM < λ1 (see H(f )1 ), it follows that −p − KM is coercive, hence 1,p surjective by Theorem 1.55. Therefore, there exists a function uw ∈ W0 () such that  −p uw = kM + θM |uw |p−1 in , uw = 0 on ∂.

144 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Next we prove that uw ≥ 0. Using as test function −uw− , we get   − p |∇uw | dx = − |∇uw |p−2 ∇uw · ∇uw− dx   = − (kM + θM |uw |p−1 )uw− dx ≤ 0, 

which implies that uw− = 0, so uw ≥ 0. Notice also that uw = 0. By nonlinear regularity theory [129] and strong maximum principle [229], we have that uw ∈ int (C01 ()+ ). Then from (4.3) we infer that uw is a supersolution of (4.2), which completes the proof. 1 Lemma 4.4. Assume that condition H(f )+ 2 holds. Then, for every w ∈ C0 (), there exists δ = δ(w) > 0 such that if 0 < ε < δ, then ε uˆ 1 is a subsolution of (4.2).

Proof. Let us fix w ∈ C01 () and set M = wC 1 () . From assumption H(f )+ 2, for σ > 0 so small that ηM − σ > λ1 , there exists γ = γ (w) > 0 such that f (x, s, ∇w(x)) ≥ (ηM − σ )s p−1 > λ1 s p−1

(4.4)

for a.e. x ∈  and all 0 < s < γ . Set δ = γ uˆ 1 −1 ∞ . Then, for every 0 < ε < δ 1,p and all ϕ ∈ W0 (), ϕ ≥ 0, from (4.4) we find that   |∇(ε uˆ 1 )|p−2 ∇(ε uˆ 1 ) · ∇ϕ dx = λ1 (ε uˆ 1 )p−1 ϕ dx    ≤ f (x, ε uˆ 1 , ∇w)ϕ dx, 

that is, ε uˆ 1 is a subsolution of (4.2). Now we are ready to prove the main result on the auxiliary problem (4.2). + 1 Theorem 4.5. Assume H(f )1 , H(f )+ 2 , and H(f )3 . Then for every w ∈ C0 () 1 there exists uw ∈ int (C0 ()+ ) which is the smallest positive solution of (4.2).

Proof. Let us fix w ∈ C01 () and set M = wC 1 () . From Lemmas 4.3 and 4.4, we get that there exist a supersolution uw and, for 0 < ε < δ(w), a subsolution ε uˆ 1 of (4.2). Since uw and ε uˆ 1 belong to int (C01 ()+ ), it is possible to choose ε small enough in order that uw − ε uˆ 1 ∈ int (C01 ()+ ). Consider now the following truncation of f : ⎧ ⎪ ⎨ f (x, ε uˆ 1 (x), ∇w(x)) if s < ε uˆ 1 (x), f+ (x, s) = f (x, s, ∇w(x)) if εuˆ 1 (x) ≤ s ≤ uw (x), ⎪ ⎩ f (x, uw (x), ∇w(x)) if s > uw (x).

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 145 1,p

Denote by + : W0 () → R the associated energy functional, that is, + (u) =

1 up − p

 

u(x)

 0

1,p

f+ (x, t) dt dx for all u ∈ W0 ().

Clearly, + is sequentially weakly lower semicontinuous, coercive, and continuously differentiable. Hence, it has a global minimizer u˜ εw which is a critical point of + , thus a weak solution of the Dirichlet problem  −p u = f+ (x, u) in , u=0 on ∂. Moreover, a standard comparison argument yields εuˆ 1 ≤ u˜ εw ≤ uw a.e. in , so that u˜ εw is a solution of problem (4.2). Also Theorem 2.19 (see also the strong maximum principle in [229]) entails u˜ εw ∈ int (C01 ()+ ). Denote by Sε the set of C01 ()-solutions of (4.2) which lie in the ordered interval [ε uˆ 1 , uw ]. As seen before, Sε = ∅. If we consider on Sε the pointwise order, through Zorn’s lemma, it has a minimal element uεw (see [165] for more details). Let us prove that uεw is the smallest solution of (4.2) in Sε . To this end, take v ∈ Sε . The function min{v, uεw } is a supersolution of (4.2) and clearly, min{v, uεw } ≥ ε uˆ 1 . Then, there exists a solution z of (4.2) such that ε uˆ 1 ≤ z ≤ min{v, uεw }. In particular, it turns out that z ∈ Sε , z ≤ uεw , and from the minimality of uεw in Sε , we conclude that z = uεw . So uεw ≤ v, as desired. Fix now a decreasing sequence {εn } of positive numbers such that εn → 0. For every n there exists uεwn which is the smallest solution of (4.2) in the ordered 1,p interval [εn uˆ 1 , uw ]. The sequence {uεwn } is bounded in W0 (), so there exists 1,p 1,p uw ∈ W0 () such that, along a subsequence, uεwn  uw in W0 (). In parεn εn p ticular, uw → uw in L () and uw (x) → uw (x) for a.e. x ∈ . Noticing that {uεwn } is decreasing by construction and invoking Proposition 2.12, we get that the convergence uεwn → uw is uniform. We want to prove that uw = 0. Assume by contradiction that uw = 0 and set zn =

uεwn . uεwn 

1,p

Hence zn ∈ W0 () and zn  = 1, thereby up to a subsequence {zn } converges 1,p 1,p weakly in W0 () to some z ∈ W0 (). On account of zn ≥ 0 a.e. in , we have z ≥ 0 a.e. in . Denote hn (x) =

f (x, uεwn (x), ∇w(x)) . (uεwn (x))p−1

146 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

The fact that uεwn is a solution of (4.2) reads as −p uεwn = f (x, uεwn (x), ∇w(x)), so we get p−1

−p zn = hn (x)zn

.

(4.5)

+ Notice also that by invoking assumptions H(f )+ 2 and H(f )3 and using the uniεn form convergence of {uw } to zero, we deduce

λ1 < ηM − σ ≤ hn (x) ≤ ζM for a.e. x ∈ , whenever n is sufficiently large. From this inequality we obtain that the sequence {hn } is bounded in L∞ (). Consequently, there exists a func  tion h ∈ Lp () such that hn  h in Lp (). By Mazur’s lemma (see, e.g., [73, Chapter II]) we derive that λ1 < ηM − σ ≤ h(x) ≤ ζM

(4.6)

for a.e. x ∈ . On the other hand, (4.5) implies that {zn } strongly converges to 1,p some z in W0 (). Here we use the boundedness of {hn } in L∞ () and the (S)+ -property of the p-Laplacian operator (see Proposition 1.56). The strong convergence ensures that z = 1, so z = 0. Passing to the limit in (4.5) shows that z verifies  −p z = h(x)zp−1 in , z=0 on ∂. Therefore 1 is an eigenvalue of this weighted eigenvalue problem. From (4.6) and the monotonicity of the first eigenvalue λˆ 1 (·) of −p with respect to the weight, we note that 1 = λˆ 1 (λ1 ) > λˆ 1 (h) which, from Proposition 1.57, implies that z changes sign, a contradiction. So, we have established that uw = 0. Nonlinear regularity theory up to the boundary (see Proposition 2.12 and Theorem 2.19) enables us to derive that uw ∈ int (C01 ()+ ). In order to conclude the proof, it remains to show that uw is the smallest positive solution of (4.2). It is enough to prove that it is the smallest positive solution in the ordered interval [0, uw ]. To this end, fix a solution v of (4.2) in [0, uw ]. In particular, v is a supersolution of (4.2), and we can choose εn > 0 such that v − εn uˆ 1 ∈ int (C01 ()+ ). Then we infer that εn uˆ 1 ≤ uεwn ≤ v ≤ uw for n ∈ N large enough. Letting n → ∞, we are led to uw ≤ v, as desired. In view of Theorem 4.5, it is well defined the map T : C01 () → C01 () given by T (w) = uw ,

(4.7)

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 147

where uw is the smallest positive solution of (4.2) as guaranteed by Theorem 4.5. It is clear that any fixed point for the map T will provide a positive solution to the original problem (4.1). In order to prove the continuity and the compactness of the map T , we will need some preliminary results. For any w ∈ C01 (), denote by Sw the set of all functions u ∈ int (C01 ()+ ) (see (1.39)) that are solutions of problem (4.2). Lemma 4.6. If {wn } is a bounded sequence in C01 () and {un } is a sequence in C01 () with un ∈ Swn for all n, then the sequence {un } is relatively compact in C01 (). Proof. Let M > 0 satisfy wn C 1 () ≤ M for all n. From assumption H(f )1 it 0 follows that  1 p f (x, un , ∇wn )un dx ≤ kM || p un p + θM λ−1 un p = 1 un  

1

−1

p ≤ kM || p λ1 p un  + θM λ−1 1 un  , 1,p

which implies that {un } is bounded in W0 () because θM < λ1 . On the basis of regularity up to the boundary in [129], we infer that {un } is bounded in C 1,α () for some α ∈ (0, 1) independent of n (due to the assumption that {wn } is bounded in C01 ()). Since C 1,α () is compactly embedded into C 1 (), we achieve the conclusion. Lemma 4.7. If {wn } is a sequence in C01 () such that wn → w in C01 () and if {un } is a sequence in C01 () with un ∈ Swn for all n, then there exist a subsequence {unk } and an element u ∈ Sw such that unk → u in C01 (). Proof. From Lemma 4.6, there exist a subsequence {unk } and some u ∈ C01 () such that unk → u in C01 (). We note that 

−p unk = f (x, unk , ∇wnk ) in , unk = 0 on ∂ 1,p

for all k. So, for every ϕ ∈ W0 () and k ∈ N we obtain 

 |∇unk |

p−2



∇unk · ∇ϕ dx =



f (x, unk , ∇wnk )ϕ dx.

Passing to the limit as k → ∞ yields   p−2 |∇u| ∇u · ∇ϕ dx = f (x, u, ∇w)ϕ dx, 



148 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

i.e., u is a solution of (4.2). In order to prove that u = 0 it is enough to make use of the same argument as in the corresponding part of the proof of Theorem 4.5. Then the regularity up to the boundary and strong maximum principle imply u ∈ int (C01 ()+ ), whence the conclusion. The following lemma is the key step in our construction. Lemma 4.8. If {wn } is a sequence in C01 () such that wn → w in C01 (), then for each v ∈ Sw there exists vn ∈ Swn for all n ∈ N such that vn → v

in C01 ().

Proof. Fix n ∈ N, and let zn0 be the unique solution of the problem  −p u = f (x, v, ∇wn ) in , u=0 on ∂. 1 Exploiting assumption H(f )+ 2 and bearing in mind that v ∈ int (C0 ()+ ), ob0 serve that f (·, v(·), ∇wn (·)) ≡ 0, which leads to zn = 0. Let us prove that zn0 lies in int (C01 ()+ ). First, it is straightforward to check 1,p that {zn0 } is bounded in W0 (), thus by Proposition 2.12 there exists a subsequence {zn0q } convergent in C01 () to a solution of the problem



−p u = f (x, v, ∇w) in , u=0 on ∂.

Taking into account that v is the unique solution of the above problem, we get lim z0 q→∞ nq

=v

in C01 ().

Actually, as readily seen, the strong convergence is true for the whole sequence lim z0 n→∞ n

=v

in C01 (),

which implies the above assertion. Next let us consider the unique (positive) solution zn1 of the following problem:  −p u = f (x, zn0 , ∇wn ) in , u=0 on ∂. As before it turns out that lim z1 n→∞ n

=v

in C01 ().

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 149

Inductively, we can define, for each n ∈ N, znk in C01 () as the unique solution of the problem  −p u = f (x, znk−1 , ∇wn ) in , u=0 on ∂, and for each k ∈ N it follows that lim zk n→∞ n

=v

in C01 ().

Let us now prove that there exists a constant c > 0 such that znk  ≤ c for all n, k ∈ N. Indeed, setting M = max{sup wn C 1 () , wC 1 () }, by hypothesis n

H(f )1 we have



znk p =



f (x, znk−1 , ∇wn )znk dx

≤ kM ||

1 p

+ θM

 

so − p1

znk p−1 ≤ λ1



(znk−1 )p dx

 p−1  − 1 p λ1 p znk , 

1

p−1

kM || p + θM znk−1 Lp ()

−1

1

k−1 p−1 ≤ λ1 p kM || p + λ−1 . 1 θM zn  −1

1

If we set c1 = λ1 p kM || p and c2 = λ−1 1 θM , it is easy to see by induction that znk p−1 ≤ c1

k−1

j

c2 + c2k zn0 p−1

j =0

for every k ≥ 1. Since c2 < 1 (according to assumption H(f )1 ) and {zn0 } is 1,p bounded in W0 (), we deduce that, for every n, k ∈ N, znk p−1 ≤ c1

∞

j

c2 + c3 ,

j =0

with a constant c3 > 0. This entails our claim. By standard arguments, we have that the net {znk }n,k is relatively compact in C01 (). Then up to a subnet, we can assume that there exists u ∈ C01 () such that lim znk = u,

n,k→∞

150 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

that is, there exists N ∈ N such that znk − uC 1 () < ε

(4.8)

for all n, k > N . At this point, for every k > N , we obtain that lim sup znk − uC 1 () ≤ ε. n→∞

We are thus able to select a subsequence {znk q } with the property lim znk q − uC 1 () =  lim znk q − uC 1 () = v − uC 1 () ≤ ε,

q→∞

q→∞

thereby u = v. Consequently, according to (4.8), for all n, k > N we have znk − vC 1 () < ε.

(4.9)

Now, for each n ∈ N, there exist a subsequence {ks = ks (n)}s ⊂ N and vn ∈ C01 () satisfying lim zks s→∞ n

= vn in C01 ().

Clearly, vn is a solution of (Pwn ). Let us prove that lim vn = v in C01 ().

n→∞

If not, we can construct a subsequence {nq } with nq > N such that vnq − vC 1 () > ε for all q ∈ N and some ε > 0. This amounts to saying that ε < vnq − vC 1 () =  lim znksq − vC 1 () = lim znksq − vC 1 () . s→∞

s→∞

In particular, for any q ∈ N there exists sq = sq (nq ) ∈ N such that ksq > N and ks

znqq − vC 1 () > ε, against (4.9). Recall that v ∈ int (C01 ()+ ). Since the convergence of {vn } to v is uniform on the compact subsets of , it follows that vn > 0 in  whenever n is suffi∂v n ciently large. Also, since { ∂v ∂ν } converges uniformly to ∂ν on ∂, we obtain that vn ∈ int (C01 ()+ ) for n large enough. This completes the proof. In the rest of this section we rely on the following well known result (see, e.g., [141, Corollary 4.4.12], [169, Theorem 4.27]).

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 151

Theorem 4.9 (Schaefer’s fixed point theorem). Let X be a Banach space and let T : X → X be a continuous and compact map. Assume that the set {u ∈ X : u = λT (u) for some λ ∈ [0, 1]} is bounded. Then T has a fixed point. In order to apply Theorem 4.9 to the map introduced in (4.7), we need to strengthen the growth condition H(f )1 by requiring 1/p 

H(f )1 there exist positive constants k0 , θ0 , θ1 with θ0 + θ1 λ1

< λ1 such that

|f (x, s, ξ )| ≤ k0 + θ0 |s|p−1 + θ1 |ξ |p−1 for a.a. x ∈ , all s ∈ R, and ξ ∈ RN . Our main result on problem (4.1) reads as follows: + Theorem 4.10. Assume that conditions H(f )1 , H(f )+ 2 , and H(f )3 hold. Then problem (4.1) has at least one (positive) solution u ∈ int (C01 ()+ ).

Proof. First, let us note that the map T in (4.7) is compact, that is, for each sequence {wn } bounded in C01 (), {T (wn )} is relatively compact in C01 (). This assertion follows readily from Lemma 4.6 applied to the sequence un = T (wn ) ∈ Swn . Next let us prove that the map T in (4.7) is continuous. Towards this let {wn } be a sequence in C01 () such that lim wn = w

n→∞

in C01 ().

For every n ∈ N, set un = T (wn ). By Lemma 4.7 there exist a subsequence {unk } and an element u ∈ Sw that fulfill lim unk = u

k→∞

in C01 ().

We claim that u is the smallest positive solution of (Pw ). In this respect, fix a positive solution v of (4.2). According to Lemma 4.8, there exists a (positive) solution vn ∈ int (C01 ()+ ) of (Pwn ) such that lim vn = v

n→∞

in C01 ().

Since unk is the smallest positive solution of (Pwnk ), we have unk ≤ vnk

for all k ∈ N.

152 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Passing to the limit yields u ≤ v, which means that lim T (wnk ) = T (w) in C01 ().

k→∞

In fact, the whole sequence {T (wn )} converges to T (w). If not, there would be a subsequence {wnq } such that T (wnq ) − T (w)C 1 () ≥ ε for every q ∈ N and for some ε > 0. Arguing as above, we find a subsequence unqr = T (wnqr ) converging to T (w), which gives rise to a contradiction. It is thus proven that the map T in (4.7) is continuous. Now we check that the set {w ∈ C01 () : w = λT (w) for some λ ∈ [0, 1]}

(4.10)

is bounded in C01 (). To this end, let w ∈ C01 () and λ ∈ [0, 1] satisfy w = λT (w). We may assume that λ > 0 (otherwise, w = 0). Then w is a solution of the Dirichlet problem  −p w = f˜λ (x, w, ∇w) in , w=0 on ∂, where f˜λ (x, s, ξ ) = λp−1 f (x, λ−1 s, ξ ). By assumption H(f )1 , the Carathéodory function f˜λ satisfies the growth condition |f˜λ (x, s, ξ )| ≤ k0 + θ0 |s|p−1 + θ1 |ξ |p−1

(4.11)

for a.a. x ∈ , all s ∈ R, and ξ ∈ RN . Notice that the coefficients in (4.11) are independent of λ ∈ (0, 1]. We claim that there is M > 0 independent of w and λ such that w ≤ M.

(4.12)

Indeed, acting with the test function w and using (4.11), we obtain the estimate  

 k0 |w| + θ0 |w|p + θ1 |∇w|p−1 |w| dx wp = f˜λ (x, w, ∇w)w dx ≤ 

≤ k0 ||

1 p

 − p1 p λ1 w + θ0 λ−1 1 w −1

−1

+ θ1 λ1 p wp .

p (see H(f )1 ) and p > 1, we arrive at (4.12). Since 1 > θ0 λ−1 1 + θ1 λ1 In view of (4.11) and (4.12), the nonlinear regularity theory up to the boundary in [129] (see also Proposition 2.12) provides a constant M  > 0 independent of w and λ such that wC 1 () ≤ M  , which justifies the boundedness of the set in (4.10) as claimed. Observe that any fixed point of the map T in (4.7) is a solution of problem (4.1) belonging to int (C01 ()+ ). Therefore, through Theorem 4.9, the proof is complete.

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 153

We set forth the counterpart of Theorem 4.10 on the negative half-line. We formulate in a symmetric way the corresponding hypotheses: H(f )− 2 for every M > 0 there exists a constant ηM > λ1 such that lim inf s→0−

f (x, s, ξ ) ≥ ηM > λ1 |s|p−2 s

uniformly for a.e. x ∈  and all ξ ∈ RN with |ξ | ≤ M; for every M > 0 there exists a constant ζM > 0 such that

H(f )− 3

lim sup s→0−

f (x, s, ξ ) ≤ ζM |s|p−2 s

uniformly for a.e. x ∈  and all ξ ∈ RN with |ξ | ≤ M. On the pattern of Theorem 4.10 we can prove the existence of a negative solution to problem (4.1). More precisely, we can show that the map assigning to every w ∈ C01 () the biggest negative solution of (4.2) (its existence can be established similarly to that of the smallest positive solution) possesses a fixed point resulting in the following statement. − Theorem 4.11. Assume that conditions H(f )1 , H(f )− 2 , and H(f )3 hold. Then 1 problem (4.1) has at least one (negative) solution v ∈ −int (C0 ()+ ).

By combining Theorems 4.10 and 4.11, we obtain the following multiplicity result. ± Theorem 4.12. Assume H(f )1 , H(f )± 2 , and H(f )3 . Then problem (4.1) has at least two solutions u and v of opposite constant sign with u ∈ int (C01 ()+ ) and v ∈ −int (C01 ()+ ).

A consequence of Theorem 4.12 is the existence of extremal constant-sign solutions for problem (4.1). Corollary 4.13. Under the assumptions of Theorem 4.12, problem (4.1) has the smallest positive solution and the biggest negative solution. Proof. From Theorem 4.10 we know that there exists a solution u∈int(C01 ()+ ) which can be regarded as a supersolution of problem (4.1). By virtue of condition H(f )+ 2 we can find ε0 > 0 such that −p (ε uˆ 1 ) = λ1 (ε uˆ 1 )p−1 ≤ f (x, ε uˆ 1 , ε∇ uˆ 1 ) provided 0 < ε ≤ ε0 . This expresses that ε uˆ 1 is a subsolution of problem (4.1) for all 0 < ε ≤ ε0 . In addition, since u ∈ int (C01 ()+ ), we may choose ε0 so small that u − ε uˆ 1 ∈ int (C01 ()+ ) for every 0 < ε ≤ ε0 .

154 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

We are thus in a position to apply [38, Theorem 3.22], which ensures the existence of the smallest positive solution uε of problem (4.1) in the ordered interval [ε uˆ 1 , u] for all 0 < ε ≤ ε0 . Set un := u 1 . Thanks to the choice of un , n

we note that the sequence {un } is decreasing. So, there exists u0 ∈ C01 ()+ such that un → u0 in C01 () and u0 is a solution of (4.1). On the basis of hypotheses + H(f )+ 2 and H(f )3 we also infer that u0 = 0 (see the proof of Theorem 4.5). This enables us to apply the strong maximum principle in [229] (or Theorem 2.19) to obtain that u0 ∈ int (C01 ()+ ). It remains to show that u0 is the smallest positive solution of problem (4.1). Let v be a positive solution of problem (4.1). The nonlinear regularity theory up to the boundary and strong maximum principle ensure that v ∈ int(C01 ()+ ). It follows that 1 uˆ 1 ≤ min{u, v} ≤ u n whenever n is sufficiently large. By applying [38, Theorem 3.22], we see that there exists a solution vn of (4.1) with vn ∈ [ n1 uˆ 1 , min{u, v}] because min{u, v} is a supersolution. Then the minimality property of un entails that un ≤ vn ≤ v. Letting n → ∞ leads to u0 ≤ v, whence the desired conclusion. In an analogous way we can proceed for the existence of the biggest negative solution of problem (4.1). Finally, we illustrate the applicability of our results with a simple example of nonlinearity f (x, s, ξ ) that fulfills all our hypotheses. Example 4.14. Let g :  × RN → R be a continuous, positive function. Then, for any continuous function f :  × R × RN → R satisfying the growth condition H(f )1 and f (x, s, ξ ) = |s|p−2 s(λ1 + g(x, ξ ))

for |s| small,

± it is readily seen that hypotheses H(f )± 2 and H(f )3 hold true, so Theorem 4.12 and Corollary 4.13 apply.

4.2 GRADIENT DEPENDENT PROBLEMS WITH (p, q)-LAPLACIAN This section is devoted to the following quasilinear elliptic problem with gradient dependence  −p u − μq u = f (x, u, ∇u) in , (4.13) u=0 on ∂ on a bounded domain  in RN (N ≥ 2) with C 2 boundary ∂, where p 1,p 1,q and q stand for the p-Laplacian and q-Laplacian on W0 () and W0 (),

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 155

respectively, with 1 < q < p < +∞, and a constant μ ≥ 0. Problem (4.13) has a strong unifying effect. If μ = 0, the equation in (4.13) is driven by the p-Laplacian, which is a (p − 1)-homogeneous operator, whereas for μ = 1 the leading operator is the (p, q)-Laplacian p + q , which is a nonhomogeneous operator. The C 2 smoothness for the boundary ∂ is necessary later on to manage appropriate regularity results. In order to get more insight on the main ideas, we suppose that N > p. The case N ≤ p is actually simpler and consequently is omitted. The right-hand side in the equation of (4.13) is determined by a Carathéodory function f :  × R × RN → R, that is, f (·, s, ξ ) is measurable for every (s, ξ ) ∈ R×RN , and f (x, ·, ·) is continuous for a.e. x ∈ , satisfying the growth condition H(f )1 there exist constants a1 , a2 ≥ 0, 0 ≤ α < p ∗ − 1, 0 ≤ β < γ

< p∗ ,

and a function σ

 ∈ Lγ (),

(p ∗ −1)p , p∗

1≤

σ ≥ 0, such that

|f (x, s, ξ )| ≤ σ (x) + a1 |s|α + a2 |ξ |β for a.e. x ∈ , all s ∈ R, ξ ∈ RN . Np is the Sobolev critical exponent. In the present secHere, as usual, p ∗ := N−p r , that is, the Hölder conjugate of r. tion for every r ≥ 1 we set r  := r−1 For a later use, we write an additional, this time unilateral growth condition:

H(f )2 there exist constants b1 , b2 ≥ 0 such that b1 λ−1 1,p + b2 < 1, and a function τ ∈ L1 () such that f (x, s, ξ )s ≤ τ (x) + b1 |s|p + b2 |ξ |p for a.e. x ∈ , all s ∈ R, ξ ∈ RN . 1,p

The notation λ1,p stands for the first eigenvalue of −p on W0 (), which is expressed by p

λ1,p =

inf

∇up

p 1,p u∈W0 ()\{0} up

(4.14)

,

where  · p denotes the usual Lp -norm (see Proposition 1.57). We emphasize that condition H(f )2 is independent of condition H(f )1 . For instance, H(f )2 is fulfilled if the sign requirement f (x, s, ξ )s ≤ 0 holds, without any growth restriction for f (x, s, ξ ). Later on we shall need the Lp -normalized, positive eigenfunction φ1,p of 1,p −p on W0 () corresponding to the eigenvalue λ1,p (see (4.14)), so that  p−1 p φ1,p dx = 1, −p φ1,p = λ1,p φ1,p in , φ1,p = 0 on ∂, 

156 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints 1,q

and the Lq -normalized positive eigenfunction φ1,q of −q on W0 () corresponding to λ1,q , that is,  q−1 q φ1,q dx = 1. −q φ1,q = λ1,q φ1,q in , φ1,q = 0 on ∂, with 

By the nonlinear regularity theory and the strong maximum principle, we know that φ1,p , φ1,q ∈ int (C01 ()+ ) and, for positive constants c1 and c2 , c1 φ1,p (x) ≤ φ1,q (x) ≤ c2 φ1,p (x) for all x ∈ . 1,p

A solution u ∈ W0 () of problem (4.13) is understood in the weak sense meaning that   (|∇u|p−2 + μ|∇u|q−2 )∇u · ∇h dx = f (x, u, ∇u)h dx (4.15) 

 1,p

for all h ∈ W0 (). The definition makes sense in view of condition H(f )1 . Later, we will need to strengthen the conditions H(f )1 and H(f )2 as follows: H(f ) f :  × R × RN → R is a Carathéodory function for which there exist p∗ constants c1 ≥ 0, c2 ≥ 0, r > max{ p−1 , N }, and a function ω ∈ L∞ (), ω ≥ 0, such that |f (x, s, ξ )| ≤ ω(x) + c1 |s|

p∗ r

p

+ c2 |ξ | r

for a.e. x ∈ , all s ∈ R, ξ ∈ RN . We point out that condition H(f ) is stronger than both H(f )1 and H(f )2 . Lemma 4.15. Hypothesis H(f ) implies H(f )1 and H(f )2 . Proof. By the choice of r in H(f ), we have that

p∗ r

< p ∗ − 1 and

Consequently, H(f ) yields H(f )1 by taking σ (x) = ω(x), α = From H(f ) and Young’s inequality we obtain that |f (x, s, ξ )s| ≤ ω(x)|s| + c1 |s| ≤ ω(x)|s| + c1 |s|

p∗ r +1 p∗ r +1

p∗ r

p r


0 and a corresponding ∗ constant c(ε). We note that the choice of r in H(f ) results in pr + 1 < p and r r−1 < p. A further application of Young’s inequality yields 

 |f (x, s, ξ )s| ≤ c(δ) ω(x)p + 1 + δ|s|p + ε|ξ |p

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 157

for a.e. x ∈  and all s ∈ R, ξ ∈ RN , with δ > 0 arbitrary and some constant c(δ). Therefore, choosing δ, ε sufficiently small, we can infer that H(f )2 holds true, which completes the proof. 1,p

As usual, the space W0 () is endowed with the norm u =



|∇u|p dx

1

p

.



The continuity of the embedding of W0 () in Lr () for 1 ≤ r ≤ p ∗ (see Theorem 1.20) guarantees the existence of a constant cr > 0 such that 1,p

1,p

ur ≤ cr u for all u ∈ W0 ().

(4.16)

In view of the growth assumption H(f )1 , we can consider the Nemytskii   1,p 1,p operator Nf : W0 () → W −1,p () (where W −1,p () = W0 ()∗ ) defined by 1,p

Nf (u) = f (·, u, ∇u) for all u ∈ W0 ().

(4.17)

Then problem (4.13) can be equivalently expressed 

−p u − μq u = Nf (u) in W −1,p (). We are going to introduce the notions of supersolution and of subsolution for problem (4.13). A function u ∈ W 1,p () is called a supersolution of problem (4.13) if it satisfies u ≥ 0 on ∂ and   (|∇u|p−2 + μ|∇u|q−2 )∇u · ∇h dx ≥ f (x, u, ∇u)h dx 

 1,p

for all h ∈ W0 (), h ≥ 0 a.e. in . A function u ∈ W 1,p () is called a subsolution of problem (4.13) if it satisfies u ≤ 0 on ∂ and   (|∇u|p−2 + μ|∇u|q−2 )∇u · ∇h dx ≤ f (x, u, ∇u)h dx 

 1,p

for all h ∈ W0 (), h ≥ 0 a.e. in . Owing to the growth condition H(f )1 , the notions of supersolution and subsolution of problem (4.13) are correctly defined. The following useful property of subsolution–supersolutions is shown in [38, Theorem 3.20]. Lemma 4.16. (a) If v1 , v2 are subsolutions of problem (4.13), then so is max{v1 , v2 }. (b) If u1 , u2 are supersolutions of problem (4.13), then so is min{u1 , u2 }.

158 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

We also quote an existence result whose proof can be found in [19, Theorem 1]. Lemma 4.17. Assume that conditions H(f )1 and H(f )2 hold. Then prob1,p lem (4.13) has at least one (weak) solution u ∈ W0 (). The regularity of solutions and a priori estimates for problem (4.13) are now addressed. Lemma 4.18. Assume that condition H(f ) holds. Then there are constants 1,p R > 0 and γ ∈ (0, 1) such that every solution u ∈ W0 () of problem (4.13) 1,γ belongs to C () and satisfies the estimate uC 1,γ () ≤ R. Proof. The fact that the solution set of problem (4.13) is nonempty is ensured 1,p by Lemmas 4.15 and 4.17. Let u ∈ W0 () be any solution of (4.13). Inserting h = u in (4.15) and using H(f )2 and (4.14), we have   (|∇u|p + μ|∇u|q ) dx = f (x, u, ∇u)u dx   ≤ (τ (x) + b1 |u|p + b2 |∇u|p ) dx   τ (x) dx + (b1 λ−1 + b ) |∇u|p dx. ≤ 2 1,p 



Since μ ≥ 0 and b1 λ−1 1,p + b2 < 1, there exists a constant c > 0 independent of the solution u such that u ≤ c.

(4.18)

Then by (4.16) and (4.18) we infer that up∗ ≤ cp∗ c and ∇up ≤ c for every solution u of problem (4.13). Using these estimates through the growth condition H(f ), it follows that f (·, u, ∇u)r ≤ c0 ,

(4.19)

with a constant c0 > 0 independent of the solution u. At this point we make use of the assumption that r > N in H(f ), which enables us to refer to the gradient bound in [53, Theorem 3.1; Remark 3.3]. Consequently, from (4.19) we infer that 1

1

∇u∞ ≤ Cf (·, u, ∇u)rp−1 ≤ C c0p−1

(4.20)

whenever u is a solution of (4.13), where the constant C depends only on p and . In particular, the solution set of (4.13) is uniformly bounded, that is,

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 159

there exists a constant C0 > 0 independent of any solution u such that u∞ ≤ C0 .

(4.21)

Due to (4.20), (4.21), the nonlinear regularity up to the boundary in [129,130] (see also Proposition 2.12) applies, which leads to the desired conclusion. Remark 4.19. A careful reading of the above proof shows that the constant c in (4.18) is explicitly given by c=

τ 1 1 − (b1 λ−1 1,p + b2 )

.

Moreover, the a priori estimate (4.18) holds under hypotheses H(f )1 and H(f )2 without need of the stronger hypothesis H(f ). We now focus on the location of nodal (that is, sign-changing) solutions for problem (4.13), which means solutions that are positive and negative on (disjoint) parts of nonzero measure. Theorem 4.20. Assume that conditions H(f )1 and H(f )2 are satisfied. (i) If f (x, 0, 0) ≥ 0 for a.e. x ∈ , then for every nodal solution u0 of problem (4.13) there exists a nontrivial solution u+ of (4.13) such that u0 ≤ u+ and u+ ≥ 0 on  . (ii) If f (x, 0, 0) ≤ 0 for a.e. x ∈ , then for every nodal solution u0 of problem (4.13) there exists a nontrivial solution u− of (4.13) such that u0 ≥ u− and u− ≤ 0 on  . (iii) If f (x, 0, 0) = 0 for a.e. x ∈ , then for every nodal solution u0 of problem (4.13) there exist two other nontrivial solutions u+ and u− of (4.13) such that u− ≤ u0 ≤ u+ , u+ ≥ 0 and u− ≤ 0 on  . Proof. (i) Let u0 be a nodal solution of problem (4.13). The assumption f (x, 0, 0) ≥ 0 for a.e. x ∈  ensures that 0 is a subsolution of problem (4.13). By Lemma 4.16 (a), we infer that u := max{0, u0 } is a subsolution of problem (4.13). 1,p

(4.22)

1,p

Let T : W0 () → W0 () be the truncation operator defined by  (T u)(x) =

u(x) u(x)

if u(x) ≥ u(x), if u(x) < u(x) 1,p

(4.23) 1,p

for a.e. x ∈ . It is clear that the operator T : W0 () → W0 () is bounded and continuous. We also consider the following cut-off function by setting for

160 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

a.e. x ∈ , every s ∈ R:  b(x, s) =

0

if s ≥ u(x),

−(u(x) − s)p−1

if s < u(x).



Let B : W0 () → W −1,p () denote the corresponding Nemytskii operator, that is, Bu(x) = b(x, u(x)), which is completely continuous in view of the com1,p pact embedding of W0 () into Lp () as stated in Theorem 1.20. Now we formulate an auxiliary problem  −p u − μq u + Bu = f (x, T u, ∇(T u)) in , (4.24) u=0 on ∂. 1,p



Using the Nemytskii operator Nf : W0 () → W −1,p () given by (4.17), problem (4.24) can be equivalently expressed as 1,p



−p u − μq u + Bu = Nf ◦ T (u) in W −1,p ().

(4.25) 1,p

By assumption H(f )1 , the operator −p − μq + B − Nf ◦ T : W0 () →  W −1,p () is bounded. We claim that −p − μq + B − Nf ◦ T is a pseudomonotone operator 1,p 1,p on W0 (). In order to show this, let a sequence {un } ⊂ W0 () be such that 1,p un  u in W0 () and lim sup−p un − μq un + Bun − Nf (T un ), un − u ≤ 0.

(4.26)

n→∞

∗

By assumption H(f )1 , we have that p∗p−α < p ∗ . Invoking the boundedness of the operator T and Rellich–Kondrachov compactness embedding theorem (see Theorem 1.20), it follows that the sequence {|T un |α } is bounded in (

p∗

p∗

)

p∗

L p∗ −α () = L α () and that un → u in L p∗ −α (). Therefore Hölder’s inequality implies  |T un |α |un − u| dx ≤ T un αp∗ un − u p∗ → 0. (4.27) 

p ∗ −α

p Also, by H(f )1 we have that p−β < p ∗ . Using again the boundedness of the operator T and Rellich–Kondrachov compactness embedding theorem (see The( p ) orem 1.20), we infer that the sequence {|∇(T un )|β } is bounded in L p−β () = p p L β () and that un → u in L p−β (). Consequently, Hölder’s inequality yields  |∇(T un )|β |un − u| dx ≤ ∇(T un )βp un − u p → 0. (4.28) 

p−β

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 161

Therefore, thanks to assumption H(f )1 and (4.27), (4.28), we derive  lim Nf (T un ), un − u = lim

n→∞

n→∞ 

f (x, T un , ∇(T un ))(un − u) dx = 0. (4.29)

Combining (4.26), (4.29), and the complete continuity of B, it turns out that lim sup−p un − μq un , un − u ≤ 0. n→∞

Using the (S)+ -property of −p − μq (see [169, Proposition 2.70]), we con1,p clude that un → u in W0 (). Thus lim −p un − μq un + Bun − Nf (T un ), un − v

n→∞

= −p u − μq u + Bu − Nf (T u), u − v 1,p

for all v ∈ W0 (), which proves that the operator −p − μq + B − Nf ◦ T is pseudomonotone. Now we verify that the operator −p − μq + B − Nf ◦ T is coercive. Let 1,p {un } be a sequence in W0 () such that un  → +∞ as n → ∞. By H(f )2 , Hölder’s inequality and the definition of T , we get the estimate (−p − μq + B − Nf ◦ T )(un ), un   q = un p + μ∇un q − (u − un )p−1 un dx {un 0, and k1 > 0 such that |f (x, s, ξ )| ≤ k0 |s|p−1 + k1 (|ξ |p−1 + μ|ξ |q−1 )

(4.37)

for a.e. x ∈ , all −δ ≤ s ≤ δ, |ξ | ≤ δ, then for every nodal solution u0 of problem (4.13) there exist a (positive) solution u+ ∈ int (C01 ()+ ) and a (negative) solution u− ∈ −int (C01 ()+ ) of problem (4.13) such that u− ≤ u0 ≤ u+ on  . Proof. (i) Owing to Lemma 4.15, we know that the present hypotheses are stronger than those in Theorem 4.20. Since assumption (4.35) entails that f (x, 0, 0) ≥ 0 for a.e. x ∈ , by Theorem 4.20 (i) there exists a nontrivial nonnegative solution u+ of (4.13) with u0 ≤ u+ on . It suffices to prove that the solution u+ fulfills the properties required in the statement. Due to assumption H(f ), we are able to invoke the regularity result in Lemma 4.18, which ensures that u+ ∈ C01 (), so in particular 0 ≤ u+ (x) ≤ M and |∇u+ (x)| ≤ M for all x ∈ , with a constant M > 1. Now we rely on the strong maximum principle in [205, Theorem 5.4.1] and on Hopf’s boundary point lemma in [205, Theorem 5.5.1] (see also [169, Section 8.2]) applied to the function u+ ∈ C01 () that satisfies −p u+ − μq u+ = f (x, u+ , ∇u+ ) in . In order to meet the needed requirements, let us denote A(s) := s p−2 + μs q−2 for s > 0. We find that c := lim

t→0+

tA (t) > −1, A(t)

since c = q − 2 if μ > 0 and c = p − 2 if μ = 0.

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 165

From assumption H(f ) and the inequality p∗ 0 and k3 = k3 (δ, M) > 0 with f (x, s, ξ ) ≥ −k2 s p−1 for a.e. x ∈ , all s ≥ δ, |ξ | ≤ M, f (x, s, ξ ) ≥ −k3 (|ξ |p−1 + μ|ξ |q−1 ) for a.e. x ∈ , all 0 ≤ s ≤ δ, δ ≤ |ξ | ≤ M. Combining with (4.35) we arrive at f (x, s, ξ ) ≥ −C1 (|ξ |) − f (s) for a.e. x ∈ , all s ≥ 0, |ξ | ≤ M, where (s) := sA(s) = s p−1 + μs q−1 and f (s) = C2 s p−1 , with positive constants C1 = C1 (δ, M) and C2 = C2 (δ, M). Taking into account that M > 1, conditions (B1) and (F2) in [205] are fulfilled. We note that  s p−1 p q −1 q (t) dt = s +μ s H (s) := s(s) − p q 0 and

 F (s) :=

s

f (t) dt =

0

C2 p s p

for all s ≥ 0. The function H is increasing on [0, +∞) and we have

1 −1  F (s) ≤ H C2p (p − 1) p s hence 1

− p1

H −1 (F (s)) ≤ C2p (p − 1) Thus the condition

 0+

ds H −1 (F (s))

s for all s > 0.

= +∞

required in (1.1.5) of [205] holds true. All the hypotheses of [205, Theorem 5.4.1] and [205, Theorem 5.5.1] are satisfied (see also [205, Remark 3, p. 117]), thereby u+ ∈ int (C01 ()+ ) as seen from formula (1.39). (ii) Observe that hypothesis (4.36) implies that f (x, 0, 0) ≤ 0 for a.e. x ∈ . Hence, by means of Lemma 4.15, we can refer to Theorem 4.20 (ii) for obtaining

166 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

a nontrivial nonpositive solution u− of (4.13) such that u0 ≥ u− on . Consequently, it is sufficient to show that the solution u− has the properties required in the statement. First, due to H(f ) and Lemma 4.18, we know that u− ∈ C01 (). Then we utilize the strong maximum principle in [205, Theorem 5.4.1] and Hopf’s boundary point lemma in [205, Theorem 5.5.1] applied to the nonnegative function −u− ∈ C01 () verifying −p (−u− ) − μq (−u− ) = f˜(x, −u− , ∇(−u− ))

in ,

where f˜(x, s, ξ ) := −f (x, −s, −ξ ). From now on the proof proceeds in a manner similar to (i) with −u− in place of u+ . (iii) We note that assumption (4.37) implies both (4.35) and (4.36). Then part (iii) can be readily obtained from assertions (i) and (ii), which completes the proof. Remark 4.22. Part (iii) of Theorem 4.21 holds more generally whenever both unilateral conditions (4.35) and (4.36) are satisfied, which is weaker than assuming the bilateral condition (4.37). Note that (4.35) and (4.36) are fulfilled by a function f that satisfies the local sign condition f (x, s, ξ )s ≥ 0 for a.e. x ∈ , all −δ ≤ s ≤ δ, |ξ | ≤ δ, for some δ > 0. 1,p

Remark 4.23. Denoting by λ2,p the second eigenvalue of −p on W0 (), one shows in [165, Theorem 2 (b)] the existence of a nodal solution u0 ∈ C01 () between a negative solution u− ∈ −int (C01 ()+ ) and a positive solution u+ ∈ int (C01 ()+ ) of problem (4.13) provided f does not depend on ξ ∈ RN and there exist η2 , θ ∈ L∞ (), with η2 (x) ≥ λ2,p for a.e. x ∈ , η2 = λ2,p on a set of positive measure, such that η2 (x) ≤ lim inf s→0

f (x, s) f (x, s) ≤ lim sup p−2 ≤ θ (x) uniformly for a.e. x ∈ . p−2 |s| s s s→0 |s|

Theorem 4.21 complements this result offering a verifiable criterion for the converse property, namely that every nodal solution of problem (4.13) is located within the ordered interval determined by a negative solution and a positive solution of problem (4.13). Remark 4.24. The result in Theorem 4.21 is sharp. For example, every nontrivial solution of the problem −p u = λ|u|p−2 u in , u = 0 on ∂, with λ > λ1,p , is nodal. Notice that in this case hypothesis H(f )2 is not fulfilled. Our next aim is to prove the existence of extremal solutions, i.e., the biggest and smallest solutions, for problem (4.13). This will permit bounding the whole

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 167

set of nodal solutions by a positive solution and a negative solution, thus complementing the location principle in Theorem 4.21 where the bounds with constantsign solutions depend on the fixed nodal solution. We start with the problem obtained for μ = 0, that is,  −p u = f (x, u, ∇u) in , (4.38) u=0 on ∂. Recall that  ⊂ RN is a bounded domain with C 2 boundary ∂. Then for any δ > 0 sufficiently small, say δ ≤ δ0 , the δ-neighborhood δ of , i.e., δ = {x ∈ RN : d(x, ) < δ}, 1,p

has a C 2 boundary ∂δ . Let λ1,p,δ be the first eigenvalue of −p on W0 (δ ) with the associated Lp -normalized positive eigenfunction φ1,p,δ , that is,  p−1 p φ1,p,δ dx = 1. −p φ1,p,δ = λ1,p,δ φ1,p,δ in δ , φ1,p,δ = 0 on ∂δ , δ

(4.39) The below result describes a continuity property of the first eigenvalue λ1,p with respect to the domain. Lemma 4.25. There holds lim λ1,p,δ = λ1,p . δ→0+

Proof. The variational characterizations of λ1,p,δ and λ1,p (see (4.14)) give u

λ1,p,δ =

p 1,p

W0 (δ ) p 1,p u∈W0 (δ )\{0} uLp (δ )

inf

≤

up p = λ1,p , 1,p u∈W0 ()\{0} up inf

whence lim sup λ1,p,δ ≤ λ1,p .

(4.40)

δ→0+

Take a sequence δn → 0+ as n → ∞. From (4.39) and (4.40) we derive   p |∇φ1,p,δn | dx = |∇φ1,p,δn |p dx = λ1,p,δn ≤ M δ0

δn

1,p

with a constant M > 0, so up to a subsequence, φ1,p,δn  u in W0 (δ0 ) and 1,p φ1,p,δn → u in Lp (δ0 ) as n → ∞ with some u ∈ W0 (δ0 ). We have that u = 0 a.e. in δ0 \  because, for any v ∈ Cc∞ (δ0 \ ), the inclusion supp v ⊂ δ0 \ δn holds provided n is sufficiently large, thus    uv dx = uv dx = lim φ1,p,δn v dx = 0. δ0 \

δ0

n→∞  δ0

168 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints 1,p

Consequently, since u ∈ W0 () and   p |∇u| dx ≤ lim inf λ1,p ≤ n→∞

δ0



p  u dx

δ0

= 1, we find that

|∇φ1,p,δn |p dx = lim inf λ1,p,δn . n→∞

In view of (4.40), the proof is complete. The next result provides subsolutions and supersolutions for problem (4.38). Proposition 4.26. Assume H(f )1 with σ ∈ L∞ (), β < p − 1, and (α = p − 1 and a1 < λ1,p ) or α < p − 1. Then there exists δˆ > 0 with the property that, ˆ there is t0 = t0 (δ) > 0 such that −tφ1,p,δ and tφ1,p,δ are for every δ ∈ (0, δ], subsolution and supersolution, respectively, of problem (4.38) for all t ≥ t0 . Proof. If a1 < λ1,p , Lemma 4.25 yields δˆ > 0 such that a1 < λ1,p,δ for all δ ∈ ˆ Because φ1,p,δ ∈ int (C 1 (δ )+ ), there exist constants ρ = ρ(δ) > 0 and (0, δ]. 0 t0 = t0 (δ) > 0 with φ1,p,δ (x) ≥ ρ and a1 a2 σ (x) + + |∇φ1,p,δ (x)|β ≤ λ1,p,δ p−1 p−α−1 p−β−1 p−1 (ρt0 ) (ρt0 ) ρ t0 for all x ∈  (this holds in both situations (α = p − 1 and a1 < λ1,p ) and (α < p − 1) of the statement). Combining with our hypothesis leads to f (x, tφ1,p,δ (x), ∇(tφ1,p,δ )(x)) ≤ σ (x) + a1 (tφ1,p,δ (x))α + a2 (t|∇φ1,p,δ (x)|)β

σ (x)  a1 a2 β ≤ + + |∇φ (x)| (tφ1,p,δ (x))p−1 1,p,δ (ρt0 )p−1 (ρt0 )p−α−1 ρ p−1 t p−β−1 0

≤ λ1,p,δ (tφ1,p,δ (x))

p−1

= −p (tφ1,p,δ )(x)

in , for all t ≥ t0 . Therefore tφ1,p,δ is a supersolution of problem (4.38). The proof for the subsolution is similar. Our existence result of extremal solutions for (4.38) is as follows. Theorem 4.27. Assume H(f ). Then: (i) Problem (4.38) has the biggest solution uˆ ∈ C01 () and the smallest solution vˆ ∈ C01 (). In particular, every solution of (4.38) is contained in the ordered interval [v, ˆ u]. ˆ (ii) If (4.35)–(4.36) hold and the set of nodal solutions of problem (4.38) is nonempty, then we have uˆ ∈ int (C01 ()+ ) and vˆ ∈ −int (C01 ()+ ). Proof. Denote by S the set of solutions of problem (4.38) endowed with the pointwise partial order ≤. Lemmas 4.15 and 4.17 guarantee that S is nonempty.

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 169

Let us show that the ordered set S is upward directed, that is, if u1 , u2 ∈ S, then there is u ∈ S such that max{u1 , u2 } ≤ u. Indeed, if u1 , u2 ∈ S, by Lemma 4.16 (a) we know that u = max{u1 , u2 } is a subsolution of problem (4.38), whereas Lemma 4.18 entails that u1 , u2 ∈ C01 (). Note that H(f ) allows us to apply Proposition 4.26, which ensures that there is δ > 0 such that tφ1,p,δ is a supersolution of problem (4.38) for t > 0 large enough. Moreover, for a possibly larger t, we can suppose that tφ1,p,δ ≥ u in  because u1 , u2 ∈ C01 () and φ1,p,δ ∈ int (C01 (δ )+ ). Now invoking the general theory of subsolutions–supersolutions (see [38, Theorem 3.17] or [169, Proposition 11.8]), the existence of u ∈ S such that u(x) ≤ u(x) ≤ tφ1,p,δ (x) for all x ∈  follows, which proves our claim. Next we show that each chain C has an upper bound in S. On the basis of [76, p. 336], there is a sequence {un } ⊂ C such that sup C = sup un = lim un . n≥1

n→∞

(4.41)

According to Lemma 4.18, the sequence {un } is bounded in C 1,γ () for some γ ∈ (0, 1). Since C 1,γ () is compactly embedded in C 1 (), along a relabeled subsequence we have that un → u in C01 () as n → ∞. In view of (4.41) it turns out that u = sup C. In addition, we can pass to the limit in the equation  −p un = f (x, un , ∇un ) in W −1,p (), obtaining that u ∈ S. The preceding discussion enables us to apply Zorn’s lemma, which provides a maximal element uˆ ∈ S. Actually, uˆ is the biggest element in S. Indeed, if u ∈ S, as shown before we can find v ∈ S such that v ≥ max{u, ˆ u}. Then the maximality of uˆ renders uˆ = v, so uˆ ≥ u in . Under the additional assumption (4.35), we may invoke Theorem 4.21 (i) ensuring that if problem (4.38) possesses nodal solutions, then uˆ ∈ int (C01 ()+ ). Arguing in the same way, we can establish the existence of the smallest solution vˆ of problem (4.38). Furthermore, through Theorem 4.21 (ii) we get that vˆ ∈ −int (C01 ()+ ) provided that (4.36) is valid and there are nodal solutions. Now we turn to extremal solutions for problem (4.13) with μ > 0, which basically is the case driven by the (p, q)-Laplacian. Here we cannot proceed as for problem (4.38), i.e., when μ = 0, because if μ > 0, the nonhomogeneity of the operator −p − μq prevents constructing subsolutions and supersolutions by means of the first eigenfunctions of −p or −q . We obtain subsolutions and supersolutions for problem (4.13) with μ > 0 relying on the following assumption: lim inf f (x, s, 0) ≤ γ1 < 0 < γ2 ≤ lim sup f (x, s, 0) s→+∞

s→−∞

uniformly for a.e. x ∈ , with constants γ1 and γ2 .

(4.42)

170 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

We have the following existence and enclosure result. Theorem 4.28. Assume that conditions H(f ) and (4.42) are verified, and that μ > 0 in problem (4.13). Then: (i) Problem (4.13) has the biggest solution uˆ ∈ C01 () and the smallest solution vˆ ∈ C01 (). Thus every solution of (4.13) is contained in the ordered interval [v, ˆ u]. ˆ (ii) If (4.35)–(4.36) hold and the set of nodal solutions of problem (4.13) is nonempty, then uˆ ∈ int (C01 ()+ ) and vˆ ∈ −int (C01 ()+ ). Proof. Denote by S the set of solutions of problem (4.13). As proven in Lemma 4.17, the set S is nonempty. Due to assumption H(f ), it is allowed to apply Lemma 4.18 for obtaining the uniform bound M := sup u∞ < +∞. u∈S

By virtue of (4.42), there exists s0 > M satisfying f (x, s0 , 0) < 0 for a.e. x ∈ . It follows that s0 is a (constant) supersolution of problem (4.13) such that s0 ≥ u for all u ∈ S. We claim that the set S endowed with the pointwise order ≤ is upward directed. To this end let u1 , u2 ∈ S. From Lemma 4.16 (a) we know that u := max{u1 , u2 } is a subsolution for problem (4.13). Since s0 ≥ u, the general theory of subsolutions–supersolutions (see [38, Theorem 3.17] or [169, Proposition 11.8]) provides a solution u ∈ S within the ordered interval [u, s0 ], which proves the claim. As in the proof of Theorem 4.27, we can check that Zorn’s lemma can be applied on the set S, thus getting a maximal element uˆ ∈ S. The fact that uˆ is the biggest element of S can be seen following the pattern of the corresponding part in the proof of Theorem 4.27. Similar arguments lead to the existence of the smallest solution of problem (4.13). Part (i) of the statement is established. Finally, combining with Theorem 4.21, we deduce part (ii) of the statement of the theorem. The proof is thus complete. Remark 4.29. Theorem 4.28 is also valid if, in place of (4.42), we assume the existence of constants c, c such that c ≤ − sup u∞ , u∈S

c ≥ sup u∞ , u∈S

f (x, c, 0) ≤ 0 ≤ f (x, c, 0) for a.e. x ∈ .

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 171

4.3 GRADIENT DEPENDENT PROBLEMS DRIVEN BY NONHOMOGENEOUS DIFFERENTIAL OPERATORS Here we study the nonlinear elliptic Dirichlet problem ⎧ N ⎪ ⎪ ⎨ − ∂ a (x, u, ∇u) = f (x, u, ∇u) in , i ∂xi i=1 ⎪ ⎪ ⎩ u=0 on ∂

(4.43)

on a bounded domain  ⊂ RN with C 2 boundary ∂. This is an extension of problem (4.1), (4.13) in the preceding two sections because the left-hand side of the equation is governed by a more general operator. Set A(x, t, ξ ) := ( a1 (x, t, ξ ), . . . , aN (x, t, ξ ) ) ∈ RN for all (x, t, ξ ) ∈  × [0, +∞) × RN . With this notation, problem (4.43) becomes 

−div A(x, u, ∇u) = f (x, u, ∇u) in , u=0 on ∂.

Let 1 < q ≤ p < +∞. We state the following hypotheses: H(A) There exist positive constants Ci (1 ≤ i ≤ 6), 0 < α ≤ 1, 0 < t0 ≤ 1 and a continuous positive function g0 on [0, +∞) such that (i) ai ∈ C( × [0, +∞) × RN , R) ∩ C 1 ( × (0, +∞) × (RN \ {0}), R) for all 1 ≤ i ≤ N ; (ii) ai (x, t, 0) = 0 for all (x, t) ∈  × [0, +∞), and 1 ≤ i ≤ N ; (iii) Dy A(x, t, y) ≤ g0 (t)(C1 |y|p−2 + C2 |y|q−2 ) for all x ∈ , t ≥ 0 and y ∈ RN \ {0}; (iv) Dy A(x, t, y)ξ · ξ ≥ (C3 |y|p−2 + C4 |y|q−2 )|ξ |2 for all x ∈ , t ≥ 0, y ∈ RN \ {0}, and ξ ∈ RN ; (v) for each M > 0 there exists CM > 0 such that   

A(x, t, y) − A(x  , t  , y) ≤ CM (1 + |y|p−1 ) |x − x  |α + |t − t  |α whenever x, x  ∈ , y ∈ RN , and t, t  ∈ [0, M];  (vi) |∂xi ai (x, t, y)| ≤ C5 t q−1 (1 + | log t|) + |y|q−1 (1 + | log |y| |) for all x ∈ , all 0 < t < t0 , 0 < |y| < t0 , and 1 ≤ i ≤ N ;  (vii) |∂t ai (x, t, y)| ≤ C6 t q−1 (1 + | log t|) + |y|q−1 (1 + | log |y| |) for all x ∈ , all 0 < t < t0 , 0 < |y| < t0 , and 1 ≤ i ≤ N .

172 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

The prototype of the differential operator constructed through the map A is the (p, q)-Laplacian p + q . Many other situations are incorporated. We provide a characteristic example. Example 4.30. Assume 1 < q < r < p and that (i) a ∈ C 1 () such that 0 < inf a ≤ sup a < +∞ and sup |a  | < +∞; (ii) ϕj

  1 ∈ C([0, +∞)) ∩ C ((0, +∞)) with



sup |ϕj (t)| < +∞ for j = p, q, r; t>0

(iii) inf ϕp (t) > 0 and inf ϕq (t) > 0; t≥0

t≥0  (iv) inf ϕp (t) + ϕr (t) > 0 and inf ϕq (t) + ϕr (t) > 0. t≥0

t≥0

Then the map A defined by

 A(x, t, y) = a(x) ϕq (t)|y|q−2 + ϕr (t)|y|r−2 + ϕp (t)|y|p−2 y satisfies assumption H(A). Remark 4.31. If H(A) holds, then the map A satisfies H(A) in Section 2.3. We will also consider the class of operators A which are asymptotically (q − 1)-homogeneous near zero. This relevant property is described in the following condition: H(A)0 There exist a0 ∈ C 1 (, (0, +∞)), A0 ∈ C( × [0, +∞) × RN , RN ), and s0 > 0 such that A(x, t, y) = a0 (x)|y|q−2 y + A0 (x, t, y)

and

for all x ∈ , t ∈ [0, +∞), y ∈ RN |A0 (x, t, y)| lim = 0 uniformly in x ∈ , t ∈ (0, s0 ]. y→0 |y|q−1

For later use, in connection with condition H(A)0 , we denote by λ1, a0 (m) the first positive eigenvalue of −div(a0 (x)|∇u|q−2 ∇u) with weight function m, namely    1,q a0 (x)|∇u|q dx : u ∈ W0 () and m|u|q dx = 1 . λ1, a0 (m) := inf 



(4.44) Now we present some useful consequences of assumption H(A). Proposition 4.32. Assume that condition H(A) holds. Then we have: (i) for every x ∈  and t ∈ [0, +∞), A(x, t, y) is maximal monotone and strictly monotone in y;

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 173



 C1 C2 p−1 q−1 (ii) |A(x, t, y)| ≤ g0 (t) + |y| |y| for all (x, t, y) ∈  × p−1 q −1 [0, +∞) × RN ; C4 C3 (iii) A(x, t, y) · y ≥ |y|p + |y|q for all (x, t, y) ∈  × p−1 q −1 [0, +∞) × RN ; (iv) there exist constants Cp > 0 and Cq > 0 such that (A(x, t, ξ ) − A(x, t, η)) · (ξ − η) ⎧ ⎪ ⎪ C3 |ξ − η|2 (|ξ | + |η|)p−2 + C4 |ξ − η|2 (|ξ | + |η|)q−2 ⎪ ⎪ ⎪ ⎪ if 1 < q ≤ p ≤ 2, |ξ | + |η| > 0, ⎪ ⎪ ⎪ ⎨ C3 Cp |ξ − η|p + C4 |ξ − η|2 (|ξ | + |η|)q−2 ≥ ⎪ ⎪ if 1 < q ≤ 2 ≤ p, |ξ | + |η| > 0, ⎪ ⎪ ⎪ ⎪ C3 Cp |ξ − η|p + C4 Cq |ξ − η|q ⎪ ⎪ ⎪ ⎩ if 2 ≤ q ≤ p for all (x, t) ∈  × [0, +∞), where Ci (1 ≤ i ≤ 4) are the positive constants in assumption H(A). Proof. Assertions (i), (ii), and (iii) are direct consequences of hypotheses H(A) (i)–(iv). We focus on the proof of assertion (iv). Since the other cases can be handled similarly, we suppose that sξ + (1 − s)η = 0 for every s ∈ [0, 1]. From hypothesis H(A) (iv), we obtain (A(x, t, ξ ) − A(x, t, η)) · (ξ − η)  1  2 p−2 2 ≥ C3 |ξ − η| |η + s(ξ − η)| ds + C4 |ξ − η| 0

1

|η + s(ξ − η)|q−2 ds.

0

If 1 < r ≤ 2, it is clear that 

1

 |η + s(ξ − η)|

r−2

1

ds ≥

0

(|ξ | + |η|)r−2 ds = (|ξ | + |η|)r−2 .

0

If r > 2, [64, Lemma 2.1] gives 

1

|η + s(ξ − η)|r−2 ds ≥ Cr (|ξ | + |η|)r−2 ,

0

with a constant Cr > 0 provided |ξ | + |η| > 0. If r ≥ 2, we also note that |ξ − η|r ≤ |ξ − η|2 (|ξ | + |η|)r−2 , so property (iv) follows.

174 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

We formulate the assumptions on the right-hand side of the equation in (4.43). H(f ) f :  × [0, +∞) × RN → R is a continuous function satisfying f (x, 0, ξ ) ≥ 0 for every (x, ξ ) ∈  × RN , as well as (i) there exist constants b1 > 0 and r1 > 1 such that |f (x, t, ξ )| ≤ b1 (1 + t r1 −1 + |ξ |p+1 )

(4.45)

for all (x, t, ξ ) ∈  × [0, +∞) × RN ; (ii) there exist constants b2 > 0 and 1 < r2 < p such that f (x, t, ξ )t ≤ b2 (1 + t r2 + |ξ |p )

(4.46)

for all (x, t, ξ ) ∈  × [0, +∞) × RN . A relevant example of such function is as follows. Example 4.33. The function f defined below verifies H(f ): f (x, t, ξ ) = t r3 −1 (2 − |ξ |p1 ) +

|ξ |p2 − t r5 −1 + |ξ |p3 1 + t r4

with 0 ≤ p1 < p + 1, 1 ≤ r3 < p, 0 ≤ p2 ≤ p, r4 ≥ 1, r5 > 1 and 0 ≤ p3 < p − 1. Our approach is based on the construction of approximate solutions, which are solutions of the auxiliary problem  −div A(x, u, ∇u) = f (x, u, ∇u) + ε in , (4.47) u=0 on ∂ corresponding to each 0 < ε ≤ 1. Fix a number β with 1 0. Then AM has the (S)+ -property. 1,p

Proof. Let a sequence {un } weakly convergent to u in W0 () with lim supAM (un ), un − u ≤ 0. n→∞

By assumption H(A) (v), Hölder’s inequality, and Lebesgue’s dominated convergence theorem, we infer that     M , ∇u) − A(x, u , ∇u)) · (∇u − ∇u) dx  (A(x, uM  n n   M α ≤ CM (1 + |∇u|p−1 )|uM n − u | |∇un − ∇u| dx 



≤ CM ∇un − ∇up

(1 + |∇u|

p−1

)

p p−1



|uM n

−u | M

pα p−1

 p−1 p

dx

→0

as n → ∞. This leads to  M lim sup (A(x, uM n , ∇un ) − A(x, un , ∇u)) · (∇un − ∇u) dx ≤ 0. n→∞

(4.50)



If p ≥ 2, from (4.50) and Proposition 4.32 (iv) we find p

lim sup ∇un − ∇up ≤ 0, n→∞

1,p

whence un → u in W0 (). If 1 < p < 2, by Hölder’s inequality we get for 1,p every v ∈ W0 () that p

∇u − ∇vp  = ≤

{|∇u|+|∇v|>0}

|∇u − ∇v|p (|∇u| + |∇v|)



{|∇u|+|∇v|>0}

p(p−2) 2

(|∇u| + |∇v|)

(|∇u| + |∇v|)p−2 |∇u − ∇v|2 dx

p(2−p) 2

dx

p 2

p

× (∇up + ∇vp )p(1− 2 ) . Setting v = un , from (4.50) and Proposition 4.32 (iv), we infer that {un } strongly 1,p converges to u in W0 ().

176 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints 

Proposition 4.35. Let BM,ε : W0 () → W −1,p () be the map defined by 1,p

 BM,ε (u), v =

 fM (x, u, ∇u)v dx + ε



v dx 

1,p

for all u, v ∈ W0 (), with M > 0 and ε ∈ (0, 1]. Then, for AM in (4.49), AM − BM,ε is a pseudomonotone operator. 1,p

Proof. Let {un } be a sequence such that un  u in W0 () and lim supAM (un ) − BM,ε (un ), un − u ≤ 0. n→∞

Since the truncated function fM is bounded, we infer that   BM,ε (un ), un − u ≤ (C˜ M + 1)un − u1 → 0, where C˜ M := max{|fM (x, t, ξ )| : (x, t, ξ ) ∈  × R × RN }, which implies lim supAM (un ), un − u ≤ 0. n→∞

Then the (S)+ -property of Proposition 4.34 implies that {un } strongly converges 1,p to u in W0 (), hence lim BM,ε (un ), un − v = BM,ε (u), u − v

n→∞ 1,p

for every v ∈ W0 (). On account of     C1 C2   p−1 q−1 , ∇u ) ≤ G(M) | + | |∇u |∇u , A(x, uM  n n n n p−1 q −1 with G(M) := max{g0 (t) : t ∈ [0, M]} (see Proposition 4.32 (ii)), Lebesgue’s dominated convergence theorem yields, as n → ∞, that  AM (un ), un − v = A(x, uM n , ∇un ) · (∇un − ∇v) dx   A(x, uM , ∇u) · (∇u − ∇v) dx = AM (u), u − v →  1,p

for all v ∈ W0 (), thus AM − BM,ε is pseudomonotone.

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 177

Proposition 4.36. For every M > 0 and ε ∈ (0, 1], the problem  −div AM (x, u, ∇u) = fM (x, u, ∇u) + ε in , u=0 on ∂

(4.51)

1,p

has a solution uM,ε ∈ W0 (). Proof. Fix M > 0 and ε ∈ (0, 1]. The operator AM − BM,ε is bounded because AM is bounded by Proposition 4.32 (ii), the function fM is bounded and 0 ≤ uM ≤ M. The boundedness of fM and Proposition 4.32 (iii) entail that AM − BM,ε is coercive, which reads as lim

u→+∞

AM (u) − BM,ε (u), u = +∞, u 1,p

where we recall that  ·  stands for the usual norm on W0 (). Recalling that AM − BM,ε is pseudomonotone (by Proposition 4.35), the main surjectivity result on pseudomonotone operators (Theorem 1.55) ensures that there exists  1,p uM,ε ∈ W0 () such that AM (uM,ε ) − BM,ε (uM,ε ) = 0 in W −1,p (), thereby uM,ε is a solution of (4.51). We further discuss certain properties of the approximate solution uM,ε . 1,p

Lemma 4.37. If uM,ε ∈ W0 () solves problem (4.51), then uM,ε (x) ≥ 0 for a.e. x ∈ . Proof. Acting with −(uM,ε )− as a test function in (4.51) shows that C3 C4 p q ∇(uM,ε )− p + ∇(uM,ε )− q p−1 q −1  ≤ A(x, 0, ∇uM,ε ) · ∇uM,ε dx 

{uM,ε ≤0}

AM (x, uM,ε , ∇uM,ε ) · (−∇(uM,ε )− ) dx   fM (x, uM,ε , ∇uM,ε ) uM,ε dx − ε (uM,ε )− dx ≤ 0 =

=



{uM,ε ≤0}



due to the fact that fM (x, t, ξ ) ≥ 0 if t ≤ 0 and Proposition 4.32 (iii). 1,p

Lemma 4.38. If uM,ε ∈ W0 () is a solution of problem (4.51), then uM,ε ∈ L∞ (). Proof. For sake of simplicity, we pose u := uM,ε . By Lemma 4.37 we know that uM = min{u, M}. Take p ∗ = Np/(N − p) if N > p, and any p ∗ > p

178 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

otherwise. Letting uL := min{u, L} and using (uL )α u as a test function, where L > M and α > 0, we obtain from Proposition 4.32 (iii) that  (fM (x, u, ∇u) + ε) (uL )α u dx   Lα A(x, uM , ∇u) · ∇u dx + (1 + α)uα A(x, uM , ∇u) · ∇u dx = {u≥L} {u 0 and D0 > 0 independent of M and ε such that uM,ε β0 +α ∗ ≤ D0 for every solution uM,ε of problem (4.51). 1,p

Proof. Set, for simplicity, u := uM,ε and use u1+α ∈ W0 (), with α > 0, as a test function in (4.51), which is possible because by Lemma 4.38 we know that u ∈ L∞ (). From assumption H(f ) (ii) (see (4.46)) and the fact that uM = min{u, M}, we see that  (fM (x, u, ∇u) + ε) u1+α dx   ≤ b2 (1 + (uM )r2 + |∇u|p )uα dx {|∇u| β0 + α ∗ p results in



∗

|∇u|p uα dx 

C∗ p p C∗ p p α ∗ +p 1+α ∗ /p p u  = up ∗ (1+α ∗ /p) ∗ p ∗ p ∗ p (p + α ) (p + α ) C∗ p p α ∗ +p uβ0 +α ∗ . ≥ (1 + ||)(p + α ∗ )p

≥

(4.56) (4.57)

From (4.55) and (4.57) we get C∗ p p (1 + α ∗ )C3 p uβ0 +α ∗ 2(p − 1)(1 + ||)(p + α ∗ )p β

≤ 4(b2 + 1)(1 + ||) max{1, uβ00 +α ∗ }. Since β0 < p, this reads as uβ0 +α ∗ ≤ max{C 1/(p−β0 ) , C 1/p } with C =

8(b2 +1)(1+||)2 (p+α ∗ )p (p−1) . C∗ p p (1+α ∗ )C3

Now we are in a position to obtain the uniform bounds. Proposition 4.40. There exists a constant D1 > 0 independent of M and ε such that for every solution u of problem (4.51) one has u∞ ≤ D1 . 1,p

Proof. Let u ∈ W0 () be a solution of (4.51) with M > 0 and ε ∈ (0, 1]. For each α > 0, we act with u1+α as a test function in problem (4.51). As in (4.55) and (4.56), we see that α+p

up ∗ (1+α/p) ≤ C where C= 8(b2 +1)(1+||) C∗ p p C3

2 (p−1)

(p + α)p β +α max{1, uβ00 +α }, 1+α

. With the number α ∗ determined in Lemma 4.39,

 we construct the sequence {αm } by α0 := α ∗ and αm+1 := p ∗ 1 + αpm − β0 .

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 181

Proceeding as in the proof of Lemma 4.38, we prove the estimate u∞ ≤ C  max{1, uβ0 +α ∗ }(p

∗ −p)(β +α ∗ )/(p ∗ α ∗ ) 0

with a positive constant C  independent of ε and M. Applying Lemma 4.39, the conclusion ensues. We turn to problem (4.47). We introduce the notation    1,q μ1 (m) = inf |∇u|q dx : u ∈ W0 () and m|u|q dx = 1 

(4.58)



corresponding to a weight function m, and Aq (t) :=

C2 q −1



C2 C4

q−1 g0 (t)q for t ≥ 0,

(4.59)

where g0 , C2 , C4 are given in hypothesis H(A). It is worth noting that Aq ≡ 1 if A(x, t, y) = |y|p−2 y + |y|q−2 y (i.e., the case of the (p, q)-Laplacian). We formulate two conditions, corresponding to two different situations, permitting us to investigate the approximate problem (4.47), and eventually (4.43). H1 (f ) There exist m ∈ L∞ () and positive constants b0 , b0 , r  with |{x ∈  : m(x) > 0}| > 0,

r > q

b0 > μ1 (m)Aq (0),

(4.60)

such that for each ρ > 0 there is Kρ > 0 satisfying 

f (x, t, ξ ) ≥ (b0 m(x) − ρ)t q−1 − b0 |ξ |r −1

(4.61)

for all x ∈ , t ∈ [0, Kρ ] and ξ ∈ RN with |ξ | ≤ Kρ . H2 (f ) There exist m ∈ L∞ () and positive constants b0 , b0 , r  with |{x ∈  : m(x) > 0}| > 0,

b0 > λ1, a0 (m),

r > q

(4.62)

(with a0 as in H(A)0 and λ1, a0 (m) as in (4.44)) such that for each ρ > 0 there is Kρ > 0 satisfying 

f (x, t, ξ ) ≥ (b0 m(x) − ρ)t q−1 − b0 |ξ |r −1

(4.63)

for all x ∈ , t ∈ [0, Kρ ] and ξ ∈ RN with |ξ | ≤ Kρ . Example 4.41. The function f in Example 4.33 satisfies conditions H1 (f ) and H2 (f ) provided 1 < r3 < q ≤ r5 . Remark 4.42. If H1 (f ) or H2 (f ) is verified, then condition H(f) in Section 2.3 holds.

182 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

First, we present an easy consequence. Lemma 4.43. Suppose H1 (f ) or H2 (f ). Then for each K > 0 there exists λ0 > 0 such that

 f (x, t, ξ ) ≥ −λ0 t q−1 + |ξ |q−1 for all x ∈ , t ∈ [0, K] and ξ ∈ RN with |ξ | ≤ K. Proof. Fix K > 0. By H1 (f ) or H2 (f ), there exist δ > 0 such that 

f (x, t, ξ ) ≥ −(b0 + 1)m∞ t q−1 − b0 δ r −q |ξ |q−1 for all x ∈ , t ∈ [0, δ], |ξ | ≤ δ. On the other hand, from (4.45) it follows that f (x, t, ξ ) ≥ −b1 (1 + t r1 −1 + |ξ |p+1 ) ≥−

b1 (1 + K r1 −1 ) q−1 t − b1 K p−q+2 |ξ |q−1 δ q−1

provided δ ≤ t ≤ K and |ξ | ≤ K. Similarly, if 0 ≤ t ≤ K and δ ≤ |ξ | ≤ K, then we have f (x, t, ξ ) ≥ −b1 (1 + t r1 −1 + |ξ |p+1 ) ≥ −

b1 (1 + K r1 −1 + K p+1 ) q−1 |ξ | . δ q−1

The desired estimates for problem (4.47) are now available. Proposition 4.44. Suppose H1 (f ) or H2 (f ). Then, for every ε ∈ (0, 1], problem (4.47) has a positive solution uε ∈ int (C01 ()+ ). Moreover, there exist 1,γ D2 > 0 and γ ∈ (0, 1) such that uε ∈ C0 () and uε C 1,γ () ≤ D2 for all 0

ε ∈ (0, 1].

Proof. Let uM,ε be a nonnegative solution for (4.51) whose existence is guaranteed by Proposition 4.36. Choose M0 > D1 , where D1 is obtained in Proposition 4.40. Then 0 ≤ uM,ε (x) ≤ D1 < M0 for every ε ∈ (0, 1] and M ≥ M0 , so uM,ε is a solution for −divA(x, u, ∇u) = fM (x, u, ∇u) + ε

in ,

u = 0 on ∂.

(4.64)

Since fM satisfies (4.45) with the same constant b1 and uM,ε ∞ ≤ D1 , the regularity result in [129] and [130] applied to (4.64) provides constants D2 > 0 1,γ and γ ∈ (0, 1) such that uM,ε ∈ C0 () and uM,ε C 1,γ () ≤ D2 for all ε ∈ (0, 1] and M ≥ M0 .

0

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 183 β

Taking M1 with M1 > max{D2 , M0 }, it holds fM1 (x, uM1 ,ε , ∇uM1 ,ε ) = f (x, uM1 ,ε , ∇uM1 ,ε ), 1,γ

whence uε := uM1 ,ε ∈ C0 () is a nonnegative solution of (4.47). Note that uε ≡ 0. Since Lemma 4.43 ensures that f(x, s, ξ ) := f (x, s, ξ ) + ε satisfies (2.49), Theorem 2.19 implies that uε ∈ int (C01 ()+ ). Some preliminaries are still necessary. Lemma 4.45. Let ϕ, u ∈ int (C01 ()+ ). Then  q   ϕ A(x, u, ∇u) · ∇ dx q−1 u    q C1 q p−q q ≤ g (u) |∇u| |∇ϕ| dx + Aq (u)|∇ϕ|q dx, 0 q−1   (p − 1)C3 with the function Aq in (4.59) and the constants C1 and C3 in hypothesis H(A). Proof. Since ϕ, u ∈ int (C01 ()+ ), there exist δ1 > δ2 > 0 such that δ1 u ≥ ϕ ≥ δ2 u in , hence u/ϕ, ϕ/u ∈ L∞ (). Therefore, by Proposition 4.32 (ii) and (iii), we find that  q  ϕ A(x, u, ∇u) · ∇ uq−1

ϕ q

ϕ q−1 A(x, u, ∇u) · ∇ϕ − (q − 1) A(x, u, ∇u) · ∇u =q u u qC1 ϕ q−1 (q − 1)C3 ϕ q ≤ g0 (u) |∇u|p−1 |∇ϕ| − |∇u|p p−1 u p−1 u

ϕ q qC2 ϕ q−1 + g0 (u) |∇u|q−1 |∇ϕ| − C4 |∇u|q . (4.65) q −1 u u Young’s inequality shows that qC1 ϕ q−1 |∇u|p−1 |∇ϕ| p−1 u q C1 (q − 1)C3 ϕ q |∇u|p + g0 (u)q |∇u|p−q |∇ϕ|q . ≤ q−1 p−1 u (p − 1)C3 g0 (u)

(4.66)

Similarly, taking into account (4.59), we get g0 (u)

ϕ q qC2 ϕ q−1 |∇u|q−1 |∇ϕ| ≤ C4 |∇u|q + Aq (u)|∇ϕ|q . (4.67) q −1 u u

Thanks to (4.65), (4.66), and (4.67), the conclusion follows.

184 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Lemma 4.46. Let a0 ∈ C(, [0, +∞)) and ϕ, u ∈ int (C01 ()+ ). Then 

ϕ q − uq  a0 (x)|∇ϕ|q−2 ∇ϕ · ∇ dx ϕ q−1  

ϕ q − uq  a0 (x)|∇u|q−2 ∇u · ∇ dx ≥ 0. − uq−1  The proof can be found in [226, Lemma 10]. Lemma 4.47. Assume H1 (f ) or H2 (f ). Then there exists a constant D4 > 0 such that 

 ϕ|∇uε | q dx ≤ D4 (|∇ϕ|q + ϕ q ) dx uε   for all ϕ ∈ C01 ()+ and uε ∈ int (C01 ()+ ) given by Proposition 4.44 with ε ∈ (0, 1]. Proof. Fix any ϕ ∈ C01 ()+ . By Proposition 4.44, {uε : ε ∈ (0, 1]} is bounded in C01 (). Then Lemma 4.43 provides a positive constant λ0 satisfying f (x, uε , ∇uε ) ≥ −λ0 (uq−1 + |∇uε |q−1 ) ε whenever ε ∈ (0, 1]. Note that ϕ/uε ∈ L∞ () because ϕ ∈ C01 ()+ and Propo1−q 1,p sition 4.44 guarantees that uε ∈ int (C01 ()+ ). Using ϕ q uε ∈ W0 () as a test function gives     f (x, uε , ∇uε ) + ε q ϕq A(x, uε , ∇uε ) · ∇ ϕ dx dx = q−1 q−1   uε uε   |∇uε |q−1 q ϕ dx ≥ − λ0 ϕ q dx − λ0 q−1   uε      C4 ϕ|∇uε | q q dx − D5 ϕ q dx, (4.68) ≥ − λ0 ϕ dx − 4  uε   q

1−q

through Young’s inequality and D5 = λ0 4q−1 C4 . On the other hand, by easy estimates involving Proposition 4.32 (ii) and (iii) (note that 1−q < 0), we obtain    ϕq A(x, uε , ∇uε ) · ∇ dx q−1  uε       C3 (q − 1) ϕ|∇uε | q ϕ|∇uε | q dx − |∇uε |p−q dx ≤ − C4 uε p−1 uε      ϕ|∇uε | q−1 qC2 g0 (uε ) |∇ϕ| dx + q −1  uε

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 185

+

qC1 p−1



 g0 (uε ) 

ϕ|∇uε | uε

q−1 |∇uε |p−q |∇ϕ| dx.

Applying Young’s inequality to the last two terms yields    ϕq A(x, uε , ∇uε ) · ∇ dx q−1  uε       C3 (q − 1) ϕ|∇uε | q ϕ|∇uε | q C4 dx − |∇uε |p−q dx ≤ − 2  uε 2(p − 1)  uε   q q + D6 g0 (uε ) |∇ϕ| dx + D7 g0 (uε )q |∇ϕ|q |∇uε |p−q dx, (4.69) 



where D6 and D7 are positive constants independent of uε and ϕ. From (4.68) and (4.69), the existence of a positive constant D8 independent of uε and ϕ such that      ϕ|∇uε | q dx ≤ D8 g0 (uε )q |∇ϕ|q (1 + |∇uε |p−q ) dx + D8 ϕ q dx uε      p−q ≤ D8 Gq (1 + D3 ) |∇ϕ|q dx + D8 ϕ q dx, 

where G := sup {g0 (t) : t ∈ (0, D3 ]}

and



  D3 := sup uε C 1 () : ε ∈ (0, 1] , 0

follows, which completes the proof. Lemma 4.48. Assume that conditions H2 (f ) and H(A)0 hold. If uε → 0 in C01 () as ε → 0+ , with uε ∈ int (C01 ()+ ) solution of (4.47), then for every ϕ ∈ int (C01 ()+ ) it is true that 

 lim

ε→0+ 

A0 (x, uε , ∇uε ) · ∇

q

ϕ q − uε



q−1

uε

dx = 0,

where A0 is the function introduced in condition H(A)0 . Proof. Fix any δ>0. Since uε → 0 in C01 (), we may admit that uε C 1 () ≤s0 0 provided ε > 0 is sufficiently small, where s0 is the constant given in H(A)0 . From assumption H(A)0 we have that |A0 (x, uε , ∇uε )| ≤ δ|∇uε |q−1 for every x ∈  and small enough ε > 0. Therefore, by Lemma 4.47 and Hölder’s inequality, we arrive at

186 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

   A0 (x, uε , ∇uε ) · ∇ 





ϕq



q−1

uε

  dx 

|A0 (x, uε , ∇uε )| |∇ϕ|ϕ q−1

≤q

dx q−1 uε |A0 (x, uε , ∇uε )| |∇uε |ϕ q dx + (q − 1) q uε  



ϕ|∇uε | q−1 ϕ|∇uε | q ≤ δqϕC 1 () dx + δ(q − 1) dx 0 uε uε 

  q ≤ δϕ 1 q(2D4 )1−1/q + 2(q − 1)D4 ||, 



C0 ()

where D4 is the positive constant obtained in Lemma 4.47. Moreover, it is clear that     q q  A0 (x, uε , ∇uε ) · ∇uε dx  ≤ δ∇uε q ≤ δs0 . 

Recalling that δ > 0 is arbitrary, the result is proven. The statement of the first main existence result for problem (4.43) is as follows. Theorem 4.49. Assume that conditions H(A), H(f ), and H1 (f ) hold. Then problem (4.43) has a (positive) solution u ∈ int (C01 ()+ ). Proof. By Proposition 4.44 there exist constants D2 > 0 and 0 < γ < 1 such that uε C 1,γ () ≤ D2 , ∀ε ∈ (0, 1]. 0

1,γ

Due to the compact embedding of C0 () into C01 (), there exist a sequence εn → 0+ and a solution u0 ∈ C01 ()+ of problem (4.43) such that un := uεn → u0 in C01 () as n → ∞. If u0 = 0 occurs, then u0 ∈ int (C01 ()+ ) by the argument carried on in the proof of Proposition 4.44, and the conclusion of the theorem is achieved. It remains to prove that u0 = 0. To this end, we argue by contradiction supposing that un → 0 in C01 (). Let ϕ ∈ int (C01 ()+ ) be an eigenfunction corresponding to the first positive eigenvalue μ1 (m) (see (4.58)). Thus ϕ is a positive solution of the Dirichlet problem  −q u = μ1 (m)m(x)|u|q−2 u in , u=0 on ∂.

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 187

Since this equation is (q − 1)-homogeneous, we may assume that  q ∇ϕq = μ1 (m) m(x)ϕ q dx = μ1 (m). 

In view of (4.60) we can choose ρ > 0 such that ρ
0 such that 

f (x, s, ξ ) ≥ (b0 m(x) − ρ)s q−1 − b0 |ξ |r −1

(4.72)

whenever x ∈ , 0 ≤ s ≤ K0 and |ξ | ≤ K0 . Since un ∞ ≤ K0 and ∇un ∞ ≤ K0 hold for sufficiently large n, by (4.72) we obtain     f (x, un , ∇un ) + εn q ϕq A(x, un , ∇un ) · ∇ ϕ dx dx = q−1 q−1   un un    |∇un |r −1 ϕ q q dx. (4.73) ≥ b0 m(x)ϕ q dx − ρϕq − b0 q−1   un Hölder’s inequality and Lemma 4.47 entail 

 



|∇un |r −1 ϕ q 

q−1 un

dx = 

|∇un |ϕ un

(q−1)/q

≤ D4

q−1 r  −q



|∇un |r −q ϕ dx q

(4.74) q

un C 1 () (∇ϕq + ϕq )(q−1)/q ϕq = o(1)

(note that r  − q > 0 and un C 1 () = o(1)). Finally, (4.71), (4.73), and (4.74) lead to  q b0 − μ1 (m)Aq (0) ≤ (Aq (un ) − Aq (0))|∇ϕ|q dx + ρϕq + o(1). 

Letting n → ∞ contradicts (4.70), which completes the proof. Now we are able to present the second main result of this section, where H(A)0 is assumed, and H2 (f ) is assumed instead of H1 (f ). Theorem 4.50. Assume that H(A), H(A)0 , H(f ), and H2 (f ) are satisfied. Then problem (4.43) has a (positive) solution u ∈ int (C01 ()+ ).

188 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Proof. At the beginning we proceed as in the proof of Theorem 4.49, finding a solution u0 ∈ C01 ()+ of problem (4.43). We note that if u0 = 0, then we are done by following the argument in the proof of Proposition 4.44. So, we need to show that u0 = 0. We argue by contradiction by supposing that un → 0 in C01 (), where {un } is the sequence considered in the proof of Theorem 4.49. Take a (positive) eigenfunction ϕ1 ∈ int (C01 ()+ ) corresponding to λ1, a0 (m), which can be obtained as minimizer of (4.44). We notice that   q 0< a0 (x)|∇ϕ1 |q dx = λ1, a0 (m) m(x)ϕ1 dx, thus





q  m(x)ϕ1



dx > 0. According to (4.62), we can choose ρ > 0 satisfying  b0 − λ1, a0 (m) q ρ< m(x)ϕ1 dx. (4.75) q ϕ1 q 

Since un ∈ int (C01 ()+ ) is a solution of problem (4.47) with ε = εn , Lemmas 4.46, 4.48, and assumption H(A)0 yield  q   q ϕ1 − un q−2 0≤ a0 (x)|∇ϕ1 | ∇ϕ1 · ∇ dx q−1  ϕ1  q   q ϕ1 − un q−2 a0 (x)|∇un | ∇un · ∇ − dx q−1  un   f (x, un , ∇un ) q q q ϕ1 dx ≤ λ1, a0 (m) m(x)(ϕ1 − un ) dx − q−1   un  q   q ϕ1 − un + A0 (x, un , ∇un ) · ∇ dx q−1  un   f (x, un , ∇un )un dx + εn un dx +    f (x, un , ∇un ) q q ϕ dx + λ (m) m(x)ϕ1 dx + o(1). (4.76) =− 1, a0 1 q−1   un Based on the proof of Theorem 4.49 with ϕ1 in place of ϕ (see (4.73) and (4.74)), through (4.63) we have   f (x, un , ∇un ) q q q ϕ dx ≥ b m(x)ϕ1 dx − ρϕ1 q + o(1). (4.77) 0 1 q−1   un Combining (4.76) and (4.77), and then letting n → ∞, it turns out that  

q q m(x)ϕ1 dx + ρϕ1 q . 0 ≤ λ1, a0 (m) − b0 

This contradicts (4.75), completing the proof.

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 189

4.4 NOTES Section 4.1 considers elliptic problems driven by the p-Laplacian, which is the prototype of quasilinear differential operator, and exhibiting gradient dependence in the right-hand side of the equation. The semilinear and quasilinear elliptic equations with gradient dependence on the reaction term occupy a distinct place in the literature. This is due to the fact that the variational methods cannot be applied. They are usually studied by means of topological degree, subsolution–supersolution method, fixed point theory and approximation techniques. For instance, in [8], assuming that f is a C 1 function with growth given by |f (x, s, ξ )| ≤ a(s)(1 + |ξ |2 ), for some increasing function a, the authors obtain a solution of problem (4.1) in an ordered interval of subsolution–supersolution. In this respect, we also mention [201] where existence results for the same problem (4.1), when p = 2, are obtained via subsolutions–supersolutions in the Sobolev space W 2,q () with q > N , provided f (x, s, ξ ) is Lipschitz continuous with respect to ξ . For the general theory of subsolutions–supersolutions for nonlinear elliptic problems depending on the gradient, we refer to [38]. Different other results based on the subsolution–supersolution method can be found in [82]. In [212], by combining Krasnoselskii’s fixed point theorem in cones with blow up techniques, the existence of a positive solution of problem (4.1) is proved when f (x, s, ξ ) is a nonnegative function and has a suitable growth with respect to s and ξ . Recently, in [81] and [226], under different growth in the gradient, the existence of a positive solution is achieved via an approximation on finite dimensional subspaces. An approximation approach in a general functional setting with a p-sublinear growth condition in the gradient can be found in [27]. A different approach is developed in [68] where the authors prove the existence of a positive and a negative solution for problem (4.1), when p = 2, through an iterative method involving mountain pass technique and assuming that f (x, s, ξ ) satisfies Lipschitz conditions in s and ξ in a neighborhood of zero and has growth |f (x, s, ξ )| ≤ a1 (1 + |s|q ), with 1 < q < 2∗ − 1. We also mention that in [93] an existence result for a positive solution of an elliptic problem with dependence on the gradient with Neumann boundary condition is given. This reference [93] has much in common with [80]. Theorem 4.12 is the main result in Faraci–Motreanu–Puglisi [80]. It establishes the existence of a positive and a negative solution for problem (4.1) by combining subsolution–supersolution techniques with Schaefer’s fixed point theorem. In addition, it is shown in Corollary 4.13 that these opposite constantsign solutions can be chosen to be extremal. The existence of a nontrivial

190 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

solution (necessarily nodal) between the extremal opposite constant-sign solutions is an open problem. The idea of the proof of Theorem 4.12 is to consider an auxiliary problem which can be studied through operator theory and subsolution–supersolution method in conjunction with a fixed point approach. It is worth mentioning that we obtain positive and negative solutions in spite of the fact that our assumptions imply that the trivial solution exists too, which is a relevant aspect for quasilinear elliptic problems with general gradient dependence. Section 4.2 investigates gradient dependent elliptic problems driven by the (p, q)-Laplacian, where 1 < q < p, which is an operator essentially different from the p-Laplacian. Theorems 4.20 and 4.21 belong to Motreanu–Motreanu– Moussaoui [164]. They study the location of nodal (that is, sign-changing) solutions for problem (4.13). Specifically, they prove that if assumptions H(f )1 , H(f )2 and f (x, 0, 0) = 0 for a.e. x ∈  are fulfilled, then every nodal solution of problem (4.13) should be between two opposite constant-sign solutions. In particular, this provides the powerful fact that the existence of a nodal solution implies under the stated hypotheses that two opposite constant-sign solutions must exist. Moreover, this phenomenon occurs even unilaterally: if we only have f (x, 0, 0) ≥ 0 (resp. f (x, 0, 0) ≤ 0) for a.e. x ∈ , then every nodal solution to problem (4.13) is bounded above by a positive solution (resp. bounded below by a negative solution). In Theorems 4.27 and 4.28 the existence of barrier solutions to problem (4.13), that is, the biggest solution and the smallest solution, is established. Taking into account what was said before about the location of nodal solutions, the biggest nontrivial solution is positive and the smallest nontrivial solution is negative, and they belong to C01 (). Our approach for obtaining the barrier solutions relies on the method of subsolutions–supersolutions related to problem (4.13). We investigate separately the cases μ = 0 and μ > 0 because there are relevant differences in the construction of subsolutions and supersolutions corresponding to the two cases. Hypothesis H(f ) enables us to apply recent estimates due to Cianchi– Maz’ya [53] ensuring the global boundedness of the gradient of any solution u to (4.13). Owing to the homogeneous boundary condition, it follows that u ∈ L∞ () and u∞ is bounded above by a constant independent of u. This is actually a substitute for the Moser iteration technique (see, e.g., [156, Theorem C]) and has to be regarded in conjunction with Lieberman’s estimates [129, 130] for the regularity up to the boundary. In particular, every solution of problem (4.13) belongs to the space C01 (). The presentation of Section 4.3 follows closely Motreanu–Tanaka [183]. The generality of problem (4.43) is twofold: (i) the left-hand side of the equation is expressed in divergence form through a possibly nonhomogeneous operator depending on the solution u and its gradient ∇u;

Nonlinear Elliptic Equations with Dependence on the Gradient Chapter | 4 191

(ii) the nonlinearity in the right-hand side of the equation is a so-called convection term, which means that it depends on the solution u and its gradient ∇u. Such a general problem is considered for the first time in [183]. Moreover, the growth allowed in H(f ) improves the corresponding conditions in the preceding works. We briefly describe some related results. The existence of positive solutions for problem (4.43) with A(x, u, ∇u) = |∇u|p−2 ∇u, that is, div A(x, u, ∇u) stands the p-Laplacian p , and with some convection term f (x, u, ∇u) are studied in [80,212,242]. Specifically, in [80] the focus is on the case of f (x, u, ∇u) with (p − 1)-sublinear growth in u and ∇u. In [212] the growth max{0, |u|r1 − C|∇u|r2 } ≤ |f (x, u, ∇u)| ≤ C(1 + |u|r1 + |∇u|r2 ) with p − 1 < r1
1 verifies the conditions required in H(g). A fundamental tool in our approach is represented by the notions of subsolution and supersolution. Definition 5.2. A function u ∈ W 1,p () is called a subsolution of problem (5.1) if u ≤ 0 on ∂, g(·, u) ∈ Lq () with some 1 < q < p ∗ and   λ|u|p−2 u − g(x, u) v dx −p u, v ≤  1,p

for all v ∈ W0 () with v ≥ 0 a.e. in . The definition of a supersolution u ∈ W 1,p () is obtained by reversing the inequalities. A function u ∈ W 1,p () is a solution to problem (5.1) if it is simultaneously a subsolution and a supersolution. First, we establish the existence of opposite constant-sign solutions to problem (5.1). To this end, as known from Proposition 1.56, the operator −p on 1,p W0 () is maximal monotone and coercive. Hence, by Theorem 1.55, there 1,p exists e ∈ W0 () such that −p e = 1. The nonlinear regularity theory and nonlinear strong maximum principle imply e ∈ int (C01 ()+ ). By λ1 we denote 1,p the first eigenvalue of −p on W0 () and by uˆ 1 we denote the corresponding positive, Lp -normalized eigenfunction, so that uˆ 1 ∈ int (C01 ()+ ) (see Proposition 1.57). Lemma 5.3. Assume H(g) (i)–(iii) and λ > λ1 . Then there exists a constant aλ > 0 such that aλ e and −aλ e are supersolution and subsolution, respectively, of problem (5.1). In addition, −ε uˆ 1 is a supersolution and ε uˆ 1 is a subsolution of problem (5.1) provided ε > 0 is sufficiently small. Proof. Fix λ > λ1 . By condition H(g) (i) there exists sλ > 0 such that g(x, s) > λ for a.a. x ∈  and all |s| > sλ . |s|p−2 s Assumption H(g) (iii) ensures the existence of a constant cλ > 0 for which one has |g(x, s) − λ|s|p−2 s| ≤ cλ for a.a. x ∈  and all |s| ≤ sλ .

Constant-Sign and Sign-Changing Solutions for Elliptic Problems Chapter | 5

195

Then we get g(x, s) ≥ λs p−1 − cλ for a.a. x ∈  and all s > 0. 1

This estimate shows that cλp−1 e is a supersolution of problem (5.1) (see Definition 5.2). Similarly, because g(x, s) ≤ λ|s|p−2 s + cλ for a.a. x ∈  and all s < 0, 1

it turns out that −cλp−1 e is a subsolution of problem (5.1), thereby the first part 1

of the lemma holds with aλ = cλp−1 . Assumption H(g) (ii) guarantees that a number δλ > 0 exists such that |g(x, s)| < λ − λ1 for a.a. x ∈  and all 0 < |s| ≤ δλ . |s|p−1

(5.3)

If ε ∈ (0, uˆδλ ], then it is straightforward to check that εuˆ 1 is a subsolution and 1 ∞ −ε uˆ 1 is a supersolution of problem (5.1). We first point out the existence of two solutions of problem (5.1) having opposite constant sign and being extremal. Theorem 5.4. Assume conditions H(g) (i)–(iii). Then for every λ > λ1 there exist a smallest positive solution u+ = u+ (λ) ∈ int (C01 ()+ ) in the ordered interval [0, aλ e] and a biggest negative solution u− = u− (λ) ∈ −int (C01 ()+ ) in the ordered interval [−aλ e, 0], with aλ > 0 determined in Lemma 5.3. Proof. Fix λ > λ1 . Lemma 5.3 ensures that u = aλ e ∈ int (C01 ()+ ) is a supersolution for problem (5.1) and u = ε uˆ 1 ∈ int (C01 ()+ ) is a subsolution for problem (5.1) if ε > 0 is small enough. For a possibly smaller ε > 0 we may assume that ε uˆ 1 ≤ aλ e. Then by the method of subsolutions–supersolutions we know that within the ordered interval [ε uˆ 1 , aλ e] there is the smallest solution uε = uε (λ) ∈ int (C01 ()+ ) of problem (5.1) (see [38]). Thus for every positive integer n sufficiently large there is the smallest solution un ∈ int (C01 ()+ ) of problem (5.1) within [ n1 uˆ 1 , aλ e]. In view of the minimality property of un , we have un ↓ u+ pointwise,

(5.4)

with some function u+ :  → R satisfying 0 ≤ u+ ≤ aλ e. We claim that u+ is a solution of problem (5.1). Taking into account that un solves (5.1), we have  p

∇un p = (λun (x)p − g(x, un (x))un (x)) dx. 

(5.5)

196 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

By H(g) (iii) and since un belongs to the ordered interval [0, aλ e], it follows 1,p that the sequence {un } is bounded in W0 (). Then due to (5.4) we derive that 1,p u+ ∈ W0 (), as well as 1,p

un  u+ in W0 () , un → u+ in Lp () and a.e. in . The fact that un is a solution of (5.1) results in  −p un , ϕ = (λ|un |p−2 un − g(x, un ))ϕ(x) dx 

(5.6)

1,p

for all ϕ ∈ W0 (). (5.7)

Inserting ϕ = un − u+ in (5.7) yields  −p un , un − u+  = (λ|un |p−2 un − g(x, un ))(un (x) − u+ (x)) dx. 

On the basis of (5.6), H(g) (iii) and the uniform boundedness of the sequence {un }, the Lebesgue’s dominated convergence theorem ensures lim −p un , un − u+  = 0.

n→∞

1,p

Then the (S)+ -property of −p on W0 () (see Proposition 1.56) implies 1,p

un → u+ in W0 () as n → ∞.

(5.8)

The strong convergence in (5.8), combined with H(g) (iii) and the uniform boundedness of {un }, allows passing to the limit in (5.7), which leads to (5.5). We know from (5.4) that u+ ∈ L∞ (). Then (5.5) and assumption H(g) (iii) entail p u+ ∈ L∞ (). The nonlinear regularity theory shows that u+ ∈ C 1,γ () for some γ ∈ (0, 1), so u+ ∈ C01 (). On the other hand, it turns out from (5.3) and hypothesis H(g) (iii) that a constant c˜λ > 0 can be found such that |g(x, s)| ≤ c˜λ s p−1 for a.a. x ∈  and all 0 ≤ s ≤ aλ e ∞ .

(5.9)

Then (5.5) and (5.9) show p−1

p u+ ≤ (λ + c˜λ )u+ .

(5.10)

Therefore Vázquez’s strong maximum principle [229] can be applied. Specifically, one takes in its statement the function β(s) = (λ + c˜λ )s p−1 for all s > 0. It is possible because  1 ds = +∞. 1 + 0 (sβ(s)) p

Constant-Sign and Sign-Changing Solutions for Elliptic Problems Chapter | 5

We are thus conducted that if u = 0, then u > 0 in  and means according to (1.39) that

∂u ∂ν

197

< 0 on ∂, which

u+ ∈ int (C01 ()+ ).

(5.11)

Arguing by contradiction, we suppose that (5.11) does not hold. In view of the above discussion, we must have u+ = 0. Then (5.4) reads as un (x) ↓ 0 for all x ∈ .

(5.12)

Setting u˜ n =

un for all n,

∇un p

we may suppose, along a relabeled subsequence, that 1,p

u˜ n  u˜ in W0 () , u˜ n → u˜ in Lp () and a.e. in ,

(5.13)

1,p

with some u˜ ∈ W0 (), and there is a function w ∈ Lp ()+ such that |u˜ n (x)| ≤ w(x) for almost all x ∈ . Notice that (5.7) can be expressed as   g(x, un ) p−1 p−1 −p u˜ n , ϕ = λ u˜ n ϕ dx − u˜ n ϕ dx p−1   un

(5.14)

1,p

for all ϕ ∈ W0 (). (5.15)

In particular, for ϕ = u˜ n − u˜ we obtain   g(x, un ) p−1 p−1 ˜ = λ u˜ n (u˜ n − u) ˜ dx − u˜ n (u˜ n − u) ˜ dx. −p u˜ n , u˜ n − u p−1   un (5.16) We see from (5.9) and (5.14) that |g(x, un (x))| p−1 u˜ n |u˜ n (x) − u(x)| ˜ ≤ c˜λ w(x)p−1 (w(x) + |u(x)|) ˜ for a.a. x ∈ . un (x)p−1 Owing to the fact that the right-hand side of the above inequality is in L1 (), on the basis of (5.13) we can apply the Lebesgue’s dominated convergence theorem to get  g(x, un ) p−1 lim u˜ n (u˜ n − u) ˜ dx = 0. n→∞  up−1 n

198 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Then (5.16) and (5.13) give lim −p u˜ n , u˜ n − u ˜ = 0,

n→∞

1,p

which in view of the (S)+ -property of −p on W0 () as stated in Proposition 1.56 results in 1,p

u˜ n → u˜ in W0 () as n → ∞,

(5.17)

so in particular ∇ u ˜ p = 1. By (5.15), (5.17), (5.12), and hypothesis H(g) (ii), we infer  1,p ˜ ϕ = λ u˜ p−1 ϕ dx for all ϕ ∈ W0 (). (5.18) −p u, 

1,p

In (5.18) it is thus expressed that u˜ ≥ 0 is an eigenfunction of −p on W0 () corresponding to the eigenvalue λ > λ1 . This is impossible according to Proposition 1.57 because u˜ must change sign. The reached contradiction proves that the claim in (5.11) holds true. We now show u+ is the smallest positive solution of (5.1) within [0, u].

(5.19)

1,p

Let u ∈ W0 () be a positive solution of (5.1) belonging to the ordered interval [0, u]. Since u ∈ L∞ (), then (5.1) and hypothesis H(g) (ii) imply p u ∈ L∞ (). The nonlinear regularity theory allows us to derive u ∈ C01 (). Then (5.10) enables us to apply Vázquez’s strong maximum principle [229], which leads to u ∈ int (C01 ()+ ). We deduce u ∈ [ n1 uˆ 1 , u] for n sufficiently large. This in conjunction with the fact that un is the smallest solution of (5.1) in [ n1 uˆ 1 , u] ensures un ≤ u if n is large enough. Making use of (5.4), we obtain u+ ≤ u, thereby (5.19) is valid. The proof of the existence of the biggest negative solution of (5.1) within the ordered interval [−aλ e, 0] can be carried out in a similar way. This completes the proof. The main result of this section as stated below provides an additional nontrivial solution to problem (5.4). The novelty is that it is nodal, which means that it changes sign in . By λ2 we denote the second eigenvalue of −p on p W01 (). Theorem 5.5. Under hypotheses H(g) (i)–(iii), for every λ > λ2 problem (5.4) admits at least a (positive) solution u+ = u+ (λ) ∈ int (C01 ()+ ), a (negative) solution u− = u− (λ) ∈ −int (C01 ()+ ), and a nontrivial sign-changing solution u0 = u0 (λ) ∈ C01 ().

Constant-Sign and Sign-Changing Solutions for Elliptic Problems Chapter | 5

199

Proof. Fix any λ > λ2 . As λ2 > λ1 , Theorem 5.4 ensures the existence of the extremal solutions u+ ∈ int (C01 ()+ ) and u− ∈ −int (C01 ()+ ). We introduce on  × R the truncation functions ⎧ ⎪ if s ≤ 0, ⎨ 0 τ+ (x, s) = s if 0 < s < u+ (x), ⎪ ⎩ u+ (x) if s ≥ u+ (x), ⎧ ⎪ ⎨ u− (x) if s ≤ u− (x), τ− (x, s) = s if u− (x) < s < 0, ⎪ ⎩ 0 if s ≥ 0, ⎧ ⎪ ⎨ u− (x) if s ≤ u− (x), τ0 (x, s) = s if u− (x) < s < u+ (x), ⎪ ⎩ u+ (x) if s ≥ u+ (x). 1,p

By means of them, the corresponding functionals E+ , E− , E0 ∈ C 1 (W0 ()) 1,p are defined as follows for every u ∈ W0 ():   u(x) 1 p (λτ+ (x, s)p−1 − g(x, τ+ (x, s)) ds dx,

∇u p − p  0 1 p E− (u) = ∇u p p   u(x) − (λ|τ− (x, s)|p−2 τ− (x, s) − g(x, τ− (x, s))) ds dx, E+ (u) =

 0

1 p E0 (u) = ∇u p p   u(x) − (λ|τ0 (x, s)|p−2 τ0 (x, s) − g(x, τ0 (x, s))) ds dx.  0

The following location property is valid: v is a critical point of E+ =⇒ 0 ≤ v(x) ≤ u+ (x) for a.a. x ∈ .

(5.20)

1,p

Indeed, if v ∈ W0 () is a critical point of E+ , then −p v + p u+ , (v − u+ )+  

 = λτ+ (x, v)p−1 − g(x, τ+ (x, v)) − λτ+ (x, u+ )p−1 + g(x, τ+ (x, u+ )) 

× (v − u+ )+ dx = 0,

200 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints 1,p

which implies v ≤ u+ . In the same way, if v ∈ W0 () is a critical point of E+ , we have 

 −p v, −v −  = − λτ+ (x, v)p−1 − g(x, τ+ (x, v)) v − dx = 0, 

so v ≥ 0. Therefore (5.20) is verified. Since the functional E+ is coercive and sequentially weakly lower semicon1,p tinuous, there exists a global minimizer v+ ∈ W0 () of E+ , that is, E+ (v+ ) =

inf

1,p

E+ .

(5.21)

W0 ()

We note from (5.20) and (5.21) that v+ is a solution of problem (5.1) and belongs to the order interval [0, u+ ]. Moreover, v+ is nontrivial which can be seen as follows. By (5.11) we can choose t > 0 sufficiently small to have t uˆ 1 ≤ u+ in  and t uˆ 1 ∞ ≤ δλ . Then (5.3) ensures that E+ (t uˆ 1 ) =

λ1 p t − p

 

t uˆ 1 (x)

(λs p−1 − g(x, s)) ds dx < 0,

 0

and in view of (5.21) we get v+ = 0. We have that v+ ∈ L∞ () and, because v+ solves problem (5.1), we derive p v+ ∈ L∞ (). This allows the nonlinear regularity theory be invoked, guaranteeing that v+ ∈ C01 (). In addition, taking p−1 into account (5.9), the estimate p v+ ≤ (λ + c˜λ )v+ ensues (see also (5.10)). Then Vázquez’s strong maximum principle [229] implies v+ ∈ int (C01 ()+ ).

(5.22)

Using (5.22) and the inequality v+ ≤ u+ in , we see that the minimality property of u+ established in Theorem 5.4 forces v+ = u+ . By (5.21) we deduce that u+ is a local minimizer of E0 on C01 (). In view of [156, Theorem 23] we may 1,p conclude that u+ is a local minimizer of E0 on the space W0 (). Proceeding in the same way with the functional E− , we show that u− is a local minimizer 1,p of E0 on W0 (). As for (5.20) we can verify v is a critical point of E0 =⇒ u− (x) ≤ v(x) ≤ u+ (x) for a.a. x ∈ . (5.23) It is readily seen that (5.23) ensures that every critical point of E0 is a solution to problem (5.1). Since the functional E0 is coercive, sequentially weakly lower semicontinuous, with inf

1,p

W0 ()

ϕ0 < 0,

Constant-Sign and Sign-Changing Solutions for Elliptic Problems Chapter | 5

201

1,p

it is clear that E0 has a global minimizer v0 ∈ W0 () satisfying v0 = 0. Then, as already observed, (5.23) implies that v0 is a nontrivial solution of problem (5.1) belonging to the ordered interval [u− , u+ ]. We may admit that v0 = u+ or v0 = u− because otherwise, due to the extremal properties of solutions u+ and u− as given by Theorem 5.4, v0 would change sign and the result be achieved setting u0 = v0 . Without loss of generality, suppose v0 = u+ . We may also assume that u− is a strict local minimizer of E0 because otherwise we would find (infinitely many) critical points of E0 that are sign changing thanks to (5.23) and the extremal properties of the solutions u− , u+ obtained in Theorem 5.4. Then there exists ρ ∈ (0, u0 − v0 ) such that E0 (u+ ) ≤ E0 (u− ) < inf{E0 (u) : u ∈ ∂Bρ (u− )},

(5.24)

1,p

where ∂Bρ (u− ) = {u ∈ W0 () : u − u− = ρ}. Owing to (5.24) and noticing that the Palais–Smale condition holds for the functional E0 , we may apply the mountain pass theorem (see Theorem 1.44) which provides a point 1,p u0 ∈ W0 () satisfying E0 (u0 ) = 0 and inf{E0 (u) : u ∈ ∂Bρ (u− )} ≤ E0 (u0 ) = inf max E0 (γ (t)), γ ∈ t∈[−1,1]

(5.25)

where 1,p

 = {γ ∈ C([−1, 1], W0 ()) : γ (−1) = u− , γ (1) = u+ }. From (5.24) and (5.25) it follows that u0 = u− and u0 = u+ . We claim E0 (u0 ) < 0.

(5.26)

To this end it suffices to construct a path γˆ ∈  such that E0 (γˆ (t)) < 0 for all t ∈ [−1, 1]. 1,p

Lp ()

(5.27)

Lp ()

, where ∂B1 = {u ∈ Lp () : u p = 1}, and Let S = W0 () ∩ ∂B1 1,p let SC = S ∩ C01 () be endowed with the topologies induced by W0 () and 1 C0 (), respectively. For later use we set 0,C = {γ ∈ C([−1, 1], SC ) : γ (−1) = −uˆ 1 , γ (1) = uˆ 1 }. Assumption H(g) (ii) provides constants μ ∈ (0, λ − λ2 ) and δ > 0 such that |g(x, s)| ≤ μ for a.a. x ∈  and all 0 < |s| ≤ δ. |s|p−1

(5.28)

202 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Then we choose ρ0 ∈ (0, λ − λ2 − μ). From Proposition 1.58 we know that there exists a path γ ∈ 0 such that p

max ∇γ (t) p < λ2 +

t∈[−1,1]

1

ρ0 . 2 1

Fix a number r with 0 < r ≤ (λ2 + ρ0 ) p − (λ2 + ρ20 ) p . The density of SC in S implies that 0,C is dense in 0 , so there is γ0 ∈ 0,C satisfying max ∇γ (t) − ∇γ0 (t) p < r ,

t∈[−1,1]

thus it turns out that p

max ∇γ0 (t) p < λ2 + ρ0 .

t∈[−1,1]

(5.29)

The boundedness of the set γ0 ([−1, 1])() in R ensures the existence of an ε1 > 0 such that ε1 |u(x)| ≤ δ for all x ∈  and all u ∈ γ0 ([−1, 1]).

(5.30)

Since u+ , −u− ∈ int (C01 ()+ ) as obtained in Theorem 5.4, for every u ∈ γ0 ([−1, 1]) and any bounded neighborhood Vu of u in C01 () we can find positive numbers hu and ju satisfying 1 1 u+ − v ∈ int (C01 ()+ ) and − u− + v ∈ int (C01 ()+ ) h j whenever h ≥ hu , j ≥ ju , v ∈ Vu . Then the compactness of γ0 ([−1, 1]) in C01 () allows us to determine a number ε0 > 0 for which one has u− (x) ≤ εu(x) ≤ u+ (x) for all x ∈ , u ∈ γ0 ([−1, 1]), ε ∈ (0, ε0 ).

(5.31)

We note that εγ0 is a continuous path in C01 () joining −ε uˆ 1 and ε uˆ 1 for each 0 < ε < min{ε0 , ε1 }. By (5.31), (5.29), (5.30), (5.28), and taking into acLp () , we obtain count the choice of ρ0 , as well as that γ0 ([−1, 1]) ⊂ ∂B1   εγ0 (t)(x) εp εp p g(x, τ0 (x, s)) ds dx

∇γ0 (t) p − λ + p p  0   εγ0 (t)(x) εp g(x, s) ds dx (5.32) ≤ (λ2 + ρ0 − λ) + p  0 εp (λ2 + ρ0 − λ + μ) < 0 for all t ∈ [−1, 1]. ≤ p

E0 (εγ0 (t)) =

Constant-Sign and Sign-Changing Solutions for Elliptic Problems Chapter | 5

203

We next apply the second deformation lemma (see Theorem 1.45) to the C 1 1,p functional E+ : W0 () → R. To this end we denote c+ = c+ (λ) = E+ (ε uˆ 1 ), m+ = m+ (λ) = E+ (u+ ), c

1,p

E++ = {u ∈ W0 () : E+ (u) ≤ c+ }. The minimality property of u+ enables us to admit that m+ < c+ . Again using the minimality of u+ in conjunction with (5.20), we see that E+ has no critical values in the interval (m+ , c+ ]. Since the functional E+ satisfies the Palais– Smale condition due to its coercivity, the second deformation lemma can be apc c plied to E+ , yielding a continuous mapping η ∈ C([0, 1] × E++ , E++ ) such that c+ η(0, u) = u and η(1, u) = u+ for all u ∈ E+ , as well as E+ (η(t, u)) ≤ E+ (u) c whenever t ∈ [0, 1] and u ∈ E++ . We are thus led to introduce the path γ+ : 1,p [0, 1] → W0 () by γ+ (t) = η(t, ε uˆ 1 )+ = max{η(t, ε uˆ 1 ), 0} for all t ∈ [0, 1]. 1,p Clearly, γ+ is continuous on W0 () and joins ε uˆ 1 and u+ . Moreover, the properties of the deformation η and (5.32) imply E0 (γ+ (t)) = E+ (γ+ (t)) ≤ E+ (ε uˆ 1 ) < 0 for all t ∈ [0, 1].

(5.33)

Now, applying the second deformation lemma to the functional E− , we con1,p struct a continuous path γ− : [0, 1] → W0 () between −ε uˆ 1 and u− such that E0 (γ− (t)) < 0 for all t ∈ [0, 1] .

(5.34)

By the concatenation of the curves γ− , εγ0 , and γ+ we obtain a path γˆ ∈ . On the basis of (5.34), (5.32), and (5.33), γˆ satisfies (5.27). Then, as noticed before, (5.26) holds, so u0 = 0. Recalling (5.23), we deduce that u0 is a nontrivial solution of (5.1) distinct from u− and u+ , with u− ≤ u0 ≤ u+ , and verifying u0 ∈ C01 () by the nonlinear regularity theory. The extremal properties of the constant sign solutions u− and u+ as described in Theorem 5.4 render u0 to be sign changing. This completes the proof.

5.2 THE CASE OF NEUMANN BOUNDARY CONDITION Let  ⊂ RN be a bounded domain with a C 2 boundary ∂. Here we study the following quasilinear elliptic problem with Neumann boundary condition  −p u = λ|u|p−2 u − f (x, u) in , (5.35) ∂u on ∂, ∂ν = 0 where λ is a positive parameter, ν stands for the outward unit normal to ∂, ∂u ∂u ∂ν denotes the normal derivative of u on ∂, i.e., ∂ν = ∇u · ν, p u := div (|∇u|p−2 ∇u) is the p-Laplacian operator with 1 < p < +∞, and f :

204 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

 × R → R is a Carathéodory function (i.e., measurable in x ∈  for all s ∈ R and continuous in s ∈ R for a.e. x ∈ ). The statement in (5.35) is the Neumann counterpart of the Dirichlet problem (5.1). By a weak solution of problem (5.35) we mean any function u ∈ W 1,p () such that  |∇u|p−2 ∇u · ∇v dx (5.36)    = λ |u|p−2 uv dx − f (x, u)v dx for all v ∈ W 1,p (). 



Here are the notions of subsolution and supersolution for problem (5.35). Definition 5.6. The function u ∈ W 1,p () ∩ L∞ () is called a subsolution for problem (5.35) and the function u ∈ W 1,p () ∩ L∞ () is called a supersolution for problem (5.35) if 

 |∇u|p−2 ∇u · ∇v − λ|u|p−2 uv + f (x, u)v dx ≤ 0 

and



 |∇u|p−2 ∇u · ∇v − λ|u|p−2 uv + f (x, u)v dx ≥ 0



for all v ∈ W 1,p () with v ≥ 0 a.e. in . In the analysis of problem (5.35), we use the Banach space C 1 () endowed with the ordered cone

C+ = u ∈ C 1 () : u(x) ≥ 0 for all x ∈  whose interior is nonempty and given by int C+ = {u ∈ C+ : u(x) > 0 for all x ∈ }. We are first going to prove the existence of two opposite constant-sign solutions u+ ∈ int C+ and u− ∈ −int C+ for problem (5.35). Given λ > 0, let us assume that the Carathéodory function f :  × R → R satisfies: H(f ) (i) there is a positive constant k such that for almost all x ∈ , one has min{f (x, k), −f (x, −k)} ≥ λk p−1 ; (ii) there are two real numbers c, α, with c ≤ α < λ such that the following inequalities hold uniformly for a.e. x ∈ : c ≤ lim inf s→0

f (x, s) f (x, s) ≤ lim sup p−2 ≤ α; p−2 |s| s s s→0 |s|

(iii) f is bounded on bounded sets.

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Remark 5.7. From (ii) it follows that f (x, 0) = 0 a.e. in , so u ≡ 0 is solution to (5.35). For the sake of simplicity, in this section, the usual norm of the space W 1,p () as given in Definition 1.14 is denoted by · . Theorem 5.8. Assume that hypothesis H(f ) holds. Then problem (5.35) admits two solutions u+ ∈ int C+ and u− ∈ −int C+ . Proof. Hypothesis H(f ) (i) entails −p (k) − λk p−1 + f (x, k) ≥ 0,

(5.37)

whereas hypothesis H(f ) (ii) guarantees that there exists δ ∈ (0, k) such that f (x, s) < λs p−1

(5.38)

for a.a. x ∈  and for all s ∈ (0, δ). Then for any ε ∈ (0, δ), we obtain from (5.38) that −p (ε) − λεp−1 + f (x, ε) < 0

(5.39)

for a.a. x ∈ . Therefore, by (5.37) and (5.39), we note that u = ε and u = k are a subsolution and a supersolution for problem (5.35), respectively. We introduce the following truncation of the identity on R: ⎧ ⎪ ⎨ k if s ≥ k, τ (s) = s if ε < s < k, ⎪ ⎩ ε if s ≤ ε, and by means of it, the nonlinear operator H : W 1,p () → W 1,p ()∗ by 

 |∇u|p−2 ∇u · ∇v + |u|p−2 uv dx H (u), v =    f (x, τ (u))v dx − (λ + 1) τ p−1 (u)v dx + 



for all u, v ∈ W 1,p (). We have      p−1 p−1 τ (u)v dx ≤ k |v| dx.   

(5.40)



From hypothesis H(f ) (iii) there is a constant L ≥ 0 such that sup (x,s)∈×R

|f (x, τ (s))| =

sup (x,t)∈×[0,k]

|f (x, t)| ≤ L.

(5.41)

206 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Through (5.40) and (5.41), we are able to show the boundedness and coercivity of the operator H . In order to show that H is also pseudomonotone, let {un } be a sequence in W 1,p () such that un  u in W 1,p () and lim supH (un ), un − u ≤ 0.

(5.42)

n→∞

Then, along a relabeled subsequence, un → u and τ (un ) → τ (u) in Lp () and a.e. in . Using the boundedness of τ , by the Lebesgue dominated convergence theorem, we obtain for every w ∈ W 1,p () that   τ p−1 (u)(u − w) dx = lim τ p−1 (un )(un − w) dx n→∞ 



and



 f (x, τ (u))(u − w) dx = lim

n→∞ 



f (x, τ (un ))(u − w) dx.

At this point, (5.42) and Proposition 1.56 enable us to conclude that H is a pseudomonotone operator. Theorem 1.55 provides some u ∈ W 1,p () such that H (u) = 0, which reads as 

  p−2 p−2 ∇u · ∇v + |u| uv dx − (λ + 1) τ p−1 (u)v dx |∇u|    (5.43) + f (x, τ (u))v dx = 0 for all v ∈ W 1,p (). 

We claim that ε ≤ u(x) ≤ k for a.e. x ∈ . Acting on (5.43) with v = (u − k)+ gives   

k p−1 (u − k) dx |∇u|p + up−1 (u − k) dx − (λ + 1) {u≥k} {u>k}  + f (x, k)(u − k) dx = 0. {u>k}

From H(f ) (i) we know that f (x, k) − λk p−1 ≥ 0 a.e. in , which results in 

 up−1 − k p−1 (u − k) dx ≤ 0 , 0≤ {u>k}

so u ≤ k a.e. in . Next, acting on (5.43) with v = (ε − u)+ leads to   |∇u|p dx + |u|p−2 u(ε − u) dx − {u≤ε} {u 0} > 0. The following minimax characterization of λ2 is given in [16] (see also Corollary 3.11). Proposition 5.11. Let S = {u ∈ W 1,p () : u p = 1} be endowed with the topology induced by W 1,p (). Then p

γ (t) p , λ2 = inf max ∇ γ ∈   −1≤t≤1

  where  = γ  ∈ C([−1, 1], S) :  γ (−1) = −uˆ 1 ,  γ (1) = uˆ 1 . In order to develop a variational approach, we set  s f (x, t) dt for all (x, s) ∈  × R. F (x, s) = 0

Under hypotheses H(f ), F :  × R → R is a Carathéodory function such that F (·, 0) = 0, the partial derivative ∂F ∂s of F (x, s) exists and is a Carathéodory ∂F function satisfying ∂s (x, s) = f (x, s) for almost all x ∈ . Using the extremal constant-sign solutions u+ and u− constructed in Theorem 5.10, we introduce the following functions on  × R:

210 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

⎧ ⎪ ⎨ 0 τ+ (x, s) = s ⎪ ⎩ v+ (x) ⎧ ⎪ ⎨ v− (x) τ− (x, s) = s ⎪ ⎩ 0 ⎧ ⎪ ⎨ v− (x) if τ0 (x, s) = s if ⎪ ⎩ v+ (x) if

if s ≤ 0, if 0 < s < v+ (x), if s ≥ v+ (x), if s ≤ v− (x), if v− (x) < s < 0, if s ≥ 0, s ≤ v− (x), v− (x) < s < v+ (x), s ≥ v+ (x),

f+ (x, s) = f (x, τ+ (x, s)), f− (x, s) = f (x, τ− (x, s)), f0 (x, s) = f (x, τ0 (x, s)). Corresponding to these functions, we define  s  s F+ (x, s) = f+ (x, t) dt, F− (x, s) = f− (x, t) dt, 0 0  s f0 (x, t) dt. F0 (x, s) = 0

The energy functionals related to the three truncated problems obtained from (5.35) are:   u(x)  1 p−1 E+ (u) := u p − (λ + 1) τ+ (x, s) ds dx + F+ (x, u(x)) dx, p  0    u(x)  1 p−1 p τ− (x, s) ds dx + F− (x, u(x)) dx, E− (u) = u − (λ + 1) p  0    u(x)  1 p−1 τ0 (x, s) ds dx + F0 (x, u)) dx. E0 (u) = u p − (λ + 1) p  0  The functionals E± , E0 are C 1 on W 1,p (). We are in a position to state the main result of this section, which is based on requiring λ > λ2 . Theorem 5.12. Assume λ > λ2 and that hypothesis H(f ) holds with α < λ−λ2 . Then problem (5.35) has at least three nontrivial solutions: the smallest positive solution v+ ∈ int C+ , the biggest negative solution v− ∈ −int C+ , and a nodal solution u0 ∈ C 1 (), with v− ≤ u0 ≤ v+ . Proof. Theorem 5.10 provides the smallest positive solution v+ ∈ int C+ and the biggest negative solution v− ∈ −int C+ of problem (5.35). We claim that v+ and v− are global minimizers for E+ and E− , respectively, as well as local minimizers of E0 .

Constant-Sign and Sign-Changing Solutions for Elliptic Problems Chapter | 5

211

To this end, we note that any critical point v ∈ W 1,p () of E+ satisfies 0 ≤ v(x) ≤ v+ (x), for a.a. x ∈ .

(5.46)

 (v) = 0 with w = (v − v )+ yields Indeed, acting on E+ +  

|∇v|p−2 ∇v · ∇(v − v+ ) + v p−1 (v − v+ ) dx {v≥v+ }   (v+ )p−1 (v − v+ ) dx + f (x, v+ )(v − v+ ) dx = 0. −(λ + 1) {v≥v+ }

{v≥v+ }

The fact that v+ is a solution for (5.35) ensures   |∇v+ |p−2 ∇v+ · ∇(v − v+ ) dx − λ (v+ )p−1 (v − v+ ) dx {v≥v+ } {v≥v+ }  f (x, v+ )(v − v+ ) dx = 0. + {v≥v+ }

Subtracting the second equality from the first one results in 

 |∇v|p−2 ∇v − |∇v+ |p−2 ∇v+ · ∇(v − v+ ) dx {v≥v+ } 

 + v p−1 − (v+ )p−1 (v − v+ ) dx = 0, {v≥v+ }

which implies that v(x) ≤ v+ (x) for a.a. x ∈ .  (v) = 0 with w = −v − has as effect On the other hand, testing E+  

 p p−1 f+ (x, v)v − dx = 0. |∇v| + |v| dx − {v≤0}

{v≤0}

Since f+ (x, s) = 0 whenever s ≤ 0, we infer that v − = 0 and so the validity of (5.46) is proven. By hypothesis H(f ) (ii), for each ε > 0 there exists δ > 0 such that F (x, s) ≤

α+ε p s for a.a. x ∈ , all s ∈ (0, δ). p

Selecting 0 < δ < min v+ , one has that F+ (x, s) = F (x, s) for all s ∈ (0, δ). 

Then for t ∈ (0, δ), by virtue of α < λ − λ2 we see that E+ (t) ≤ (−λ + α + ε)

tp tp || < (−λ2 + ε) ||. p p

If ε < λ2 , we get E+ (t) < 0. The functional E+ has a global minimizer z ∈ W 1,p () because it is coercive and sequentially weakly lower semicontinuous. According to what was

212 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

said before, it holds E+ (z) < 0, so z = 0. Owing to (5.46), it follows that 0 ≤ z(x) ≤ v+ (x) for a.e. x ∈ . Therefore z is a solution for Eq. (5.35). In view of the nonlinear regularity theory and strong maximum principle, we have that z ∈ int C+ . Due to the minimality property of v+ , actually we obtain that z = v+ . Since E+ and E0 coincide on a C 1 ()-neighborhood of v+ , we see that v+ is a local C 1 ()-minimizer of E0 , hence it is also a local minimizer of E0 in W 1,p () (see [156, Proposition 24]). Proceeding in a similar way with E− in place of E+ , we obtain the corresponding result for v− . Arguing exactly as for showing (5.46) we can establish that any critical point v of E0 satisfies v− (x) ≤ v(x) ≤ v+ (x), for a.a. x ∈ .

(5.47)

We may assume that v+ and v− are strict local minimizers of E0 because otherwise handling (5.47) and the extremal properties of v+ and v− we find infinitely many sign-changing solutions for problem (5.35). Since E0 is coercive, it satisfies the Palais–Smale condition. Therefore we can apply the mountain pass theorem (see, e.g., [207], [169, Theorem 5.40] or Theorem 1.44) to E0 to obtain a critical point u0 of E0 satisfying max{E0 (v+ ), E0 (v− )} < E0 (u0 ) = c := inf max E0 (γ (t)), γ ∈ −1≤t≤1

with

(5.48)



 = γ ∈ C([−1, 1], W 1,p ()) : γ (−1) = v− , γ (1) = v+ .

We claim that E0 (u0 ) < 0

(5.49)

To this end, we construct a path γ ∈  along which E0 is negative. Towards this, we make use of the characterization of the second eigenvalue of the negative p-Laplacian given in Proposition 5.11. In addition to S endowed with the topology induced by W 1,p () (see Proposition 5.11), we consider SC = S ∩ C 1 () endowed with the topology induced by C 1 (). Then, besides    =  γ ∈ C([−1, 1], S) :  γ (−1) = −uˆ 1 ,  γ (1) = uˆ 1 , we also take    C = γ  ∈ C([−1, 1], SC ) :  γ (−1) = −uˆ 1 ,  γ (1) = uˆ 1 .  . Given any μ ∈ (0, λ − It is worthwhile to point out that  C is dense in  λ2 − α), hypothesis H(f ) (ii) yields δ > 0 such that F (x, s)
0 defined above there is an ε > 0 such that, for all x ∈ , v− (x) < εu(x) < v+ (x) and ε|u(x)| < δ

for all u = γ0 (t) ∈ γ0 ([−1, 1]). (5.52)

On the basis of (5.50), (5.51), and (5.52), we infer that  εp p F (x, εu(x)) dx E0 (εγ0 (t)) = ( ∇u p − λ) + p  εp ≤ (λ2 + κ − λ + α + μ) < 0 for all t ∈ [−1, 1]. p

(5.53)

Notice that E+ (v+ ) =: m+ < c+ := E+ (ε uˆ 1 ). Due to the minimality property of v+ and (5.46), there are no critical values of E+ in the interval (m+ , c+ ]. As a consequence we can apply the second deformation lemma (see, e.g., [169, Theorem 5.34] or Theorem 1.45) to the functional E+ . We obtain a continuous c c c −1 mapping η : [0, 1] × E++ → E++ , where E++ = E+ ((−∞, c+ ]), such that η(0, u) = u, η(1, u) = v+ , c

E+ (η(t, u)) ≤ E+ (u) for all t ∈ [0, 1], all u ∈ E++ . We observe that the continuous path γ+ (t) = η(t, ε uˆ 1 )+ for all t ∈ [0, 1] joins ε uˆ 1 and v+ , and for all t ∈ [0, 1], E0 (γ+ (t)) = E+ (γ+ (t)) = E+ (η(t, ε uˆ 1 )) −

η(t, ε uˆ 1 )− p ≤ E+ (ε uˆ 1 ) < 0. p (5.54)

For the functional E− , we can similarly construct a continuous path γ− joining v− and −ε uˆ 1 such that E0 (γ− (t)) < 0 for all t ∈ [0, 1].

(5.55)

Concatenating γ− , εγ0 and γ+ produces a path γ belonging to . Taking into account (5.53), (5.54), and (5.55), we derive that E0 (γ (t)) < 0 for all t ∈ [0, 1]. Hence (5.49) holds true. It turns out that u0 is a nontrivial critical point of E0

214 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

and by (5.47) that u0 solves (5.46). The L∞ -boundedness of u0 , together with H(f ) (iii), enables us to apply [127, Theorem 4.9] to deduce that u0 ∈ C 1 (). It remains only to show that u0 must change sign. Arguing indirectly, suppose that u0 is of constant sign, say u0 ≥ 0. By (5.47) we have that u0 = v+ , which comes in contradiction with the fact that E0 (v+ ) < E0 (u0 ) as known from (5.48). Through the same argument we can show that u0 ≤ 0 cannot happen. This amounts to saying that u0 changes sign, which completes the proof.

5.3 PROBLEMS WITH MULTIVALUED TERMS This section deals with the following inclusion problem involving the Dirichlet boundary condition:  −p u ∈ λ∂j (x, u) in , (5.56) u=0 on ∂. Here  ⊂ RN (N ≥ 3) is a bounded domain with a C 2 boundary ∂, 1 < p < +∞, λ > 0 is a parameter and p denotes the p-Laplacian operator

 1,p for all u ∈ W0 (), p u := div |∇u|p−2 ∇u that is, (5.2) holds true. As usual, p ∗ will denote the Sobolev critical exponent. The multivalued reaction term ∂j (x, s) is the generalized gradient of a nonsmooth potential s → j (x, s), which is subjected to the following conditions: H(j ) j :  × R → R is a Carathéodory function for which there exist constants a1 > 0, 1 < q < p ∗ , 0 < a2 ≤ a3 such that: (i) j (x, ·) is locally Lipschitz for a.a. x ∈  and j (·, 0) ∈ L1 (); q−1 (ii) |ξ | ≤ a1 1 + |s| for a.e. x ∈  and for all s ∈ R, ξ ∈ ∂j (x, s); j (x, s) (iii) lim sup ≤ 0 uniformly for a.e. x ∈ ; p |s|→+∞ |s| min ∂j (x, s) max ∂j (x, s) (iv) a2 ≤ lim inf ≤ lim sup ≤ a3 uniformly for p−1 s s p−1 s→0+ s→0+ a.e. x ∈ ; max ∂j (x, s) min ∂j (x, s) ≤ lim sup ≤ a3 uniformly for (v) a2 ≤ lim inf p−2 − |s| s |s|p−2 s s→0 s→0− a.e. x ∈ . The functions (x, s) → min ∂j (x, s), max ∂j (x, s) are well defined because ∂j (x, s) is a compact interval. Moreover, they are measurable. 1,p The functional space that fits problem (5.56) is the Sobolev space W0 () equipped with the norm u := ∇u p (see Remark 1.18). In addition, we uti-

Constant-Sign and Sign-Changing Solutions for Elliptic Problems Chapter | 5

215

lize C01 (), which turns out to be an ordered Banach space with the order cone

C01 ()+ := u ∈ C01 () : u(x) ≥ 0 for all x ∈  , whose interior is nonempty and characterized in (1.39) as int (C01 ()+ )   ∂u (x) < 0, ∀x ∈ ∂ , = u ∈ C01 ()+ : u(x) > 0, ∀x ∈  and ∂ν where, as usual, ν(x) denotes the outward unit normal to ∂ at any point x. 1,p Given λ > 0, we say that u ∈ W0 () is a (weak) solution of problem (5.56) if p u ∈ Lζ () for some ζ > 1 and −p u(x) ∈ λ∂j (x, u(x)) a.e. in . From H(j ) (iii) it follows that (5.56) admits the zero solution for all λ > 0. We are interested in finding nontrivial solutions. Now we can state our main result on problem (5.56). Hereafter by λ1 and λ2 1,p we denote the first two eigenvalues of −p on W0 () (see Propositions 1.57 and 1.58). Theorem 5.13. Assume that hypotheses H(j ) hold. Then, for every λ > λ2 /a2 + 1, problem (5.56) possesses at least three nontrivial solutions, the smallest positive, the biggest negative, and a nodal one, lying in C01 (). We provide a simple example for which Theorem 5.13 applies. For simplicity, we drop the dependence of j (x, s) with respect to x ∈ . Example 5.14. The nonsmooth potential j : R → R defined for every s ∈ R by  log(1 + |s|)|s|p−1 if |s| ≤ 1, j (s) = log(2)|s|q if |s| > 1, for 1 < q < p, is locally Lipschitz and satisfies hypotheses H(j ) with a1 = log(2) and a2 = a3 = 1. Theorem 5.13 is going to be proven through a nonsmooth variational ap1,p proach. The nonsmooth energy functional ϕλ : W0 () → R associated to the inclusion (5.56) is defined by 

u p 1,p − λ j (x, u) dx for all u ∈ W0 (). ϕλ (u) = p  1,p

Proposition 5.15. The functional ϕλ : W0 () → R is locally Lipschitz. 1,p Moreover, if u ∈ W0 () is a critical point of ϕλ , then u ∈ C01 () and u solves (5.56).

216 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Proof. The function u → 1,p W0 ()

u p 1 p is of class C whose differential  W −1,p (). Aubin–Clarke’s theorem

→ cian −p : Theorem 2.181]) ensures that the function  u → j (x, u) dx

is the p-Lapla(see, e.g., [38,



is Lipschitz continuous on any bounded subset of Lq () and its generalized gradient is included in the set

 N (u) := w ∈ Lq () : w(x) ∈ ∂j (x, u(x)) for a.e. x ∈  , (5.57) where 1 < q < p ∗ and q1 + q1 = 1. Since, according to Theorem 1.20, W0 () is continuously embedded in Lq (), the functional ϕλ turns out to be locally 1,p Lipschitz on W0 (). We have 1,p

∂ϕλ (u) ⊂ −p (u) − λN(u). So, if 0 ∈ ∂ϕλ (u), then −1,p 

−p u = λw in W0

(5.58)

() 

with some w ∈ N (u). Hence, we obtain that p u ∈ Lq () and u solves (5.56). Combining H(j ) (ii) with (5.58) yields the estimate −up u ≤ a1 (|u| + |u|q ) for a.e. x ∈ , which on account of [95, Theorem 1.5.5] implies u ∈ L∞ (). From H(j ) (ii) and (5.58) it follows that p u ∈ L∞ (). So, by the nonlinear regularity theory, the function u belongs to C01 (). We complement Proposition 5.15 with a strong maximum principle-type result for problem (5.56). Proposition 5.16. If u ∈ C01 ()+ is a solution of problem (5.56) satisfying u = 0, then u ∈ int (C01 ()+ ). Proof. Fix 0 < θ < a2 . Through H(j ) (iii) one can find δ > 0 such that min ∂j (x, s) ≥ θ s p−1 for a.e. x ∈  and all 0 < s < δ. Since in (5.58) we now have w ∈ L∞ (), there exists a constant c > 0 such that p u ≤ cup−1 a.e. in . Vázquez’s maximum principle [229, Theorem 5] directly gives u ∈ int (C01 ()+ ).

Constant-Sign and Sign-Changing Solutions for Elliptic Problems Chapter | 5

217

Definition 5.17. A function u ∈ W 1,p () is called a subsolution of problem (5.56) provided u ≤ 0 on ∂ and there exists w ∈ N (u) (see (5.57) with some 1 < q < p ∗ ) such that  1,p −p u, v ≤ λ wv dx for all v ∈ W0 () with v(x) ≥ 0 a.e. in . 

The definition of a supersolution is obtained by reversing the inequalities involving u. The following comparison principle for differential inclusions in [41, Corollary 4.1] will be employed later on. Proposition 5.18. If u, u ∈ W 1,p () are a subsolution and a supersolution, respectively, of problem (5.56) and u(x) ≤ u(x) for a.a. x ∈ , then the set

1,p H := u ∈ W0 () : u solves (5.56) and u(x) ≤ u(x) ≤ u(x) for a.e. x ∈  is nonempty, compact, and downward and upward directed with respect to the pointwise order. Moreover, H has the smallest and the biggest element. We proceed by obtaining a minimal positive solution. Theorem 5.19. If λ > λ1 /a2 then the inclusion problem (5.56) possesses the smallest positive solution uˆ λ ∈ int (C01 ()+ ). Proof. Fix λ > λ1 /a2 . We split the proof in several steps. Claim 1. Problem (5.56) has a positive solution uˆ ∈ int (C01 ()+ ). Consider the truncation j+ :  × R → R defined by j+ (x, s) = j (x, s + )

for all (x, s) ∈  × R,

where, as before, s + := max{0, s}. Its generalized gradient with respect to s is equal to ⎧ ⎪ if s < 0, ⎨ {0} ∂j+ (x, s) = {τ ξ : ξ ∈ ∂j (x, 0), τ ∈ [0, 1]} if s = 0, ⎪ ⎩ ∂j (x, s) if s > 0. The related functional ϕλ+ : W0 () → R given by 

u p 1,p ϕλ+ (u) = − λ j+ (x, u) dx for all u ∈ W0 () p  1,p

1,p

is weakly sequentially lower semicontinuous and locally Lipschitz on W0 (). We are going to prove that ϕλ+ is coercive. Let 0 < ε < λ1 /(pλ). By H(j ) (iii)

218 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

there exists M > 0 such that j+ (x, s) < εs p for a.a. x ∈  and all s > M, while H(j ) (ii) and Proposition 1.40 provide j+ (x, s) ≤ j (x, 0) + a1 (M + M q ) for almost every x ∈  and every 0 ≤ s ≤ M. Using the variational characterization of λ1 in Proposition 1.57, we obtain  

u p p j (x, 0) + a1 (M + M q ) dx − λε u p −λ ϕλ+ (u) ≥ p {u≤M}   1 ελ ≥ −

u p − M  , p λ1 with a constant M  > 0. Since p > 1, the choice of ε allows us to conclude that ϕλ+ is coercive. Now the Weierstrass theorem (see, e.g., [152, Theorem 1.1]) yields uˆ ∈ 1, p W0 () fulfilling ˆ = ϕλ+ (u)

inf ,

1 p

u∈W0

()

ϕλ+ (u).

(5.59)

As in the proof of Proposition 5.15, we see that uˆ ∈ C01 () and there exists  wˆ ∈ Lq () with the properties w(x) ˆ ∈ ∂j+ (x, u(x)) ˆ

for a.e. x ∈ 

and 

−p uˆ = λwˆ in W −1,p ().

(5.60)

Acting on (5.60) with the test function uˆ − leads to   − p − |∇ uˆ | dx = λ wˆ uˆ − dx = 0 



because w(x) ˆ = 0 whenever u(x) ˆ < 0, thus uˆ ∈ C01 ()+ . Next we prove that uˆ = 0. Let us observe that due to our hypothesis we can choose θ < a2 with λ > λ1 /θ . By assumption H(j ) (iii) there exists δ > 0 such that min ∂j+ (x, s) > θ s p−1 for a.e. x ∈  and every 0 < s < δ. Proposition 1.40 implies j+ (x, s) ≥ j (x, 0) +

θ sp . p

Constant-Sign and Sign-Changing Solutions for Elliptic Problems Chapter | 5

219

Since the function uˆ 1 given by Proposition 1.57 is bounded, we can find τ > 0 such that τ uˆ 1 (x) < δ for all x ∈ , which enables us to write    p θ τ p uˆ 1 τ p uˆ 1 p + ϕλ (τ uˆ 1 ) ≤ −λ dx j (x, 0) + p p  ≤

τp (λ1 − λθ ) + ϕλ+ (0) < ϕλ+ (0). p

On account of (5.59) this renders uˆ = 0. Hence, we are in a position to invoke Proposition 5.16, from which we infer that uˆ ∈ int (C01 ()+ ). In view of H(j ) (i), we derive that ∂j+ (x, s) ⊂ ∂j (x, s) for a.e. x ∈  and all s > 0, so uˆ solves (5.56). Claim 2. Problem (5.56) possesses the smallest positive solution uˆ λ ∈ int (C01 ()+ ). Let ε0 > 0 be such that ε uˆ 1 < uˆ in  for all ε ∈ (0, ε0 ), which exists because uˆ ∈ int (C01 ()+ ). Thanks to H(j ) (iii) and the inequality λ > λ1 /a2 , one has for ε > 0 sufficiently small that p−1

−p (ε uˆ 1 ) = λ1 ε p−1 uˆ 1

≤ λ min ∂j+ (x, εuˆ 1 ) a.e. in .

Therefore, according to Definition 5.17, ε uˆ 1 turns out to be a subsolution of problem (5.56). Since uˆ is a solution of (5.56), it can be regarded as a supersolution. Proposition 5.18 guarantees that the ordered set 1,p Hε := u ∈ W0 () : ε uˆ 1 (x) ≤ u(x) ≤ u(x) ˆ for a.e. x ∈ , and

u solves (5.56) is nonempty and contains a solution of (5.56) which is the smallest one with respect to the pointwise order. If, for n large enough, we denote by un , with n large enough, the smallest solution of (5.56) lying in H1/n then 

−p un = λwn in W −1,p ()

(5.61)

with some wn ∈ N (un ) (see (5.57)). By the minimality property of un , we get un ≥ un+1 . So, there exists uˆ λ :  → (0, +∞) such that lim un (x) = uˆ λ (x) pointwise in .

n→∞

1,p

(5.62)

The sequence {un } is bounded in W0 () because (5.61), H(j ) (ii), and Proposition 1.40 entail    q

un p = −p un , un  = λ wn un dx ≤ λa1 un + un dx     ≤ λa1 uˆ + uˆ q dx. 

220 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

From (5.62), passing to a relabeled subsequence if necessary, it follows that 1,p 1,p uˆ λ ∈ W0 (), un  uˆ λ in W0 (), as well as un → uˆ λ in Lq (). Again through (5.61), H(j ) (ii), and Proposition 1.40 we achieve  −p un , un − uˆ λ  = λ wn (un − uˆ λ ) dx   q−1 ≤ λa1 (1 + un )(un − uˆ λ ) dx ≤ M un − uˆ λ q , 

with some M > 0. This implies lim sup−p un , un − uˆ λ  ≤ 0. n→∞

1,p

Since −p has the (S)+ -property (see Proposition 1.56), un → uˆ λ in W0 (). By assumption H(j ) (ii), the sequence {wn } turns out to be bounded  in W −1,p (). Thus, possibly along a relabeled subsequence, wn  wˆ λ in  −1,p W (). At this point, Proposition 1.39, in conjunction with (5.61), ensures that wˆ λ ∈ N (uˆ λ ) and 

−p uˆ λ = λwˆ λ in W −1,p (),

(5.63)

that is, uˆ λ solves problem (5.56). The reasoning developed in the proof of Proposition 5.15 demonstrates that uˆ λ ∈ C01 ()+ . Next we verify that uˆ λ = 0. Arguing by contradiction, suppose uˆ λ = 0. Notice that vn := un / un is a weak solution of the Dirichlet problem ⎧ ⎪ wn (x) p−1 ⎨ v in , −p v = λ (5.64) un (x)p−1 ⎪ ⎩ v=0 on ∂. The weight function in (5.64) belongs to L∞ (). Indeed, choosing θ ∈ (λ1 /λ, a2 ), then (5.62) with uˆ λ = 0 and H(j ) (iii) show for any sufficiently large n that wn (x) min ∂j (x, un (x)) ≥ >θ p−1 un (x) un (x)p−1 and wn (x) max ∂j (x, un (x)) ≤ < 2a3 p−1 un (x) un (x)p−1 a.e. in . Through (5.65) and the last part of Proposition 1.57 we obtain

w  λ1 n λ1 p−1 < λ1 (θ ) = < λ, θ un

(5.65)

Constant-Sign and Sign-Changing Solutions for Elliptic Problems Chapter | 5

221

which expresses that λ > 0 is an eigenvalue of the weighted problem (5.64) greater than the first one. Hence, by Proposition 1.57, the corresponding eigenfunction vn should be nodal. However, this comes in contradiction with the definition of vn , whence uˆ λ = 0. On the basis of Proposition 5.16, it turns out that uˆ λ ∈ int (C01 ()+ ). In order to prove that uˆ λ is the smallest among the positive solutions 1,p of (5.56), let u ∈ W0 () be a positive solution of (5.56). Applying again Proposition 5.16, we know that u ∈ int (C01 ()+ ), whence uˆ 1 (x) ≤ u(x) n

for all x ∈ 

provided n is large enough. The function n1 uˆ 1 is a subsolution and min{u(x), u(x)} ˆ is a supersolution of (5.56) (see Definition 5.17). Consequently, by Propo1,p sition 5.18 there exists a solution v ∈ W0 () of (5.56) such that uˆ 1 (x) ≤ v(x) ≤ min{u(x), u(x)} ˆ for a.e. x ∈ . n Then the minimality of un in H1/n entails un (x) ≤ v(x) ≤ u(x) for a.e. x ∈ . Letting n → ∞ provides uˆ λ ≤ u, which completes the proof. The proof of the below result parallels the proof of Theorem 5.19. Theorem 5.20. If λ > λ1 /a2 then problem (5.56) possesses the biggest negative solution uˇ λ ∈ −int (C01 ()+ ). Fix λ > λ1 /a2 and consider the extremal solutions uˆ λ and uˇ λ obtained in Theorems 5.19 and 5.20, respectively. Now we deal with the existence of solutions to problem (5.56) that lie in the interval [uˇ λ , uˆ λ ], which will eventually lead to a nodal solution. To this end, we introduce a truncation-perturbation of the nonsmooth potential j (x, s) as follows: ⎧ ⎪ ⎨j (x, uˇ λ (x)) + max ∂j (x, uˇ λ (x))(s − uˇ λ (x)) if s < uˇ λ (x), j˜(x, s) := j (x, s) if uˇ λ (x) ≤ s ≤ uˆ λ (x), ⎪ ⎩ j (x, uˆ λ (x)) + min ∂j (x, uˆ λ (x))(s − uˆ λ (x)) if s > uˆ λ (x). It is straightforward to show that the function j˜ :  × R → R satisfies the Carathéodory condition with j˜(x, ·) locally Lipschitz, and hypotheses H(j ) hold true for j˜ in place of j . Hence Proposition 5.15 guarantees that the functional 1,p ϕ˜λ : W0 () → R defined by 

u p 1,p − λ j˜(x, u) dx for all u ∈ W0 () ϕ˜ λ (u) = p  is locally Lipschitz. Moreover, the below result is valid.

222 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Proposition 5.21. If u is a critical point of ϕ˜ λ , then u ∈ C01 (), u solves (5.56) and has the location property uˇ λ ≤ u ≤ uˆ λ in . Proof. According to Definition 1.41, we have 0 ∈ ∂ ϕ˜λ (u). As in the proof of Proposition 5.15, we get that u ∈ C01 () and there exists w ∈ N˜ (u), where

 N˜ (u) := w ∈ Lq () : w(x) ∈ ∂ j˜(x, u(x)) for a.e. x ∈  , such that 

−p u = λw in W −1,p ().

(5.66)

Using (5.66), (5.63) with the test function (u − uˆ λ )+ ∈ W0 () and the definition of the potential j˜ yield  −p u + p uˆ λ , (u − uˆ λ )+  = λ (w − wˆ λ )(u − uˆ λ )+ dx   min ∂j (x, uˆ λ ) − wˆ λ (u − uˆ λ )+ dx ≤ 0. =λ 1,p



This is true because   ∂ j˜(x, s) = min ∂j (x, uˆ λ (x)) for a.e. x ∈  and all s > uˆ λ (x), whereas wˆ λ ∈ N (uˆ λ ). The strict monotonicity of −p implies u ≤ uˆ λ . A similar argument provides u ≥ uˇ λ , thus uˇ ≤ u ≤ uˆ in . Taking into account that ∂ j˜(x, uˆ λ (x)) ⊂ ∂j (x, uˆ λ (x))

and

∂ j˜(x, uˇ λ (x)) ⊂ ∂j (x, uˇ λ (x)),

we arrive at w ∈ N (u). Therefore, by (5.66), u solves (5.56). The next result deals with the existence of a sign-changing solution, which in addition satisfies an important location property. Theorem 5.22. If λ > λ2 /a2 +1, then problem (5.56) possesses a nodal solution u˜ λ ∈ C01 () such that uˇ λ (x) ≤ u˜ λ (x) ≤ uˆ λ (x) for all x ∈ . Proof. Since λ2 /a2 + 1 > λ1 /a2 , Theorems 5.19 and 5.20 provide the smallest positive solution uˆ λ ∈ int (C01 ()+ ) and the biggest negative solution uˇ λ ∈ −int (C01 ()+ ) of (5.56). Set j˜+ (x, s) := j˜(x, s + ). The functional ϕ˜λ+ : 1,p W0 () → R given by ϕ˜λ+ (u) :=

u p −λ p

 

j˜+ (x, u) dx

1,p

for all u ∈ W0 ()

Constant-Sign and Sign-Changing Solutions for Elliptic Problems Chapter | 5

223

turns out to be locally Lipschitz, weakly sequentially lower semicontinuous, and 1,p coercive. So, as in the proof of Theorem 5.19, there exists u˜ ∈ W0 () fulfilling ˜ = ϕ˜ λ+ (u)

inf

1,p

u∈W0 ()

ϕ˜λ+ (u) < ϕ˜λ+ (0).

(5.67)

We claim that u˜ = uˆ λ . By Proposition 5.16 we know that u˜ ∈ int (C01 ()+ ). Let us note that the restrictions to C01 ()+ of the functionals ϕ˜λ and ϕ˜λ+ coincide. Hence, u˜ is a C01 ()-local minimizer for the functional ϕ˜λ . Thanks 1,p ˜ to [175], u˜ is also a W0 ()-local minimizer for ϕ˜λ , whence 0 ∈ ∂ ϕ˜λ (u). Through Proposition 5.16 we get 0 < u(x) ˜ ≤ uˆ λ (x) for all x ∈ . The minimality of uˆ λ forces u˜ = uˆ λ , and the claim is verified. Accordingly, (5.67) reads as ϕ˜λ (uˆ λ ) =

inf ,

1 p

u∈W0

()

ϕ˜λ+ (u) < ϕ˜ λ (0).

(5.68)

Observe that uˆ λ turns out to be a local minimizer of ϕ˜λ because uˆ λ ∈ int (C01 ()+ ). The same holds true for uˇ λ ∈ −int (C01 ()+ ), with ϕ˜λ (uˇ λ ) < ϕ˜λ (0).

(5.69)

We examine two situations that may occur. Case 1. The functional ϕ˜λ has not a strict local minimum at uˆ λ or uˇ λ . Conse1,p quently, there exists another local minimizer u˜ λ ∈ W0 () \ {uˆ λ , uˇ λ } satisfying ϕ˜λ (u˜ λ ) = ϕ˜λ (uˆ λ )

or

ϕ˜λ (u˜ λ ) = ϕ˜λ (uˇ λ ).

Proposition 5.16 ensures that u˜ λ ∈ C01 (), u˜ λ solves (5.56), and uˇ λ (x) ≤ u˜ λ (x) ≤ uˆ λ (x)

for all x ∈ .

(5.70)

From (5.68) or (5.69) it clearly follows u˜ λ = 0. Let us show that the solution u˜ λ is nodal. On the contrary, without loss of generality suppose that u˜ λ ∈ C01 ()+ , then thanks to Proposition 5.16, u˜ λ ∈ int (C01 ()+ ). In view of (5.70) and the minimality of uˆ λ , this is impossible. Case 2. The functional ϕ˜ λ has a strict local minimum at both uˆ λ and uˇ λ . In this situation we can find r ∈ (0, uˆ λ − uˇ λ ) such that max{ϕ˜λ (uˆ λ ), ϕ˜λ (uˇ λ )} < σr :=

inf

u∈∂Br (uˆ λ )

ϕ˜ λ (u).

(5.71)

224 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

As in the proof of Theorem 5.19 we can show that ϕ˜ λ is coercive. In order to prove that the nonsmooth condition (P S) in Definition 1.42 is fulfilled, let a 1,p sequence {un } in W0 () be such that {ϕ˜ λ (un )} is bounded and 1,p

(ϕ˜λ )0 (un ; v − un ) + εn v − un ≥ 0 for all v ∈ W0 (), all n ∈ N, (5.72) with εn → 0+ . By the coercivity of ϕ˜λ , the sequence {un } must be bounded. So, 1,p passing to a subsequence if necessary, we may assume that un  u in W0 () and un → u in Lq (). Through (5.72) and hypothesis H(j ) (ii) one has  0 ≤ −p un , u − un  + λ a1 (1 + |un |q−1 )|u − un | dx + εn u − un 

≤ −p un , u − un  + λM u − un q + εn u − un , with a constant M > 0. This amounts to saying that lim sup−p un , un − u ≤ 0. n→∞

1,p

Since, by Proposition 1.56, −p has the (S)+ -property, un → u in W0 (), proving the nonsmooth condition (P S) in Definition 1.42, as desired. We are in a position to apply Theorem 1.44. Specifically, setting

1, p  := γ ∈ C([−1, 1], W0 ()) : γ (−1) = uˇ λ , γ (1) = uˆ λ and c˜ := inf max ϕ˜λ (γ (t)), γ ∈ t∈[−1,1] 1,p

there exists a critical point u˜ λ ∈ W0 () of ϕ˜λ such that ϕ˜λ (u˜ λ ) = c˜ ≥ σr . By (5.71) this makes clear that u˜ λ = uˆ λ and u˜ λ = uˇ λ . Proposition 5.21 ensures that u˜ λ ∈ C01 (), u˜ λ solves (5.56), and uˇ λ (x) ≤ u˜ λ (x) ≤ uˆ λ (x) for all x ∈ . The main point for proving that the solution u˜ λ is nodal consists in the following. Claim. u˜ λ = 0. Choose θ ∈ (0, a2 ) such that λ > λ2 /θ + 1. Proposition 1.58 provides a path γ0 ∈ 0 satisfying

γ0 (t) p < λ2 + θ

for all t ∈ [−1, 1].

(5.73)

Constant-Sign and Sign-Changing Solutions for Elliptic Problems Chapter | 5

225

Without loss of generality, we may suppose γ0 ∈ C([−1, 1], C01 ()) because 1,p S ∩ C01 () is dense in S (endowed with the induced W0 ()-topology) while γ0 ([−1, 1]) is compact. Through H(j ) (i), (iv), (v), and Proposition 1.40, we find δ > 0 such that j (x, s) > j (x, 0) +

θ |s|p for a.e. x ∈  and every |s| ≤ δ. p

(5.74)

Finally, since γ0 ([−1,1]) is compact in C01 () while uˆ λ ,− uˇ λ ∈ int(C01 ()+ ), there exists ε > 0 such that ε|u(x)| < min{δ, uˆ λ (x), −uˇ λ (x)}

for all x ∈ , all u ∈ γ0 ([−1, 1]). (5.75)

Gathering (5.73)–(5.75) and recalling that u p = 1 results in  εp p ϕ˜λ (εu) = u − λ j˜(x, εu) dx p     θ ε p |u|p εp dx j (x, 0) + ≤ (λ2 + θ ) − λ p p  εp ≤ (λ2 + θ − λθ ) + ϕ˜ λ (0) for all u ∈ γ0 ([−1, 1]). p By the choice of λ this entails ϕ˜ λ (εγ0 (t)) < ϕ˜λ (0)

for all t ∈ [−1, 1].

(5.76)

Let us now apply the deformation assertion in Theorem 1.45 to the functional ϕ˜λ+ posing a := ϕ˜λ+ (uˆ λ ) and b := ϕ˜λ+ (ε uˆ 1 ). As in the proof of Theorem 5.19 we show a < b < ϕ˜λ+ (0). One has Kc (ϕ˜λ+ ) ⊂ {uˆ λ } for any c < ϕ˜λ+ (0), where Kc (ϕ˜λ+ ) stands for the set of critical points of ϕ˜λ+ with critical value c. Indeed, if u ∈ Kc (ϕ˜λ+ ) \ {uˆ λ }, then thanks to Propositions 5.16 and 5.21 we have that u ∈ int (C01 ()+ ), u is a positive solution of (5.56), u ≤ uˆ λ , and u = uˆ λ , against the minimality of uˆ λ . Consequently, Ka (ϕ˜λ+ ) = {uˆ λ }, while Kc (ϕ˜λ+ ) = ∅ for each c ∈ (a, b]. Due to Theorem 1.45, there exists a continuous mapping h : [0, 1] × Z → Z, where

1,p Z := u ∈ W0 () : ϕ˜λ+ (u) ≤ b , with the properties h(0, u) = u, h(1, u) = uˆ λ for all u ∈ Z, and ϕ˜λ+ (h(t, u)) ≤ ϕ˜λ+ (u)

for all (t, u) ∈ [0, 1] × Z.

226 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints 1,p

Then the path γ+ : [0, 1] → W0 () defined by γ+ (t) := (h(t, ε uˆ 1 ))+

for all t ∈ [0, 1],

satisfies: γ+ (0) = ε uˆ 1 , γ+ (1) = uˆ λ , ϕ˜ λ = ϕ˜λ+ on the set γ+ ([0, 1]), and ϕ˜ λ (γ+ (t)) < ϕ˜λ (0)

for all t ∈ [0, 1].

(5.77) 1,p

Likewise, we construct a continuous path γ− : [0, 1] → W0 () such that γ− (0) = uˇ λ , γ− (1) = −ε uˆ 1 , and ϕ˜ λ (γ− (t)) < ϕ˜λ (0)

for all t ∈ [0, 1].

(5.78)

Concatenating γ− , εγ0 , and γ+ determines a path γ ∈ . In view of (5.76)–(5.78), γ fulfills ϕ˜λ (γ (t)) < ϕ˜λ (0)

for all t ∈ [−1, 1].

It turns out that c˜ < ϕ˜λ (0) and, thus the claim u˜ λ = 0 ensues. Finally, we prove that u˜ λ is nodal. If not, u˜ λ would be of constant sign. In the case where u˜ λ ∈ C01 ()+ , by Proposition 5.16 we deduce that u˜ λ ∈ int (C01 ()+ ), 0 < u˜ λ ≤ uˆ λ . Then the minimality of uˆ λ ensures u˜ λ = uˆ λ , which is impossible according / −C01 ()+ . Consequently, to (5.71). A similar reasoning prevents to have u˜ λ ∈ u˜ λ must be nodal, which completes the proof.

5.4 NOTES Section 5.1 treats multiple solutions with sign information for a quasilinear elliptic problem governed by the p-Laplacian operator with 1 < p < +∞. There are many results studying the existence of multiple solutions for boundary value problems involving nonlinear elliptic equations with or without parameters. A large category of them concerns problems driven by the p-Laplacian or (p, q)-Laplacian operators. In these results, one imposes on the involved nonlinearities convenient assumptions describing the behavior at infinity and at zero. A prototype is represented by the classical work of Ambrosetti, Brézis, and Cerami [9], which investigates the semilinear case (i.e., the case p = 2) with a positive parameter λ > 0 and the concave–convex nonlinearities s → λ|s|q−2 s + |s|r−2 s with 1 < q < 2 < r < 2∗ . The result has been extended by Ambrosetti, García Azorero, and Peral [10] to the case of the p-Laplacian with 1 < p < +∞.

Constant-Sign and Sign-Changing Solutions for Elliptic Problems Chapter | 5

227

Among the papers devoted to the study of multiple solutions, a distinct class is formed by those which seek solutions with prescribed qualitative properties. For instance, a challenging goal is to obtain solutions of required sign. In this respect, we mention that, for quasilinear elliptic equations with Dirichlet boundary condition, the existence of multiple solutions with precise sign information is studied in [40,42,49,156]. Section 5.1 addresses exactly this topic. It is based on [40] showing the existence of multiple solutions to problem (5.1) for all 1,p values λ > λ2 , where λ2 denotes the second eigenvalue of −p on W0 (), and guaranteeing that for any such λ there exist at least three nontrivial solutions so that two are of opposite constant sign whereas the third is nodal (i.e., sign changing). This is the content of Theorem 5.5. In particular, it extends the main result in [11] obtained for p = 2 and, a fortiori, the corresponding results from [12] and [220], by dropping certain requirements on the nonlinearity g in problem (5.1). The most significant feature of Theorem 5.5 is that it sets forth the significant new information of the existence of a sign-changing solution of problem (5.1). The topic of searching a sign-changing solution between extremal constant-sign solutions originates in the work of Carl–Perera [49], where the study is related to the Fuˇcík spectrum of the negative p-Laplacian on the 1,p space W0 () by exploiting subsolution–supersolution techniques to get both extremal constant-sign and nodal solutions for a problem without parameters. It is also worth to point out that the approach here is completely different from the one in [11] where the Morse theory and Lyapunov–Schmidt reduction are used. Specifically, in the case of Theorem 5.5 the approach relies on the existence of extremal constant-sign solutions whose existence is presented in Theorem 5.4. Other main tools are the method of subsolutions–supersolutions, truncation, strong maximum principle and variational arguments. For the use of these techniques we also refer to results ensuring the existence of three nontrivial solutions in the works of Carl–Motreanu [43], Jin [114], and Motreanu– Motreanu–Papageorgiou [166] provided the relevant positive parameter λ is sufficiently small. We also refer to the monograph [200] for quasilinear elliptic problems involving p-Laplacian type operators. The investigation of multiple solutions for nonlinear elliptic problems with a more general differential operator, possibly nonhomogeneous, in place of the p-Laplacian has been recently considered in Miyajima–Motreanu–Tanaka [156], Motreanu– Tanaka [180–182]. We also mention that the nonlinear elliptic problems with nonhomogeneous Dirichlet boundary condition were treated through a nonsmooth variational approach in Goldshtein–Motreanu–Motreanu [106]. In the case of nonlinear Neumann problems, there are many results ensuring the existence of multiple solutions (see [2,23,28,156,168,209]), but few results involving the study of the sign of multiple solutions. We refer to [178,179,232, 240] for the special case of jumping nonlinearities. The existence of multiple solutions for problem (5.35) with a complete study of their sign was done in [22], on which is based the presentation in Section 5.2. The main result is stated as Theorem 5.12, which provides two

228 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

opposite constant-sign solutions and besides a sign-changing solution between them. Our approach relies on variational methods combined with subsolution– supersolution and truncation techniques. We emphasize that we do not impose any growth condition for the nonlinearity f (x, u) in problem (5.35). The reason is that we are dealing with truncated functionals for which the subcritical growth condition is automatically valid. There are essential differences between the results and tools used for the Dirichlet and Neumann boundary value problems as can be seen from Theorems 5.5 and 5.12. The technical difficulties are overcome thanks to specific contributions on Neumann problems involving the p-Laplacian (see [2,23]). Section 5.3 studies a differential inclusion problem of elliptic type driven by the p-Laplacian operator with a set-valued reaction term in the form of the generalized gradient of a locally Lipschitz function (see Section 1.2). Such problems cover discontinuous single-valued nonlinearities (see Chang [51]). The presence of the set-valued reaction term ∂j (x, s) requires completely different ideas in order to handle truncations and verify the appropriate Palais–Smale condition. As required in hypotheses H(j ) (iii), (iv), the locally Lipschitz potential j (x, ·) is p-sublinear at infinity and p-superlinear at zero. It is worthwhile pointing out that in this multivalued setting, the variational methods and subsolutions–supersolutions can still be employed to achieve multiplicity results. The comparison principle for differential inclusions in Proposition 5.18 is taken from Carl–Motreanu [41, Corollary 4.1] (see also Carl–Le–Motreanu [38, Corollary 4.24]). The main result is formulated as Theorem 5.13, which extends to a nonsmooth inclusion setting the results obtained in the C 1 case in [144]. More precisely, for problem (5.56) depending on a parameter λ > 0 it is proven the existence of a smallest positive, a biggest negative, and a nodal (i.e., signchanging) solution provided λ > 0 is large enough. By minimizing a truncated energy functional we first find a positive solution of (5.56). Subsolution– supersolution arguments then provide a smallest positive solution uˆ λ ∈ C01 (). Analogously, a biggest negative solution uˇ λ is obtained. A third solution u˜ λ ∈ C01 () is next obtained through the nonsmooth mountain pass-like theorem while the nonsmooth second deformation lemma ensures that it is not zero. Finally, due to the extremality of the constant-sign solutions uˆ λ and uˇ λ , the solution u˜ λ must be nodal. Results of the same nature in a nonsmooth setting are given in Averna–Marano–Motreanu [18], Carl–Motreanu [41], Iannizzotto– Papageorgiou [113]. An open problem is to find a second sign-changing solution for the nonsmooth problem (5.56). In the smooth case, a result of this type can be found in [165].

Chapter 6

Nonlinear Elliptic Systems 6.1 ELLIPTIC SYSTEMS FULLY DEPENDING ON THE GRADIENT OF THE SOLUTION In this section we deal with the following homogeneous Dirichlet boundary value problem formed by a system of nonlinear elliptic equations: ⎧ ⎪ ⎨ −p1 u1 − μ1 q1 u1 = f1 (x, u1 , u2 , ∇u1 , ∇u2 ) in , −p2 u2 − μ2 q2 u2 = f2 (x, u1 , u2 , ∇u1 , ∇u2 ) in , ⎪ ⎩ on ∂. u1 = u2 = 0

(6.1)

Here  ⊂ RN is a bounded domain with the boundary ∂, μ1 , μ2 ≥ 0 are parameters, pi and qi are the pi -Laplacian and qi -Laplacian, respectively, with 1 < qi < pi < +∞, and fi :  × R × R × RN × RN → R is a Carathéodory function, which means that x → f (x, s1 , s2 , ξ1 , ξ2 ) is measurable for all (s1 , s2 , ξ1 , ξ2 ) ∈ R × R × RN × RN and (s1 , s2 , ξ1 , ξ2 ) → f (x, s1 , s2 , ξ1 , ξ2 ) is continuous for a.a. x ∈ , i = 1, 2. We recall that the 1,p pi -Laplacian is the operator pi : W0 i () → W −1,pi () given by pi u = div(|∇u|pi −2 ∇u). For simplicity we suppose that N > max{p1 , p2 }, so the Sobolev critical Np1 Np2 exponents corresponding to p1 and p2 are p1∗ = N−p and p2∗ = N−p , respec1 2 tively. In the case where N ≤ max{p1 , p2 } the treatment is simpler, and thus it is omitted. Accordingly, for every r ∈ [1, +∞] we denote by r its Hölder conjugate, i.e., r is determined by 1r + r1 = 1. In the sequel we denote by λ1,pi the first eigenvalue of the negative pi -Laplacian (i = 1, 2). According to Proposition 1.57, it has the following variational characterization p

λ1,pi =

inf

∇u pii

pi 1,p u∈W0 i ()\{0} u pi

,

hence i u pii ≤ λ−1 1,pi ∇u pi

p

p

1,pi

for all u ∈ W0

().

(6.2)

Nonlinear Differential Problems with Smooth and Nonsmooth Constraints. DOI: 10.1016/B978-0-12-813386-6.00006-7 229 Copyright © 2018 Elsevier Inc. All rights reserved.

230 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

We consider the following conditions on the functions f1 and f2 . H(f1 , f2 )1 There exist constants ai ≥ 0, bi ≥ 0, αi ∈ [0, pi∗ − 1), βi ∈ [0, (pp∗i ) ) i



and functions σi ∈ Lγi (), with γi ∈ [1, pi∗ ), such that |f1 (x, s1 , s2 , ξ1 , ξ2 )| ≤ σ1 (x) + a1 (|s1 |

α1

+ b1 (|ξ1 |β1 + |ξ2 | |f2 (x, s1 , s2 , ξ1 , ξ2 )| ≤ σ2 (x) + a2 (|s1 | + b2 (|ξ1 |

β2 p1 p2

+ |s2 | β1 p2 p1

α2 p1∗ p2∗

α1 p2∗ p1∗

)

),

+ |s2 |α2 )

+ |ξ2 |β2 )

for a.a. x ∈  and all s1 , s2 ∈ R. H(f1 , f2 )2 The following unilateral growth condition holds: f1 (x, s1 , s2 , ξ1 , ξ2 )s1 + f2 (x, s1 , s2 , ξ1 , ξ2 )s2 ≤ ω(x) + c(|s1 |p1 + |s2 |p2 ) + d(|ξ1 |p1 + |ξ2 |p2 ) for a.a. x ∈  and all s1 , s2 ∈ R, ξ1 , ξ2 ∈ RN , withω ∈ L1 (), and posi−1 tive constants c and d satisfying max λ−1 1,p1 , λ1,p2 c + d < 1. The solutions of system (6.1) are understood in the weak sense. Precisely, 1,p 1,p a weak solution of problem (6.1) is a pair (u1 , u2 ) ∈ W0 1 () × W0 2 () such that   |∇u1 |p1 −2 ∇u1 · ∇v1 dx + μ1 |∇u1 |q1 −2 ∇u1 · ∇v1 dx   = f1 (x, u1 , u2 , ∇u1 , ∇u2 )v1 dx, (6.3)   |∇u2 |p2 −2 ∇u2 · ∇v2 dx + μ2 |∇u2 |q2 −2 ∇u2 · ∇v2 dx    f2 (x, u1 , u2 , ∇u1 , ∇u2 )v2 dx (6.4) =  1,p

1,p

for all (v1 , v2 ) ∈ W0 1 () × W0 2 (). It is easy to check that, assuming H(f1 , f2 )1 , the integrals in (6.3) and (6.4) exist. We also consider the related Dirichlet problems driven by the pi -Laplacian operators (i = 1, 2): ⎧ ⎪ ⎨ −p1 u1 = f1 (x, u1 , u2 , ∇u1 , ∇u2 ) in , (6.5) −p2 u2 = f2 (x, u1 , u2 , ∇u1 , ∇u2 ) in , ⎪ ⎩ u1 = u2 = 0 on ∂,

Nonlinear Elliptic Systems Chapter | 6



−p1 u = f1 (x, u, 0, ∇u, 0) u=0

231

in , on ∂,

(6.6)

−p2 u = f2 (x, 0, u, 0, ∇u) in , u=0 on ∂.

(6.7)

We can state the following existence theorem for solutions of problem (6.1). 1,p 1,p We set W = W0 1 () × W0 2 (). Theorem 6.1. Assume that the Carathéodory functions fi :  × R × R × RN × RN → R (i = 1, 2) satisfy hypotheses H(f1 , f2 )1 and H(f1 , f2 )2 . Then problem (6.1) has a solution for all μ1 , μ2 ≥ 0. Proof. Fix μ1 , μ2 ≥ 0. We associate to problem (6.1) the nonlinear operator A : W → W ∗ = W −1,p1 () × W −1,p2 () defined by A(u1 , u2 ) = (−p1 u1 − μ1 q1 u1 − N1 (u1 , u2 ), −p2 u2 − μ2 q2 u2 − N2 (u1 , u2 )),

(6.8)



where Ni : W → W −1,pi () denotes the Nemytskii operator determined by fi , that is, Ni (u1 , u2 ) = fi (·, u1 (·), u2 (·), ∇u1 (·), ∇u2 (·)). It is straightforward to verify that due to hypothesis H(f1 , f2 )1 the operator A is well defined. We first prove that the operator A in (6.8) is pseudomonotone. To this end, consider a sequence {(u1,n , u2,n )} ⊂ W weakly converging to (u1 , u2 ) in W with lim sup A(u1,n , u2,n ), (u1,n − u1 , u2,n − u2 ) ≤ 0.

(6.9)

n→∞

We claim that for i = 1, 2 we have  fi (x, u1,n , u2,n , ∇u1,n , ∇u2,n )(ui,n − ui ) dx = 0. lim n→∞ 

(6.10)

In order to show (6.10), on the basis of the growth condition in hypothesis H(f1 , f2 )1 it suffices to note that  





 

|σi ||ui,n − ui | dx ≤ σi γi ui,n − ui γi → 0,

|ui,n |αi |ui,n − ui | dx ≤ ui,n αpi∗ ui,n − ui i

pi∗ pi∗ −αi

|∇ui,n |βi |ui,n − ui | dx ≤ ∇ui,n βpii ui,n − ui

→ 0,

pi pi −βi

→ 0,

232 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

 |uj,n |

αi pj∗ pi∗

|ui,n − ui | dx ≤ uj,n



 |∇uj,n |

αi pj∗ pi∗ pj∗

ui,n − ui

βi pj p

βi pj pi

pi∗ pi∗ −αi

|ui,n − ui | dx ≤ ∇uj,n pj i ui,n − ui



→ 0, i = j,

pi pi −βi

→ 0, i = j.

These convergence relations are consequences of Hölder’s inequality and Rellich–Kondrachov compactness embedding theorem (see Theorem 1.20). Therefore, in view of hypothesis H(f1 , f2 )1 , (6.10) holds true. Inserting (6.10) in (6.9) leads to

lim sup −p1 u1,n − μ1 q1 u1,n , u1,n − u1  n→∞ (6.11)  + −p2 u2,n − μ2 q2 u2,n , u2,n − u2  ≤ 0. We claim that lim sup −pi ui,n − μi qi ui,n , ui,n − ui  ≤ 0, i = 1, 2.

(6.12)

n→∞

If not, we may admit that along relabeled subsequences one has lim −p1 u1,n − μ1 q1 u1,n , u1,n − u1  > 0,

n→∞

lim −p2 u2,n − μ2 q2 u2,n , u2,n − u2  < 0.

n→∞

(6.13)

The second inequality in (6.13) enables us to utilize the (S)+ -property of the 1,p map −p2 − μ2 q2 on W0 2 () (see [169, p. 39], also Proposition 1.56) to in1,p fer the strong convergence u2,n → u2 in W0 2 (). Then a contradiction arises between (6.11) and (6.13), thereby (6.12) holds true. Then on the basis of (6.12) and invoking again the (S)+ -property of the 1,p map −pi − μi qi on W0 i (), we are able to infer the strong convergence 1,pi ui,n → ui in W0 () for i = 1, 2. Once this strong convergence is achieved, it is a straightforward matter to prove that for each (v1 , v2 ) ∈ W we have lim A(u1,n , u2,n ), (u1,n − v1 , u2,n − v2 ) = A(u1 , u2 ), (u1 − v1 , u2 − v2 ).

n→∞

The operator A is thus pseudomonotone. Notice that the operator A is bounded, in the sense that it maps bounded sets into bounded sets. This is because the operator −pi − μi qi is bounded on 1,p W0 i () for i = 1, 2, and assumption H(f1 , f2 )1 is supposed to be valid. The next step in the proof is to show that the operator A in (6.8) is coercive, which means lim

u →+∞

Au, u = +∞, u

(6.14)

Nonlinear Elliptic Systems Chapter | 6

233

where for all u = (u1 , u2 ) ∈ W we set u = ∇u1 p1 + ∇u2 p2 . Indeed, arguing by means of hypothesis H(f1 , f2 )2 gives p

p

q

q

A(u1 , u2 ), (u1 , u2 ) = ∇u1 p11 + ∇u2 p22 + μ1 ∇u1 q11 + μ2 ∇u2 q22   − f1 (x, u1 , u2 , ∇u1 , ∇u2 )u1 dx − f2 (x, u1 , u2 , ∇u1 , ∇u2 )u2 dx    

p1 p2 −1 ≥ 1 − max λ−1 1,p1 , λ1,p2 c − d ( ∇u1 p1 + ∇u2 p2 ) − ω 1 .   −1 Thanks to 1 − max λ−1 , λ 1,p1 1,p2 c − d > 0 as posed in assumption H(f1 , f2 )2 , it turns out that (6.14) is satisfied. Since A : W → W ∗ introduced in (6.8) is a pseudomonotone, bounded, and coercive operator, we may apply the main theorem on pseudomonotone operators (see Theorem 1.55), which ensures that a solution of the equation Au = 0 exists. According to (6.8), this implies that (u1 , u2 ) is a weak solution of system (6.1). A natural topic is the uniqueness of solution for problem (6.1). It is convenient to introduce the vector field f :  × R2 × (RN )2 → R2 given by f (x, s, ξ ) = (f1 (x, s, ξ ), f2 (x, s, ξ )) for a.e. x ∈ , every s ∈ R2 , ξ ∈ (RN )2 .

(6.15)

A new assumption is now formulated. (U ) There exist constants a, b ≥ 0 and a function τ = (τ1 , τ2 ) ∈ L1 (, R2 ) such that f (x, s, ·) − τ (x) is linear on (RN )2 , and (f (x, s, ξ ) − f (x, t, ξ )) · (s − t) ≤ a|s − t|2 for a.e. x ∈ , every s, t ∈ R2 , ξ ∈ (RN )2 , |f (x, s, ξ ) − τ (x)| ≤ b|ξ | for a.e. x ∈ , every (s, ξ ) ∈ R2 × (RN )2 .

(6.16) (6.17)

Theorem 6.2. Assume that conditions H(f1 , f2 )1 , H(f1 , f2 )2 , and (U ) are satisfied. −1 2 (i) When p1 = p2 = 2 and aλ−1 1,2 + b(2λ1,2 ) < 1, the solution of problem (6.1) is unique for every μ1 , μ2 ≥ 0. (ii) When q1 = q2 = 2, the solution of problem (6.1) is unique for every −1 12 μ1 , μ2 > aλ−1 1,2 + b(2λ1,2 ) . 1

Proof. Theorem 6.1 guarantees the existence of solutions of problem (6.1). Let 1,p us admit that there are two solutions u = (u1 , u2 ), v = (v1 , v2 ) ∈ W0 1 () × 1,p2 W0 () for (6.1) with a fixed (μ1 , μ2 ).

234 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints 1,p1

Testing the first equation in the system (6.1) with u1 − v1 ∈ W0 1,p the second equation with u2 − v2 ∈ W0 2 () entails

() and

−p1 u1 + p1 v1 , u1 − v1  + μ1 −q1 u1 + q1 v1 , u1 − v1  + −p2 u2 + p2 v2 , u2 − v2  + μ2 −q2 u2 + q2 v2 , u2 − v2   = (f (x, u, ∇u) − f (x, v, ∇u)) · (u − v) dx   + (f (x, v, ∇u) − τ (x) − f (x, v, ∇v) + τ (x)) · (u − v) dx. (6.18) 

(i) If p1 = p2 = 2, by (6.18), (6.15), (6.16), (6.17), the monotonicity of −q1 and −q2 in conjunction with Cauchy–Schwarz inequality, we get the estimate ∇(u1 − v1 ) 22 + ∇(u2 − v2 ) 22 ≤ a( u1 − v1 22 + u2 − v2 22 )  + (f1 (x, v1 , v2 , (u1 − v1 )∇(u1 − v1 ), (u1 − v1 )∇(u2 − v2 )) − τ1 (x)) dx  + (f2 (x, v1 , v2 , (u2 − v2 )∇(u1 − v1 ), (u2 − v2 )∇(u2 − v2 )) − τ2 (x)) dx  2 2 ≤ aλ−1 1,2 ( ∇(u1 − v1 ) 2 + ∇(u2 − v2 ) 2 )  1 + b (|u1 − v1 | + |u2 − v2 |)(|∇(u1 − v1 )|2 + |∇(u2 − v2 )|2 ) 2 dx  1 −1 2 ≤ (aλ1,2 + b(2λ−1 1,2 ) )( ∇(u1

− v1 ) 22 + ∇(u2 − v2 ) 22 ).

−1 2 Since aλ−1 1,2 + b(2λ1,2 ) < 1, we derive that ui = vi for i = 1, 2. (ii) In the case of q1 = q2 = 2, we can proceed as in part (i) to obtain 1

μ1 ∇(u1 − v1 ) 22 + μ2 ∇(u2 − v2 ) 22 −1 2 2 2 ≤ (aλ−1 1,2 + b(2λ1,2 ) )( ∇(u1 − v1 ) 2 + ∇(u2 − v2 ) 2 ). 1

−1 2 The conclusion ensues by choosing μ1 , μ2 > aλ−1 1,2 + b(2λ1,2 ) . 1

We point out a useful a priori estimate for the solutions of problem (6.1). Lemma 6.3. If the Carathéodory functions fi :  × R × R × RN × RN → R, i = 1, 2, satisfy assumptions H(f1 , f2 )1 , H(f1 , f2 )2 , then the set of solutions of 1,p 1,p problem (6.1) is bounded in W0 1 () × W0 2 () uniformly with respect to μ1 , μ2 ≥ 0. 1,p

1,p

Proof. If (u1 , u2 ) ∈ W0 1 () × W0 2 () is a solution of problem (6.1), by (6.3), (6.4), (6.2), and hypothesis H(f1 , f2 )2 , we get the estimate

Nonlinear Elliptic Systems Chapter | 6 p

235

p

∇u1 p11 + ∇u2 p22  ≤ (f1 (x, u1 , u2 , ∇u1 , ∇u2 )u1 + f2 (x, u1 , u2 , ∇u1 , ∇u2 )u2 ) dx  ≤ (|ω(x)| + c( |u1 |p1 + |u2 |p2 ) + d( |∇u1 |p1 + |∇u2 |p2 )) dx   

p p −1 ≤ ω 1 + max λ−1 , λ c + d ( ∇u1 p11 + ∇u2 p22 ). 1,p1 1,p2   −1 , λ Since it was supposed that 1−max λ−1 1,p1 1,p2 c −d > 0, we obtain the bound p

p

∇u1 p11 + ∇u2 p22 ≤



ω 1

 =: c0 , −1 1 − max λ−1 , λ c − d 1,p1 1,p2

(6.19)

with the constant c0 independent of (μ1 , μ2 ), which yields the desired conclusion. Using that μ1 , μ2 are parameters, we can examine the asymptotic property involving the limit as μ1 → 0 and μ2 → 0. Theorem 6.4. Assume conditions H(f1 , f2 )1 and H(f1 , f2 )2 . Given a sequence of parameters {(μ1,n , μ2,n )} with μ1,n → 0+ and μ2,n → 0+ , and a sequence {(u1,n , u2,n )} of solutions of the corresponding problems (6.1) (with (μ1 , μ2 ) = (μ1,n , μ2,n )), one can find a relabeled subsequence of {(u1,n , u2,n )} 1,p 1,p such that u1,n → u1 in W0 1 () and u2,n → u2 in W0 2 (), with (u1 , u2 ) ∈ 1,p 1,p W0 1 () × W0 2 () weak solution of problem (6.5). Proof. The fact that (u1,n , u2,n ) is a weak solution of problem (6.1) (with (μ1 , μ2 ) = (μ1,n , μ2,n )) permits applying Lemma 6.3 and deducing that the 1,p 1,p sequence {(u1,n , u2,n )} is bounded in W0 1 () × W0 2 (). Then a relabeled 1,p subsequence can be found to have the weak convergence u1,n u1 in W0 1 () 1,p 1,p 1,p and u2,n u2 in W0 2 () for some (u1 , u2 ) ∈ W0 1 () × W0 2 (). As in the proof of Theorem 6.1, hypothesis H(f1 , f2 )1 implies that (6.10) holds true. This essentially relies on Rellich–Kondrachov compactness embedding theorem. Then, using that μ1,n → 0+ and μ2,n → 0+ , we note that the equations in (6.1) (with (μ1 , μ2 ) = (μ1,n , μ2,n )) yield in the limit lim −pi ui,n , ui,n − ui  = 0

n→∞

for i = 1, 2.

At this point we can refer to the (S)+ -property of the operators −pi : 1,p W0 i () → W −1,pi () (i = 1, 2) in Proposition 1.56 for obtaining the strong 1,p 1,p convergence u1,n → u1 in W0 1 () and u2,n → u2 in W0 2 (). Once the strong convergence is obtained, we can pass to the limit in the equations of (6.1)

236 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

(with (μ1 , μ2 ) = (μ1,n , μ2,n )) because −pi ui,n → −pi ui and fi (·, u1,n (·), u2,n (·), ∇u1,n (·), ∇u2,n (·))

→ fi (·, u1 (·), u2 (·), ∇u1 (·), ∇u2 (·)) in W −1,pi ()

(6.20)

for i = 1, 2. Then due to the fact that μi,n → 0+ , (6.20) and the boundedness of the operators −qi for i = 1, 2, letting n → ∞ in (6.1) (with (μ1 , μ2 ) = (μ1,n , μ2,n )) results in the fact that (u1 , u2 ) becomes a weak solution of problem (6.5). This completes the proof. Next we focus on the asymptotic study as (μ1 , μ2 ) → (0, +∞) (the situation where (μ1 , μ2 ) → (+∞, 0) is symmetric). Theorem 6.5. Assume H(f1 , f2 )1 and H(f1 , f2 )2 , and in addition that β1 ≤ q2 p1 + (p1∗ ) p2 . For any sequence {(μ1,n , μ2,n )} of parameters with μ1,n → 0 and μ2,n → +∞, and any sequence {(u1,n , u2,n )} of solutions of the corresponding problems (6.1) (with (μ1 , μ2 ) = (μ1,n , μ2,n )), one can find a relabeled subse1,p quence {(u1,n , u2,n )} such that u1,n → u1 strongly in W0 1 () and u2,n → 0 1,q2 1,p1 strongly in W0 (), with u1 ∈ W0 () weak solution of problem (6.6). 1,p

Proof. By Lemma 6.3 we know that {(u1,n , u2,n )} is bounded in W0 1 () × 1,p W0 2 (), thus it contains a relabeled subsequence with the weak conver1,p 1,p gence u1,n u1 in W0 1 () and u2,n u2 in W0 2 () for some (u1 , u2 ) ∈ 1,p1 1,p2 W0 () × W0 (). Actually, (6.1) (with (μ1 , μ2 ) = (μ1,n , μ2,n )) renders ⎧ ⎪ in , ⎪ ⎨ −p1 u1,n − μ1,n q1 u1,n = f1 (x, u1,n , u2,n , ∇u1,n , ∇u2,n ) 1 1 − μ2,n p2 u2,n − q2 u2,n = μ2,n f2 (x, u1,n , u2,n , ∇u1,n , ∇u2,n ) in , ⎪ ⎪ ⎩ u =u =0 on ∂. 1,n

2,n

(6.21) Taking into account that μ1,n → 0 and μ2,n → +∞, we infer from (6.21) that lim −p1 u1,n , u1,n − u1  = lim −q2 u2,n , u2,n − u2  = 0.

n→∞

n→∞

Here it was also used the boundedness of the operators −p2 , −q1 as well 1,p as (6.10). Through the (S)+ -property of the operators −p1 : W0 1 () → 1,q W −1,p1 () and −q2 : W0 2 () → W −1,q2 () in Proposition 1.56, it turns 1,p1 1,q out that u1,n → u1 in W0 () and u2,n → u2 in W0 2 (). Then, recalling that μ2,n → +∞, the second equation in (6.21) provides in the limit that q2 u2 = 0, thereby u2 = 0. Now, recalling that μ1,n → 0, we can pass to the limit as n → ∞ in the first equation of (6.21) on the basis of H(f1 , f2 )1 with β1 ≤ (pq∗2)p p1 . We conclude that u1 is a solution of problem (6.6). 1

2

Nonlinear Elliptic Systems Chapter | 6

237

Finally, we consider the limiting process as μ1 → +∞ and μ2 → +∞. Theorem 6.6. Assume that conditions H(f1 , f2 )1 and H(f1 , f2 )2 hold. Given a sequence {(μ1,n , μ2,n )} with μ1,n → +∞ and μ2,n → +∞, every sequence {(u1,n , u2,n )} of solutions of the corresponding problems (6.1) (with (μ1 , μ2 ) = 1,q (μ1,n , μ2,n )) satisfies u1,n → 0 strongly in W0 1 () and u2,n → 0 strongly in 1,q W0 2 (). 1,p

Proof. By Lemma 6.3, the sequence {(u1,n , u2,n )} is bounded in W0 1 () × 1,p W0 2 (), so we can extract a relabeled subsequence satisfying the weak 1,p 1,p convergence u1,n u1 in W0 1 () and u2,n u2 in W0 2 () with some 1,p 1,p (u1 , u2 ) ∈ W0 1 () × W0 2 (). Writing (6.1) (with (μ1 , μ2 ) = (μ1,n , μ2,n )) in the form ⎧ 1 1 ⎪ ⎪ ⎨ − μ1,n p1 u1,n − q1 u1,n = μ1,n f1 (x, u1,n , u2,n , ∇u1,n , ∇u2,n ) in , − μ12,n p2 u2,n − q2 u2,n = ⎪ ⎪ ⎩ u1,n = u2,n = 0

1 μ2,n f2 (x, u1,n , u2,n , ∇u1,n , ∇u2,n )

in , on ∂, (6.22)

we derive on the same pattern as before that lim −q1 u1,n , u1,n − u1  = lim −q2 u2,n , u2,n − u2  = 0.

n→∞

n→∞

The validity of these equalities is basically the consequence of (6.22), where μ1,n → +∞ and μ2,n → +∞. We are thus in a position to invoke the 1,q (S)+ -property of the operators −q1 : W0 1 () → W −1,q1 () and −q2 : 1,q W0 2 () → W −1,q2 () in Proposition 1.56, which entails that strongly u1,n → 1,q 1,q u1 in W0 1 () and u2,n → u2 in W0 2 (). Then letting n → ∞ in (6.22) leads to q1 u1 = 0 and q2 u2 = 0, which reads as u1 = u2 = 0. Since this occurs for every convergent subsequence of {(u1,n , u2,n )}, we have the strong convergence for the whole sequence, which completes the proof.

6.2 SUBSOLUTIONS–SUPERSOLUTIONS FOR NONLINEAR ELLIPTIC SYSTEMS In this section, we investigate the system (6.1) of Section 6.1 through a different method, namely by means of subsolution–supersolution. In comparison with equations, the subsolution–supersolution method for systems has specific features. We suppose that N > max{p1 , p2 }, which implies that the Sobolev Npi , i = 1, 2. The case N ≤ critical exponent corresponding to pi is pi∗ = N−p i max{p1 , p2 } is basically simpler and can be carried out along the same lines. We also assume in this section that the bounded domain  ⊂ RN has a C 2 boundary ∂.

238 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

We say that (u1 , u2 ), (u1 , u2 ) ∈ W 1,p1 () × W 1,p2 () is a subsolution– supersolution of problem (6.1) if ui ≤ ui a.e. in , ui ≤ 0 ≤ ui on ∂ for i = 1, 2, and  |∇u1 |p1 −2 ∇u1 · ∇v1 + μ1 |∇u1 |q1 −2 ∇u1 · ∇v1 

− f1 (x, u1 , w2 , ∇u1 , ∇w2 )v1  + |∇u2 |p2 −2 ∇u2 · ∇v2 + μ2 |∇u2 |q2 −2 ∇u2 · ∇v2 

− f2 (x, w1 , u2 , ∇w1 , ∇u2 )v2 ≤ 0 and 

|∇u1 |p1 −2 ∇u1 · ∇v1 + μ1 |∇u1 |q1 −2 ∇u1 · ∇v1 

− f1 (x, u1 , w2 , ∇u1 , ∇w2 )v1  + |∇u2 |p2 −2 ∇u1 · ∇v2 + μ2 |∇u2 |q2 −2 ∇u1 · ∇v2 

− f2 (x, w1 , u2 , ∇w1 , ∇u2 )v2 ≥ 0 1,p

1,p

for all (v1 , v2 ) ∈ W0 1 () × W0 2 () with v1 , v2 ≥ 0 a.e. in , and all (w1 , w2 ) ∈ W 1,p1 () × W 1,p2 () such that ui ≤ wi ≤ ui a.e. in , i = 1, 2. In the sequel, we will require the existence of a subsolution–supersolution (u1 , u2 ), (u1 , u2 ) ∈ W 1,p1 () × W 1,p2 () of problem (6.1) satisfying: H(f1 , f2 ) There exist constants bi ≥ 0, βi ∈ [0, (pp∗i ) ) and functions σi ∈

Lγi (), with γi ∈ [1, pi∗ ), such that

i

|f1 (x, s1 , s2 , ξ1 , ξ2 )| ≤ σ1 (x) + b1 (|ξ1 |β1 + |ξ2 | |f2 (x, s1 , s2 , ξ1 , ξ2 )| ≤ σ2 (x) + b2 (|ξ1 |

β2 p1 p2

β1 p2 p1

),

+ |ξ2 |β2 )

for a.a. x ∈ , all s = (s1 , s2 ) ∈ [u1 (x), u1 (x)] × [u2 (x), u2 (x)], ξ1 , ξ2 ∈ RN , i = 1, 2. Under H(f1 , f2 ), the integrals in the definition of subsolution–supersolution exist. Fix μ1 , μ2 ≥ 0 and according to H(f1 , f2 ) let (u1 , u2 ), (u1 , u2 ) ∈ W 1,p1 () × W 1,p2 () be a subsolution–supersolution of problem (6.1). For i = 1, 2, we consider the truncation operators Ti : W 1,pi () → W 1,pi () de-

Nonlinear Elliptic Systems Chapter | 6

239

fined by ⎧ ⎪ ⎨ ui (x) if u(x) > ui (x), (Ti u)(x) = u(x) if ui (x) ≤ u(x) ≤ ui (x), ⎪ ⎩ ui (x) if u(x) < ui (x). The operators Ti are continuous and bounded. With the constants βi in H(f1 , f2 ), we set for a.a. x ∈ , all s ∈ R, i = 1, 2: ⎧ βi ⎪ ⎪ pi −βi ⎪ u (x)) if s > ui (x), (s − i ⎨ (6.23) πi (x, s) = 0 if ui (x) ≤ s ≤ ui (x), ⎪ ⎪ β ⎪ i ⎩ −(ui (x) − s) pi −βi if s < ui (x). It is clear that πi is a Carathéodory function satisfying the estimate βi

|πi (x, s)| ≤ ρi (x) + ci |s| pi −βi

(6.24) pi

for a.a. x ∈ , all s ∈ R, with a constant ci ≥ 0 and a function ρi ∈ L βi (). We pi

∗

have that ρi ∈ L βi () because ui , ui ∈ W 1,pi (), so ui , ui ∈ Lpi () by the Sobolev embedding theorem (see Theorem 1.20), and βi < (pp∗i ) by hypothesis i H(f1 , f2 ). Moreover, a direct reasoning based on (6.23) shows the existence of (i) (i) positive constants r1 and r2 such that 

(i)



πi (x, u)u dx ≥ r1 u

pi pi −βi pi pi −βi

(i)

− r2

for all u ∈ W 1,pi (), i = 1, 2. (6.25)

On the basis of (6.24), the corresponding Nemytskii operators i : W 1,pi () → pi

L βi () given by i u(x) = πi (x, u(x)) are completely continuous due to the compact embedding of W 1,pi () into Lpi () (see Theorem 1.20). For any λ > 0, consider the auxiliary problem ⎧ ⎪ −p1 u1 − μ1 q1 u1 + λ1 (u1 ) ⎪ ⎪ ⎪ ⎪ ⎪ = f1 (x, T1 u1 , T2 u2 , ∇(T1 u1 ), ∇(T2 u2 )) in , ⎨ (6.26) −p2 u2 − μ2 q2 u2 + λ2 (u2 ) ⎪ ⎪ ⎪ = f2 (x, T1 u1 , T2 u2 , ∇(T1 u1 ), ∇(T2 u2 )) in , ⎪ ⎪ ⎪ ⎩ u1 = u2 = 0 on ∂. Theorem 6.7. Let (u1 , u2 ), (u1 , u2 ) ∈ W 1,p1 () × W 1,p2 () be a subsolution– supersolution of problem (6.1) such that condition H(f1 , f2 ) is satisfied. Then, for every λ > 0 sufficiently large, problem (6.26) has a (weak) solution 1,p 1,p (u1 , u2 ) ∈ W0 1 () × W0 2 ().

240 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Proof. With the subsolution–supersolution (u1 , u2 ), (u1 , u2 ), we introduce the ordered intervals [ui , ui ] = {u ∈ W 1,pi () : ui ≤ u ≤ ui a.e. in }, i = 1, 2.

(6.27)

By (6.27) and hypothesis H(f1 , f2 ), the Nemytskii operator N : [u1 , u1 ] × [u2 , u2 ] ⊂ W 1,p1 () × W 1,p2 () p1

p2





→ L β1 () × L β2 () → W −1,p1 () × W −1,p2 () defined through the functions fi by N (u1 , u2 ) = (f1 (x, u1 , u2 , ∇u1 , ∇u2 ), f2 (x, u1 , u2 , ∇u1 , ∇u2 ))

(6.28)

is bounded and completely continuous thanks to Rellich–Kondrachov compactness embedding theorem (see Theorem 1.20). 1,p For our fixed λ > 0, we define the nonlinear operator A : W0 1 () × 1,p W0 2 () → W −1,p1 () × W −1,p2 () by A(u1 , u2 ) = (A1 (u1 , u2 ), A2 (u1 , u2 )) := (−p1 u1 − μ1 q1 u1 + λ1 u1 , −p2 u2 − μ2 q2 u2 + λ2 u2 ) − N (T1 u1 , T2 u2 ). According to (6.24), (6.27), (6.28) and hypothesis H(f1 , f2 ), the operator A is well defined, bounded, and continuous. Now we show that the operator A is pseudomonotone. Let (u1,n , u2,n ) 1,p 1,p (u1 , u2 ) in W0 1 () × W0 2 () and lim sup A(u1,n , u2,n ), (u1,n − u1 , u2,n − u2 ) ≤ 0.

(6.29)

n→∞

Since

pi pi −βi

< pi∗ as known from assumption H(f1 , f2 ), it follows the strong pi

convergence ui,n → ui in L pi −βi (). Taking into account (6.24), we get  lim πi (x, ui,n (x))(ui,n − ui ) dx = 0, i = 1, 2. (6.30) n→∞ 

Similarly, by Hölder’s inequality and Rellich–Kondrachov compactness embedding theorem in Theorem 1.20, the weak convergence (u1,n , u2,n ) (u1 , u2 ) in 1,p 1,p W0 1 () × W0 2 () enables us to derive that  

|σi | |ui,n − ui | dx ≤ σi γi ui,n − ui γi → 0 as n → +∞.

(6.31)

Nonlinear Elliptic Systems Chapter | 6

241

We claim that  |∇(Ti ui,n )|βi |ui,n − ui | dx → 0 as n → +∞.

(6.32)



From the expression of the truncation operators, it turns out that   |∇(Ti ui,n )|βi |ui,n − ui | dx = |∇ui |βi |ui,n − ui | dx 

{ui,n ui }

|∇ui |βi |ui,n − ui | dx.

< pi∗ , we have

 

{ui,n ui }

|∇ui |βi |ui,n − ui | dx ≤ ∇ui βpii ui,n − ui

→ 0,

pi pi −βi

pi pi −βi

→ 0,

→ 0,

whence (6.32). Similarly, for i = j , we can prove  |∇(Tj uj,n )|

βi pj pi

|ui,n − ui | dx ≤ ∇(Tj uj,n )



βi pj pi pj

ui,n − ui

pi pi −βi

→ 0. (6.33)

Combining (6.31), (6.32), and (6.33) with hypothesis H(f1 , f2 ) yields  lim fi (x, T1 u1,n , T2 u2,n , ∇(T1 u1,n ), ∇(T2 u2,n ))(ui,n − ui ) dx = 0, n→∞ 

i = 1, 2.

(6.34)

If we insert (6.30) and (6.34) into (6.29), we arrive at

lim sup −p1 u1,n − μ1 q1 u1,n , u1,n − u1  n→∞

 + −p2 u2,n − μ2 q2 u2,n , u2,n − u2  ≤ 0.

(6.35)

We show that (6.35) implies lim sup −pi ui,n − μi qi ui,n , ui,n − ui  ≤ 0, i = 1, 2. n→∞

(6.36)

242 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Arguing by contradiction, assume that lim −p1 u1,n − μ1 q1 u1,n , u1,n − u1  > 0,

n→∞

lim −p2 u2,n − μ2 q2 u2,n , u2,n − u2  < 0.

n→∞

1,p

The second inequality and the (S)+ -property of −p2 − μ2 q2 on W0 2 () 1,p (see [169, p. 39]) yield u2,n → u2 in W0 2 (), which contradicts (6.35), thus proving (6.36). 1,p By (6.36) and the (S)+ -property of −pi − μi qi on W0 i (), we derive 1,p the strong convergence ui,n → ui in W0 i () as n → ∞, i = 1, 2. Conse1,p 1,p quently, for every element (v1 , v2 ) ∈ W0 1 () × W0 2 (), one gets lim A(u1,n , u2,n ), (u1,n − v1 , u2,n − v2 ) = A(u1 , u2 ), (u1 − v1 , u2 − v2 ),

n→∞

establishing that the operator A is pseudomonotone. Next we show that the operator A is coercive, which means that for every se1,p 1,p quence {(u1,n , u2,n )} ⊂ W0 1 () × W0 2 () such that (u1,n , u2,n ) → +∞, we have lim

n→∞

A(u1,n , u2,n ), (u1,n , u2,n ) = +∞. (u1,n , u2,n )

(6.37)

By (6.25) and hypothesis H(f1 , f2 ), we note that p

q

A1 (u1,n , u2,n ), u1,n  = ∇u1,n p11 + μ1 ∇u1,n q11  + λ π1 (x, u1,n )u1,n dx   f1 (x, T1 u1,n , T2 u2,n , ∇(T1 u1,n ), ∇(T2 u2,n ))u1,n dx −  p1

 p −β p (1) (1) σ1 (x)|u1,n (x)| dx ≥ ∇u1,n p11 + λ r1 u1,n 1p1 1 − r2 − p1 −β1    β1 p2 − b1 |∇(T1 u1,n )|β1 |u1,n | dx − b1 |∇(T2 u2,n )| p1 |u1,n | dx. 



Let us estimate separately the three integral terms. First, observe that  σ1 (x)|u1,n (x)| dx ≤ σ1 γ1 u1,n γ1 ≤ C1 ∇u1,n p1 , 

with a constant C1 > 0. Second, using Young’s inequality with any ε > 0, we get  p1 p1 −β1 p1 β1 |∇(T1 u1,n )| |u1,n (x)| dx ≤ ε ∇(T1 u1,n ) p1 + C1 (ε) u1,n p1 

p1 −β1

Nonlinear Elliptic Systems Chapter | 6

p

p

p

≤ ε ∇u1,n p11 + ε ∇u1 p11 + ε ∇u1 p11 + C1 (ε) u1,n

243

p1 p1 −β1 p1 p1 −β1

and third,  |∇(T2 u2,n )|

β1 p2 p1

|u1,n (x)| dx

 p

p

p ≤ ε ∇(T2 u2,n ) p22

+ C2 (ε) u1,n

p

≤ ε ∇u2,n p22 + ε ∇u2 p22 + ε ∇u2 p22 + C2 (ε) u1,n

p1 p1 −β1 p1 p1 −β1

p1 p1 −β1 p1 p1 −β1

for some constants C1 (ε), C2 (ε) > 0. These estimates render p

A1 (u1,n , u2,n ), u1,n  ≥ (1 − ε) ∇u1,n p11 − C1 ∇u1,n p1 (1) + (λr1

− C(ε)) u1,n

p1 p1 −β1 p1 p1 −β1

p

− ε ∇u2,n p22 − D(ε),

with positive constants C(ε), D(ε). Similarly, we obtain

A2 (u1,n , u2,n ), u2,n  ≥ (1 − ε) ∇u2,n p22 − C1 ∇u2,n p2 p

+ (λr1(2) − C (ε)) u2,n

p2 p2 −β2 p2 p2 −β2

− ε ∇u1,n p11 − D (ε), p

with positive constants C1 , C (ε), D (ε). From the last two inequalities, by choosing ε > 0 sufficiently small and λ > 0 sufficiently large, we notice that (6.37) holds true because p1 , p2 > 1. 1,p 1,p Therefore the operator A : W0 1 () × W0 2 () → W −1,p1 () × W −1,p2 () is pseudomonotone, bounded, and coercive. Consequently, we may apply the main theorem on pseudomonotone operators as stated in Theorem 1.55, so a 1,p 1,p solution (u1 , u2 ) ∈ W0 1 () × W0 2 () of the equation A(u1 , u2 ) = 0

(6.38)

exists. Taking into account that Eq. (6.38) reads as system (6.26), we can thus conclude that the solution (u1 , u2 ) of (6.38) is a solution of the auxiliary problem (6.26), which completes the proof. Here is our main result on the existence of a solution to problem (6.1) within the ordered rectangle determined by a subsolution–supersolution. Theorem 6.8. Let (u1 , u2 ), (u1 , u2 ) ∈ W 1,p1 () × W 1,p2 () be a subsolution– supersolution of problem (6.1) such that condition H(f1 , f2 ) is fulfilled. Then 1,p 1,p there exists a solution (u1 , u2 ) ∈ W0 1 () × W0 2 () of problem (6.1) satisfying the enclosure property ui ≤ ui ≤ ui a.e. in , i = 1, 2.

244 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Proof. Assumption H(f1 , f2 ) enables us to apply Theorem 6.7, which provides a solution (u1 , u2 ) = (u1 (λ), u2 (λ)) to the auxiliary problem (6.26) for every λ > 0 sufficiently large. We claim that (u1 , u2 ) ∈ [u1 , u1 ] × [u2 , u2 ] (see (6.27)). Let us check that u1 ≤ u1 a.e. in . To this end, we test with the function 1,p (u1 − u1 )+ = max{u1 − u1 , 0} ∈ W0 1 () in problem (6.26) and in the definition of supersolution to problem (6.1), which gives  +

−p1 u1 − μ1 q1 u1 , (u1 − u1 )  + λ π1 (x, u1 )(u1 − u1 )+ dx   f1 (x, T1 u1 , T2 u2 , ∇(T1 u1 ), ∇(T2 u2 ))(u1 − u1 )+ dx (6.39) = 

and

−p1 u1 − μ1 q1 u1 , (u1 − u1 )+   ≥ f1 (x, u1 , w2 , ∇u1 , ∇w2 )(u1 − u1 )+ dx

(6.40)



whenever w2 ∈ W 1,p2 () with u2 ≤ w2 ≤ u2 a.e. in . Thanks to the definition of the truncation operator T2 , u2 ≤ T2 u2 ≤ u2 , so we can insert w2 = T2 u2 in (6.40), resulting in

−p1 u1 − μ1 q1 u1 , (u1 − u1 )+   ≥ f1 (x, u1 , T2 u2 , ∇u1 , ∇(T2 u2 ))(u1 − u1 )+ dx.

(6.41)



Subtracting (6.41) from (6.39) provides

−p1 u1 − μ1 q1 u1 − (−p1 u1 − μ1 q1 u1 ), (u1 − u1 )+   + λ π1 (x, u1 )(u1 − u1 )+ dx   (f1 (x, T1 u1 , T2 u2 , ∇(T1 u1 ), ∇(T2 u2 )) ≤ 

− f1 (x, u1 , T2 u2 , ∇u1 , ∇(T2 u2 )))(u1 − u1 )+ dx = 0, where the last equality holds because T1 u1 = u1 on the set {u1 > u1 }. This amounts to saying that  (|∇u1 |p1 −2 ∇u1 − |∇u1 |p1 −2 ∇u1 ) · (∇u1 − ∇u1 ) dx {u1 >u1 }  (|∇u1 |q1 −2 ∇u1 − |∇u1 |q1 −2 ∇u1 ) · (∇u1 − ∇u1 ) dx + μ1 {u1 >u1 }

Nonlinear Elliptic Systems Chapter | 6

 +λ

{u1 >u1 }

245

p1

(u1 − u1 ) p1 −β1 dx ≤ 0.

Then, the inequalities (|ζ |p1 −2 ζ − |η|p1 −2 η) · (ζ − η) > 0, (|ζ |q1 −2 ζ − |η|q1 −2 η) · (ζ − η) > 0 (6.42) for all ζ, η ∈ RN with ζ = η, ensure that u1 ≤ u1 a.e. in . On the same pattern, making use of adequately chosen test functions, we can show that u1 ≤ u1 and u2 ≤ u2 ≤ u2 a.e. in . Hence the claim (u1 , u2 ) ∈ [u1 , u1 ] × [u2 , u2 ] is verified. Due to the claim, we have that Ti ui = ui and i ui = 0 for i = 1, 2. Therefore problem (6.26) reduces to (6.1), thus (u1 , u2 ) solves problem (6.1). As an application of Theorem 6.8 we show the existence of at least one positive solution of problem (6.1) under the following hypothesis: H(f1 , f2 ) For i = 1, 2, there exist ai ∈ Lαi (), with αi > N and ai (x) > 0 on a set of positive measure, bi ≥ 0, βi ∈ [0, (pp∗i ) ), σi ∈ Lγi (), with i

γi ∈ [1, pi∗ ), and a Carathéodory function gi : ×R×RN → R satisfying 0 ≤ ai (x) ≤ fi (x, s1 , s2 , ξ1 , ξ2 ) ≤ gi (x, si , ξi ) ≤ σi (x) + bi |ξi |βi for a.a. x ∈ , all s1 , s2 > 0, ξ1 , ξ2 ∈ RN , with bi , βi , σi as in H(f1 , f2 ), and such that the Dirichlet problem −pi u − μi qi u = gi (x, u, ∇u) in , u=0 on ∂

has a supersolution ui ∈ W 1,pi () ∩ C(). Theorem 6.9. Assume that condition H(f1 , f2 ) is satisfied. Then there exists 1,p 1,p a solution (u1 , u2 ) ∈ W0 1 () × W0 2 () of problem (6.1), which is positive, meaning that ui > 0 a.e. in , i = 1, 2. Proof. In view of assumption H(f1 , f2 ) , for i = 1, 2 it holds that ai ∈ W −1,pi (), hence the Dirichlet problem −pi u − μi qi u = ai in , u=0 on ∂ 1,p

possesses a unique solution ui ∈ W0 i (). The assumption ai ∈ Lαi () allows us to invoke [53, Theorem 3.1] for deducing that ui is bounded. Then,

246 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

through the nonlinear regularity theory (see Proposition 2.12), it follows that ui ∈ C 1 (). Let us prove that ui > 0 in , i = 1, 2.

(6.43)

Acting with the test function −u− i produces

−pi ui − μi qi ui , −u− i =−

 

ai (x)u− i (x) ≤ 0,

which yields  

− qi pi (|∇u− i | + μi |∇ui | ) dx ≤ 0,

thereby ui ≥ 0 a.e. in . Taking into account that ui is nontrivial, the strong maximum principle in Theorem 2.19 implies (6.43). We further claim that ui ≤ ui a.e. in , i = 1, 2.

(6.44)

Arguing indirectly, assume, for instance, that the open set {u1 > u1 } is nonempty. Using hypothesis H(f1 , f2 ) and acting with (u1 − u1 )+ imply 

−p1 u1 − μ1 q1 u1 , (u1 − u1 )+  = a1 (x)(u1 − u1 )+ dx   + g1 (x, u1 (x), ∇u1 (x))(u1 − u1 ) dx ≤ 

≤ −p1 u1 − μ1 q1 u1 , (u1 − u1 )+ , hence  {u1 >u1 }

+ μ1

(|∇u1 |p1 −2 ∇u1 − |∇u1 |p1 −2 ∇u1 ) · (∇u1 − ∇u1 ) dx  {u1 >u1 }

(|∇u1 |q1 −2 ∇u1 − |∇u1 |q1 −2 ∇u1 ) · (∇u1 − ∇u1 ) dx ≤ 0.

From the strict inequalities in (6.42), we infer that the function u1 − u1 is locally constant on {u1 > u1 }. Taking a path γ : [0, 1] →  such that γ (0) ∈ {u1 > u1 } and γ (1) ∈ ∂, and noting that u1 ≤ u1 on ∂, we find t0 ∈ (0, 1] such that γ (t0 ) ∈ / {u1 > u1 } and γ (t) ∈ {u1 > u1 } for all t ∈ [0, t0 ). The continuity of u1 , u1 and the fact that u1 − u1 is locally constant on {u1 > u1 } yield a contradiction. So (6.44) ensues.

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Next we prove that (u1 , u2 ), (u1 , u2 ) is a subsolution–supersolution for problem (6.1). By hypothesis H(f1 , f2 ) we derive that  a1 (x)v(x) dx

−p1 u1 − μ1 q1 u1 , v =  f1 (x, u1 , w2 , ∇u1 , ∇w2 )v dx ≤ 

and 

 f1 (x, u1 , w2 , ∇u1 , ∇w2 )v dx ≤ 

g1 (x, u1 , ∇u1 )v dx 

≤ −p1 u1 − μ1 q1 u1 , v 1,p

1,p

for all v ∈ W0 1 () with v ≥ 0 a.e. in  and all w2 ∈ W0 2 () with u2 ≤ w2 ≤ u2 a.e. in  (notice that due to (6.44), it makes sense to consider 1,p w2 ∈ W0 2 () such that u2 ≤ w2 ≤ u2 a.e. in ). Similarly, from hypothesis H(f1 , f2 ) it is seen that  a2 (x)v(x) dx

−p2 u2 − μ2 q2 u2 , v =  f2 (x, w1 , u2 , ∇w1 , ∇u2 )v dx ≤ 

and 

 f2 (x, w1 , u2 , ∇w1 , ∇u2 )v dx ≤



g2 (x, u2 , ∇u2 )v dx 

≤ −p2 u2 − μ2 q2 u2 , v 1,p

(6.45)

1,p

for all v ∈ W0 2 () with v ≥ 0 a.e. in  and all w1 ∈ W0 1 () with u1 ≤ w1 ≤ u1 a.e. in . Altogether, (u1 , u2 ), (u1 , u2 ) is a subsolution–supersolution for (6.1). Now we are in a position to apply Theorem 6.8, which ensures the existence 1,p 1,p of a solution (u1 , u2 ) ∈ W0 1 () × W0 2 () of problem (6.1) satisfying ui ≤ ui ≤ ui a.e. in , i = 1, 2. Then, in view of (6.43), we infer that ui > 0 a.e. in , i = 1, 2. Example 6.10. Hypothesis H(f1 , f2 ) is fulfilled whenever there exist ai , ai ∈ Lαi (), with αi > N and ai (x) > 0 on a set of positive measure, such that 0 ≤ ai (x) ≤ fi (x, s1 , s2 , ξ1 , ξ2 ) ≤ ai (x) for a.a. x ∈  and all s1 , s2 > 0, ξ1 , ξ2 ∈ RN .

248 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

6.3 ELLIPTIC SYSTEMS WITH VARIATIONAL STRUCTURE We turn to the study of the following system of quasilinear elliptic equations with homogeneous Dirichlet boundary condition in the variational form ⎧ ⎪ ⎨ −p1 u1 = fs1 (x, u1 (x), u2 (x)) in , (6.46) −p2 u2 = fs2 (x, u1 (x), u2 (x)) in , ⎪ ⎩ u1 = u2 = 0, on ∂, on a bounded domain  ⊂ RN with a C 2 boundary ∂, where p1 > 1 and p2 > 1 are fixed numbers. In (6.46) we have the pi -Laplacian operator pi u = div(|∇u|pi −2 ∇u) on 1,p 1,p the space W0 i () for i = 1, 2. The first eigenvalue of −pi on W0 i () as given in Proposition 1.57 is denoted by λ1,pi , while by uˆ 1,pi we denote the corresponding positive eigenfunction satisfying uˆ 1,pi pi = 1. The second 1,p eigenvalue of −pi on W0 i () is denoted by λ2,pi . The right-hand sides of the equations in (6.46) are expressed through the partial derivatives fs1 and fs2 of a function f (x, s1 , s2 ) with f :  × R × R → R that is assumed to be Carathéodory (i.e., measurable in x ∈  and continuous in (s1 , s2 ) ∈ R2 ), twice differentiable with respect to (s1 , s2 ) ∈ R2 and the partial derivatives fs1 , fs2 , fs1 s1 , fs1 s2 , fs2 s2 are Carathéodory functions on  × R2 , with fs1 , fs2 bounded on bounded sets. We suppose without any loss of generality that f (x, 0, 0) = 0 for a.e. x ∈ . To the system (6.46) we associate the following elliptic equations with homogeneous Dirichlet boundary condition −p1 u1 = fs1 (x, u1 (x), 0) in , (6.47) u1 = 0, on ∂ and



−p2 u2 = fs2 (x, 0, u2 (x)) u2 = 0,

in , on ∂. 1,p

(6.48) 1,p2

A weak solution of (6.46) is any pair (u1 , u2 ) ∈ W0 1 () × W0 that   p1 −2 |∇u1 | ∇u1 · ∇v1 dx = fs1 (x, u1 , u2 )v1 dx,   |∇u2 |p2 −2 ∇u2 · ∇v2 dx = fs2 (x, u1 , u2 )v2 dx 

() such



1,p

1,p

for all (v1 , v2 ) ∈ W0 1 () × W0 2 (). Moreover, we say that the solution (u1 , u2 ) is positive (resp. negative) if u1 , u2 are both positive (resp. negative).

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The next definition introduces the notion of subsolution–supersolution (or trapping region) in the case of the system (6.46). It is a particular case of (6.1) considered in Sections 6.1 and 6.2. Definition 6.11. We say that (u1 , u2 ), (u1 , u2 ) ∈ W 1,p1 () × W 1,p2 () form a subsolution–supersolution for problem (6.46) if ui ≤ ui a.e. in , ui ≤ 0 ≤ ui a.e. on ∂ for i = 1, 2, 

|∇u1 |p1 −2 ∇u1 · ∇v1 − fs1 (x, u1 , w2 )v1 dx  

+ |∇u2 |p2 −2 ∇u2 · ∇v2 − fs2 (x, w1 , u2 )v2 dx ≤ 0, 

and



|∇u1 |p1 −2 ∇u1 · ∇v1 − fs1 (x, u1 , w2 )v1 dx

 

|∇u2 |p2 −2 ∇u2 · ∇v2 − fs2 (x, w1 , u2 )v2 dx ≥ 0 + 

1,p

1,p

for all (v1 , v2 ) ∈ W0 1 () × W0 2 () such that v1 , v2 ≥ 0 a.e. in , and all (w1 , w2 ) ∈ W 1,p1 () × W 1,p2 () with ui ≤ wi ≤ ui for i = 1, 2. We formulate the hypotheses on the function f (x, s1 , s2 ) in (6.46): H(f )1 There are constants k1 > 0, k2 > 0, d1 < 0, d2 < 0 such that fs1 (x, k1 , s2 ) ≤ 0 fs1 (x, d1 , s2 ) ≥ 0 fs2 (x, s1 , k2 ) ≤ 0 fs2 (x, s1 , d2 ) ≥ 0

for a.e. x ∈  and every s2 ∈ [0, k2 ], for a.e. x ∈  and every s2 ∈ [d2 , 0], for a.e. x ∈  and every s1 ∈ [0, k1 ], for a.e. x ∈  and every s1 ∈ [d1 , 0].

H(f )2 There are constants c1 > λ1,p1 and c2 > λ1,p2 such that fs1 (x, s1 , s2 )

≥ c1 uniformly for a.e. x ∈  and every 0 < s2 ≤ k2 , p −1 s1 1 fs (x, s1 , s2 ) lim inf 1 p −2 ≥ c1 uniformly for a.e. x ∈  and every d2 ≤ s2 < 0, − |s1 | 1 s1 s1 →0 fs (x, s1 , s2 ) lim inf 2 p −1 ≥ c2 uniformly for a.e. x ∈  and every 0 < s1 ≤ k1 , + s2 →0 s2 2 fs (x, s1 , s2 ) ≥ c2 uniformly for a.e. x ∈  and every d1 ≤ s1 < 0. lim inf 2 p −2 s2 →0− |s2 | 2 s2 lim inf s1 →0+

We recall that the positive cone C01 ()+ ⊂ C01 () has a nonempty interior characterized in (1.39).

250 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Theorem 6.12. Assume that hypotheses H(f )1 , H(f )2 are satisfied. Then the following assertions hold: (a) Problem (6.46) has a positive solution (u1 , u2 ) ∈ int (C01 ()+ ) ×

 int (C01 ()+ ) and a negative solution (v1 , v2 ) ∈ −int (C01 ()+ ) × 

−int (C01 ()+ ) with ui ≤ ki and vi ≥ di , i = 1, 2. (b) Problem (6.47) admits a positive solution u1 ∈ int (C01 ()+ ) with u1 ≤ k1 and a negative solution v1 ∈ −int (C01 ()+ ) with v1 ≥ d1 . (c) Problem (6.48) admits a positive solution u2 ∈ int (C01 ()+ ) with u2 ≤ k2 and a negative solution v2 ∈ −int (C01 ()+ ) with v2 ≥ d2 . Proof. (a) Assumption H(f )1 implies that −p1 (k1 ) − fs1 (x, k1 , s2 ) ≥ 0 for a.e. x ∈  and every s2 ∈ [0, k2 ], −p2 (k2 ) − fs2 (x, s1 , k2 ) ≥ 0 for a.e. x ∈  and every s1 ∈ [0, k1 ]. (6.49) Hypothesis H(f )2 guarantees that there exists δ ∈ (0, min{k1 , k2 }) such that ⎧ ⎨ f (x, s , s ) > λ s p1 −1 for a.e. x ∈ , every s ∈ (0, δ), s ∈ (0, k ], s1 1 2 1,p1 1 1 2 2 ⎩ f (x, s , s ) > λ s p2 −1 for a.e. x ∈ , every s ∈ (0, k ], s ∈ (0, δ). s2

1

2

1,p2 2

1

1

2

(6.50) Choose ε > 0 sufficiently small such that ε uˆ 1,pi (x) ≤ δ for all x ∈ , i = 1, 2. Then from (6.50) we infer that −p1 (ε uˆ 1,p1 ) − fs1 (x, ε uˆ 1,p1 , s2 ) ≤ 0 for a.e. x ∈  and every s2 ∈ [0, k2 ], −p2 (ε uˆ 1,p2 ) − fs2 (x, s1 , ε uˆ 1,p2 ) ≤ 0 for a.e. x ∈  and every s1 ∈ [0, k1 ]. (6.51) By (6.49) and (6.51) we infer that (u1 ,u2 ) := (ε uˆ 1,p1 ,ε uˆ 1,p2 ) and (u1 ,u2 ):= (k1 , k2 ) form a pair of subsolution and supersolution in the sense of Definition 6.11 for problem (6.46) provided ε > 0 is sufficiently small. Now we apply the general result in Theorem 6.8 ensuring the existence of a positive solution (u1 , u2 ) ∈ int (C01 ()+ ) × int (C01 ()+ ) of problem (6.46) satisfying ui ≤ ki , i = 1, 2 (see also [38, Section 5.5]). Following the same reasoning, we show that (d1 , d2 ) and (−ε uˆ 1,p1 , −ε uˆ 1,p2 ) constitute a pair of subsolution–supersolution in the sense of Definition 6.11 for problem (6.46) provided ε > 0 is sufficiently small. Consequently, as before,



 we obtain a negative solution (v1 , v2 ) ∈ −int (C01 ()+ ) × −int (C01 ()+ ) with vi ≥ di , i = 1, 2.

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(b) From (6.49) we derive −p1 (k1 ) − fs1 (x, k1 , 0) ≥ 0 for a.e. x ∈ , whereas (6.51) implies −p1 (ε uˆ 1,p1 ) − fs1 (x, ε uˆ 1,p1 , 0) ≤ 0 for a.e. x ∈ . Thus u1 := ε uˆ 1,p1 and u1 := k1 constitute a pair of subsolution and supersolution for problem (6.47) provided ε > 0 is sufficiently small. This results in the existence of a positive solution u1 ∈ int (C01 ()+ ) of (6.47) with u1 ≤ k1 . On the same pattern, we construct an ordered pair of subsolution and supersolution for problem (6.47), which leads to a negative solution v1 ∈ −int (C01 ()+ ) of (6.47) with v1 ≥ d1 . (c) The proof is similar to that of (b). Next we are concerned with the existence of a minimal positive solution and of a maximal negative solution of (6.46), (6.47), (6.48) in prescribed trapping regions away from zero. Theorem 6.13. Assume that hypotheses H(f )1 , H(f )2 are satisfied. Then: (a) Given a solution (u1 , u2 ) of problem (6.46) in [ε uˆ 1,p1 , k1 ] × [εuˆ 1,p2 , k2 ] for some ε > 0, there exists a minimal solution (uε1 , uε2 ) of (6.46) in [ε uˆ 1,p1 , k1 ] × [ε uˆ 1,p2 , k2 ] such that uεi ≤ ui , i = 1, 2. Similarly, given a solution (v1 , v2 ) of problem (6.46) in [d1 , −ε uˆ 1,p1 ] × [d2 , −ε uˆ 1,p2 ] for some ε > 0, there exists a maximal solution (v1ε , v2ε ) of (6.46) in [d1 , −ε uˆ 1,p1 ] × [d2 , −ε uˆ 1,p2 ] such that viε ≥ vi , i = 1, 2. (b) There exist a minimal solution uε1 of (6.47) in [εuˆ 1,p1 , k1 ] and a maximal solution v1ε of (6.47) in [d1 , −ε uˆ 1,p1 ]. (c) There exist a minimal solution uε2 of (6.48) in [εuˆ 1,p2 , k2 ] and a maximal solution v2ε of (6.48) in [d2 , −ε uˆ 1,p2 ]. Proof. (a) We provide the proof only for the first part of this assertion because the second part can be verified analogously. Theorem 6.12 ensures that the solution (u1 , u2 ) exists. For a fixed ε > 0 let us denote by Sε the set of all (v1 , v2 ) ∈ [ε uˆ 1,p1 , k1 ] × [ε uˆ 1,p2 , k2 ] solving (6.46) and vi ≤ ui , i = 1, 2. We prove that there is a minimal element of Sε , which will be done by applying Zorn’s lemma. Consider a chain C in Sε . Then there is a sequence {(uk1 , uk2 )}k≥1 ⊂ C, with uk+1 ≤ uki , i = 1, 2, for all k ≥ 1, such that i inf C = inf{(uk1 , uk2 ) : k ≥ 1} (see, e.g., [112, Lemma 3.10]). Using that (uk1 , uk2 ) is a solution of (6.46), we can 1,p easily establish the boundedness of the sequence {(uk1 , uk2 )}k≥1 in W0 1 () × 1,p2 1,p W0 (). Along a subsequence, we may suppose that uki uˆ i in W0 i ()

252 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

and a.e. in  as k → ∞, for i = 1, 2, thus (uˆ 1 , uˆ 2 ) ∈ [ε uˆ 1,p1 , k1 ] × [εuˆ 1,p2 , k2 ] and uˆ i ≤ ui , for i = 1, 2. Moreover, through the (S)+ -property of the operators −p1 and −p2 given in Proposition 1.56, we infer the strong convergence 1,p uki → uˆ i in W0 i () as k → ∞, for i = 1, 2. Consequently, (uˆ 1 , uˆ 2 ) is a solution of (6.46) that belongs to Sε and inf C = (uˆ 1 , uˆ 2 ) ∈ Sε . Then Zorn’s lemma can be applied, which yields the existence of (uε1 , uε2 ) as required. Assertions (b) and (c) can be proven along the same lines. In order to obtain the existence of a minimal positive solution and of a maximal negative solution for (6.46), (6.47), and (6.48), we need a new hypothesis: H(f )3 There are constants α1 ≥ c1 and α2 ≥ c2 such that fs1 (x, s1 , s2 )

≤ α1 uniformly for a.e. x ∈  and every 0 < s2 ≤ k2 , p −1 s1 1 fs (x, s1 , s2 ) ≤ α1 uniformly for a.e. x ∈  and every d2 ≤ s2 < 0, lim sup 1 p −2 1 s1 s1 →0− |s1 |

lim sup s1

→0+

fs2 (x, s1 , s2 )

≤ α2 uniformly for a.e. x ∈  and every 0 < s1 ≤ k1 , p −1 s2 2 fs (x, s1 , s2 ) ≤ α2 uniformly for a.e. x ∈  and every d1 ≤ s1 < 0. lim sup 2 p −2 2 s2 s2 →0− |s2 |

lim sup s2

→0+

Theorem 6.14. Assume that hypotheses H(f )1 –H(f )3 hold. Then: (a) Problem (6.46) admits a positive solution (u1,+ , u2,+ ) ∈ int (C01 ()+ ) × int (C01 ()+ ) with ui,+ ≤ ki , for i = 1, 2, which is minimal among the solutions

positive  of (6.46), and a negative solution (u1,− , u2,− ) ∈ −int (C01 ()+ ) × −int (C01 ()+ ) with ui,− ≥ di , for i = 1, 2, which is maximal among the negative solutions of (6.46). (b) Problem (6.47) admits the smallest positive solution u1 ∈ int (C01 ()+ ) with u1 ≤ k1 and the biggest negative solution v1 ∈ −int (C01 ()+ ) with v 1 ≥ d1 . (c) Problem (6.48) admits the smallest positive solution u2 ∈ int (C01 ()+ ) with u2 ≤ k2 and the biggest negative solution v2 ∈ −int (C01 ()+ ) with v 2 ≥ d2 . Proof. (a) We only prove the first part of this assertion because the second part can be proven through a similar reasoning. The successive application of The1,p 1,p orem 6.13 produces a sequence {(un1 , un2 )}n≥n0 in W0 1 () × W0 2 (), with n0 sufficiently large, such that for every integer n ≥ n0 we have that (un1 , un2 ) is a solution of problem (6.46) that is minimal (in the sense of minimal element in a partially ordered set) in the trapping region [ n1 uˆ 1,p1 , k1 ] × [ n1 uˆ 1,p2 , k2 ] and

Nonlinear Elliptic Systems Chapter | 6

253

un+1 ≤ uni , i = 1, 2. This enables us to get i 1,pi

uni ui,+ in W0

() and uni ↓ ui,+ pointwise in  for i = 1, 2, 1,p

(6.52)

1,p

with some (u1,+ , u2,+ ) ∈ W0 1 () × W0 2 (). By the same reasoning as in the proof of Theorem 6.13 based on the (S)+ -property of the operators −p1 and −p2 given in Proposition 1.56, we can show that (u1,+ , u2,+ ) is a solution for (6.46). We claim that ui,+ = 0 for i = 1, 2. Arguing by contradiction, let us suppose that u2,+ = 0. For every n ≥ n0 , we set u˜ n =

un1 fs1 (x, un1 , un2 ) p1 −1 = u˜ n . and h n un1 (un1 )p1 −1

Along a relabeled subsequence, on the basis of H(f )2 , H(f )3 we may assume that 1,p1

u˜ n u˜ in W0

p1

() a.e. in , and hn h in L p1 −1 (), p1

1,p

with some u˜ ∈ W0 i () and h ∈ L p1 −1 (). The fact that (un1 , un2 ) is a solution of problem (6.46) entails   |∇ u˜ n |p1 −2 ∇ u˜ n · ∇(u˜ n − u) ˜ dx = hn (u˜ n − u) ˜ dx. 

(6.53)



Then (6.53) gives  lim

n→∞ 

|∇ u˜ n |p1 −2 ∇ u˜ n · ∇(u˜ n − u) ˜ dx = 0.

The (S)+ -property of the operator −p1 in Proposition 1.56 implies that 1,p u˜ n → u˜ in W0 1 (), which renders u˜ = 0. Hypotheses H(f )2 and H(f )3 ensure that for any ε > 0 and a.a. x ∈  there exists an integer n(x) such that for every n ≥ n(x), (c1 − ε)u˜ n (x)p1 −1 ≤ hn (x) ≤ (α1 + ε)u˜ n (x)p1 −1 . Letting n → ∞ and because ε > 0 is arbitrary, we arrive through Mazur’s lemma at c1 u(x) ˜ p1 −1 ≤ h(x) = g(x)u(x) ˜ p1 −1 ≤ α1 u(x) ˜ p1 −1 , with c1 ≤ g(x) ≤ α1 a.e. in . Consequently, u˜ is an eigenfunction associated to the eigenvalue 1 for the weighted problem (with the weight g(x) > 0) −p1 u(x) ˜ = g(x)u(x) ˜ p1 −1 in , (6.54) u˜ = 0 on ∂.

254 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Since the eigenfunction u˜ of (6.54) does not change sign, it follows that 1 = λ1 (g), that is, the first eigenvalue for

−p1 u(x) ˜ = λg(x)u(x) ˜ p1 −1 u˜ = 0

in , on ∂

(see Proposition 1.57). On the other hand we know that λ1 (g) ≤ λ1 (c1 ) < λ1 (λ1 ) = 1, which is a contradiction. Therefore ui,+ = 0 for i = 1, 2. Applying the strong maximum principle in Theorem 2.19 leads to (u1,+ , u2,+ ) ∈ int (C01 ()+ ) × int (C01 ()+ ). It remains to show that (u1,+ , u2,+ ) is a minimal positive solution of prob1,p 1,p lem (6.46). Let (v1 , v2 ) ∈ W0 1 () × W0 2 () be a positive solution of (6.46) such that v1 ≤ u1,+ and v2 ≤ u2,+ . By the strong maximum principle in Theorem 2.19, we know that (v1 , v2 ) ∈ int (C01 ()+ ) × int (C01 ()+ ). This in conjunction with the construction of the solution (u1,+ , u2,+ ) guarantees that 1 uˆ i,pi ≤ vi ≤ ui,+ ≤ uni ≤ ki , i = 1, 2, n

(6.55)

provided n is sufficiently large. Since (un1 , un2 ) is a minimal solution of (6.46) in [ n1 uˆ 1,p1 , k1 ] × [ n1 uˆ 1,p2 , k2 ], from (6.55) we derive that uni ≤ vi , i = 1, 2. By simple comparison with (6.55), it follows that u1,+ = v1 and u2,+ = v2 , which completes the proof. Assertions (b) and (c) can be proven along the same lines. Remark 6.15. The results of this section presented until now do not require a variational structure for problem (6.46). They hold for Carathéodory functions g1 (x, s1 , s2 ) and g2 (x, s1 , s2 ) in place of the partial derivatives fs1 (x, s1 , s2 ) and fs2 (x, s1 , s2 ), respectively, provided hypotheses H(f )1 –H(f )3 are satisfied by g1 and g2 . Theorem 6.14 provides the existence of a minimal positive solution (u1,+ , u2,+ ) of problem (6.46) with ui,+ ≤ ki for i = 1, 2. Corresponding to it, we introduce the truncation τ+ :  × R2 → R2 in the following way: if x ∈  and (s1 , s2 ) ∈ R2 , we define τ+ (x, s1 , s2 ) to be the projection of (s1 , s2 ) on the closed convex subset [0, u1,+ (x)] × [0, u2,+ (x)] of R2 . Similarly, using the maximal negative solution (u1,− , u2,− ) of problem (6.46) with ui,− ≥ di for i = 1, 2, which is provided by Theorem 6.14, we introduce the truncation τ− :  × R2 → R2 by assigning to every x ∈  and (s1 , s2 ) ∈ R2 the projection τ− (x, s1 , s2 ) of (s1 , s2 ) on the closed convex subset [u1,− (x), 0] × [u2,− (x), 0] of R2 . Finally, we define the truncation τ0 :  × R2 → R2 by letting τ0 (x, s1 , s2 ) be the projection of (s1 , s2 ) on the closed convex subset [u1,− (x), u1,+ (x)] × [u2,− (x), u2,+ (x)] of R2 for all x ∈  and (s1 , s2 ) ∈ R2 .

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255

On the basis of these truncations, let us define the following truncationsperturbations on  × R2 for f (x, s1 , s2 ) in (6.46): f+ (x, s1 , s2 ) = f (x, τ+ (x, s1 , s2 )) + (s1 − u1,+ (x))+ fs1 (x, τ+ (x, s1 , s2 )) + (s2 − u2,+ (x))+ fs2 (x, τ+ (x, s1 , s2 )) − s1− fs1 (x, τ+ (x, s1 , s2 )) − s2− fs2 (x, τ+ (x, s1 , s2 )), f− (x, s1 , s2 ) = f (x, τ− (x, s1 , s2 )) − (s1 − u1,− (x))− fs1 (x, τ− (x, s1 , s2 )) − (s2 − u2,− (x))− fs2 (x, τ− (x, s1 , s2 )) + s1+ fs1 (x, τ− (x, s1 , s2 )) + s2+ fs2 (x, τ− (x, s1 , s2 )), f0 (x, s1 , s2 ) = f (x, τ0 (x, s1 , s2 )) − (s1 − u1,− (x))− fs1 (x, τ0 (x, s1 , s2 )) − (s2 − u2,− (x))− fs2 (x, τ0 (x, s1 , s2 )) + (s1 − u1,+ (x))+ fs1 (x, τ0 (x, s1 , s2 )) + (s2 − u2,+ (x))+ fs2 (x, τ0 (x, s1 , s2 )). It is clear that f+ , f− , f0 :  × R2 → R are Carathéodory functions, which are locally Lipschitz with respect to the variables (s1 , s2 ) ∈ R2 . Moreover, by means of a careful calculation we can show that their generalized gradients with respect to the variables (s1 , s2 ) ∈ R2 reduce to the ordinary differential of f on the rectangles [0, u1,+ (x)] × [0, u2,+ (x)], [u1,− (x), 0] × [u2,− (x), 0] and [u1,− (x), u1,+ (x)] × [u2,− (x), u2,+ (x)], respectively. Precisely, denoting by f the differential of f with respect to the variables (s1 , s2 ) ∈ R2 , we have that ∂(s1 ,s2 ) f+ (x, s1 , s2 ) = {f (x, s1 , s2 )} for a.e. x ∈ , every (s1 , s2 ) ∈ [0, u1,+ (x)] × [0, u2,+ (x)],

(6.56)

∂(s1 ,s2 ) f− (x, s1 , s2 ) = {f (x, s1 , s2 )} for a.e. x ∈ , every (s1 , s2 ) ∈ [u1,− (x), 0] × [u2,− (x), 0],

(6.57)

∂(s1 ,s2 ) f0 (x, s1 , s2 ) = {f (x, s1 , s2 )} for a.e. x ∈ , every (s1 , s2 ) ∈ [u1,− (x), u1,+ (x)] × [u2,− (x), u2,+ (x)]. (6.58) These calculations are based on [55, Theorem 2.5.1]. Using the truncations-perturbations f+ , f− , f0 , we introduce three function1,p 1,p als on W0 1 () × W0 2 () by: 1 1 p p ∇u1 p11 + ∇u2 p22 − E+ (u1 , u2 ) = p1 p2

 

f+ (x, u1 (x), u2 (x)) dx,

256 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

 1 1 p1 p2 E− (u1 , u2 ) = ∇u1 p1 + ∇u2 p2 − f− (x, u1 (x), u2 (x)) dx, p1 p2  1 1 p p E0 (u1 , u2 ) = ∇u1 p11 + ∇u2 p22 − f0 (x, u1 (x), u2 (x)) dx. p1 p2  1,p

The functionals E+ , E− , E0 are locally Lipschitz continuous on W0 1 () × 1,p W0 2 (), so the notion of critical point as considered in Definition 1.41 makes sense for them. This means that (w1 , w2 ) is a critical point of E+ if (0, 0) ∈ ∂E+ (w1 , w2 ).

(6.59)

We present a location result for the critical points of the functionals E+ , E− , and E0 . To this end we need a new hypothesis: H(f )4 (a) for a.e. x ∈ , the function fs1 (x, s1 , ·) is nondecreasing on the interval [d2 , k2 ] whenever s1 ∈ [d1 , k1 ]; (b) for a.e. x ∈ , the function fs2 (x, ·, s2 ) is nondecreasing on the interval [d1 , k1 ] whenever s2 ∈ [d2 , k2 ]. We illustrate the compatibility of hypotheses H(f )1 –H(f )4 by an example. Example 6.16. Hypotheses H(f )1 –H(f )4 are satisfied for the system: ⎧ p1 +q1 −2 u + βp (u+ )p1 −1 (u+ )p2 ⎪ ⎪ 1 1 1 2 ⎪ −p1 u1 = −α(p1 + q1 )|u1 | ⎪ ⎪ ⎪ p −2 1 ⎪ + γp1 |u1 | u1 in , ⎨ p2 −1 (u+ )p1 −p2 u2 = −α(p2 + q2 )|u2 |p2 +q2 −2 u2 + βp2 (u+ 2) 1 ⎪ ⎪ ⎪ p2 −2 u ⎪ ⎪ + γp |u | 2 2 2 ⎪ ⎪ ⎩ u1 = u2 = 0

in , on ∂,

with constants p1 ≥ 2, p2 ≥ 2, α, β, q1 , q2 > 0, and γ > max{λ1,p1 /p1 , λ1,p2 /p2 }. Here we have the potential f (x, s1 , s2 ) = −α(|s1 |p1 +q1 + |s2 |p2 +q2 ) + β(s1+ )p1 (s2+ )p2 + γ (|s1 |p1 + |s2 |p2 ). Actually, for any constants k1 , k2 > 0 and d1 , d2 < 0, hypotheses H(f )1 –H(f )4 are fulfilled provided α > 0 is sufficiently large. We can describe the location of the critical points of the functionals E+ , E− , and E0 . Lemma 6.17. Assume H(f )1 –H(f )4 . Then: 1,p

1,p

if (v1 , v2 ) ∈ W0 1 () × W0 2 () is a critical point of E+ then 0 ≤ v1 (x) ≤ u1,+ (x) and 0 ≤ v2 (x) ≤ u2,+ (x) for a.e. x ∈ ;

(6.60)

Nonlinear Elliptic Systems Chapter | 6 1,p

1,p

if (v1 , v2 ) ∈ W0 1 () × W0 2 () is a critical point of E− then u1,− (x) ≤ v1 (x) ≤ 0 and u2,− (x) ≤ v2 (x) ≤ 0 for a.e. x ∈ ; 1,p

257

(6.61)

1,p

if (v1 , v2 ) ∈ W0 1 () × W0 2 () is a critical point of E0 then u1,− (x) ≤ v1 (x) ≤ u1,+ (x) and u2,− (x) ≤ v2 (x) ≤ u2,+ (x) for a.e. x ∈ . (6.62) Proof. In order to check the claim in (6.60), let (v1 , v2 ) be a critical point of E+ , i.e., (6.59) is fulfilled with (w1 , w2 ) = (v1 , v2 ). This reads as ⎧ ⎪ ⎨ −p1 v1 = g1 (x) in , (6.63) −p2 v2 = g2 (x) in , ⎪ ⎩ v 1 = v2 = 0 on ∂, with (g1 (x), g2 (x)) ∈ ∂(s1 ,s2 ) f+ (x, v1 (x), v2 (x)) for a.e. x ∈ ,

(6.64)

where [55, Theorem 2.7.5] has been applied. Since (u1,+ , u2,+ ) is a solution of (6.46), from (6.63) and (6.64) we deduce

−p1 v1 + p1 u1,+ , (v1 − u1,+ )+   = (g1 (x) − fs1 (x, u1,+ (x), u2,+ (x)))(v1 (x) − u1,+ (x)) dx  =

{v1 >u1,+ }

{v1 >u1,+ ; v2 u1,+ ; 0≤v2 ≤u2,+ }

(fs1 (x, u1,+ , v2 ) − fs1 (x, u1,+ , u2,+ ))(v1 − u1,+ ) dx,

where we essentially use that ∂(s1 ,s2 ) f+ (x, s1 , s2 ) = {f (x, s1 , s2 )} for a.e. x ∈ . H(f )4 (a) implies that

−p1 v1 + p1 u1,+ , (v1 − u1,+ )+  ≤ 0. On the basis of the strict monotonicity of the function ξ → |ξ |p1 −2 ξ on RN , we derive that v1 ≤ u1,+ . In the same way, through H(f )4 (b), we prove that v2 ≤ u2,+ . By H(f )2 and H(f )3 , we know that fs1 (x, 0, s2 ) = 0 for a.e. x ∈ , every s2 ∈ [0, k2 ]. Then from (6.63) and (6.64) it turns out that  

−p1 v1 , −v1−  = − g1 v1 dx = − fs1 (x, 0, 0)v1 dx {v1 0 such that p

f (x, s1 , s2 ) − f (x, s1 , 0) > λ1,p2

s2 2 p2

Nonlinear Elliptic Systems Chapter | 6

259

for a.e. x ∈ , every 0 < s1 ≤ k1 and 0 < s2 ≤ δ. Then, for t > 0 small enough, we find t p2 E+ (w1 , t uˆ 1,p2 ) = E+ (w1 , 0) + λ1,p2 p2  − (f (x, w1 (x), t uˆ 1,p2 (x)) − f (x, w1 (x), 0))dx 

< E+ (w1 , 0). This contradicts that (w1 , 0) is a global minimizer of E+ , so (6.67) is verified. Writing the explicit expression of (6.59) in the form of (6.63) and (6.64), we obtain on the basis of (6.60) and (6.56) that (w1 , w2 ) is a solution of problem (6.46). Hypothesis H(f )2 and (6.65) allow us to apply the strong maximum principle in Theorem 2.19 to both equations in (6.46) with (w1 , w2 ) in place of (u1 , u2 ). Thanks to (6.67) and (6.65), this entails that (w1 , w2 ) ∈ int (C01 ()+ ) × int (C01 ()+ ). Recalling (6.60) and that (u1,+ , u2,+ ) is a minimal positive solution of (6.46), we find that (w1 , w2 ) = (u1,+ , u2,+ ). In this way we can infer that (u1,+ , u2,+ ) is a local minimizer of E0 on C01 () × C01 () because the functions E+ and E0 coincide on int (C01 ()+ ) × int (C01 ()+ ). Then 1,p 1,p it follows that (u1,+ , u2,+ ) is a local minimizer of E0 on W0 1 () × W0 2 () due to the property of Sobolev versus Hölder local minimizers for quasilinear elliptic equations as shown in [91]. Arguing similarly with the functional E− in place of E+ and utilizing (6.57) instead of (6.56), we get the desired conclusion for (u1,− , u2,− ). Here is a proposition revealing how Eqs. (6.47) and (6.48) are useful for studying system (6.46). Proposition 6.19. Assume H(f )1 –H(f )4 . Then there exists (u+ , v+ ) ∈ int (C01 ()+ ) × int (C01 ()+ ) such that u+ is a solution of (6.47), v+ is a solution of (6.48), u+ ≤ u1,+ , v+ ≤ u2,+ , E+ (u+ , 0) = inf E+ (·, 0), E+ (0, v+ ) = inf E+ (0, ·).



 Similarly, there exists (u− , v− ) ∈ −int (C01 ()+ ) × −int (C01 ()+ ) such that u− is a solution of (6.47), v− is a solution of (6.48), u− ≥ u1,− , v− ≥ u2,− , E− (u− , 0) = inf E− (·, 0), E− (0, v− ) = inf E− (0, ·). 1,p

Proof. Since E+ (·, 0) : W0 1 () → R is coercive and sequentially weakly 1,p lower semicontinuous, there exists u+ ∈ W0 1 () such that E+ (u+ , 0) = inf E+ (·, 0).

260 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

This ensures that u+ is a critical point of E+ (·, 0), which means that 0 ∈ ∂E+ (·, 0)(u+ ). By hypothesis H(f )4 we have −p1 u1,+ = fs1 (x, u1,+ (x), u2,+ (x)) ≥ fs1 (x, u1,+ (x), 0), so u1,+ is a supersolution of (6.47). It follows that

−p1 u+ + p1 u1,+ , (u+ − u1,+ )+   ≤ ((f+ )s1 (x, u+ , 0) − fs1 (x, u1,+ , u2,+ ))(u+ − u1,+ ) dx = 0. {u+ >u1,+ }

From this direct comparison we obtain that 0 ≤ u+ (x) ≤ u1,+ (x) for a.e. x ∈  and u+ = 0. This ensures that u+ is solution of problem (6.47). Through the strong maximum principle (see Theorem 2.19) it follows that u+ ∈ int (C01 ()+ ). The assertions related to v+ , u− , and v− can be shown to be true along the same lines. For the proof of our main result we require the following subhomogeneous conditions: H(f )5 For any t ∈ [0, 1] we have (i) fs1 (·, ts1 , 0) ≤ t p1 −1 fs1 (·, s1 , 0) for all d1 ≤ s1 ≤ 0, (ii) fs1 (·, ts1 , 0) ≥ t p1 −1 fs1 (·, s1 , 0) for all 0 ≤ s1 ≤ k1 . H(f )6 For any t ∈ [0, 1] we have (i) fs2 (·, 0, ts2 ) ≤ t p2 −1 fs2 (·, 0, s2 ) for all d2 ≤ s2 ≤ 0, (ii) fs2 (·, 0, ts2 ) ≥ t p2 −1 fs2 (·, 0, s2 ) for all 0 ≤ s2 ≤ k2 . Remark 6.20. One can check that H(f )5 and H(f )6 are verified for Example 6.16. We are in a position to state the main result regarding problem (6.46). Theorem 6.21. Assume H(f )1 –H(f )6 and that H(f )2 holds (in a stronger form) with constants c1 > λ2,p1 and c2 > λ2,p2 . Then problem (6.46) admits at least three nontrivial solutions: a minimal positive solution (u1,+ , u2,+ ) ∈ int (C01 ()+ ) × int (C01 ()+ ), a maximal negative solution (u1,− , u2,− ) ∈ (−int (C01 ()+ )) × (−int (C01 ()+ )), and a third solution (u1,0 , u2,0 ) ∈ C01 () × C01 () such that ui,− ≤ ui,0 ≤ ui,+ in , for i = 1, 2, which is neither positive nor negative.

Nonlinear Elliptic Systems Chapter | 6

261

Proof. By Lemma 6.18 we know that (u1,+ , u2,+ ) and (u1,− , u2,− ) are local minimizers of the functional E0 . Due to (6.62), they are strict local minimizers being extremal positive and negative solutions of (6.46), respectively. Since the functional E0 is coercive, it satisfies the nonsmooth Palais–Smale condition in Definition 1.42, so we can apply to the locally Lipschitz functional E0 the nonsmooth version of the mountain pass theorem in Theorem 1.44. This yields a 1,p 1,p critical point (u1,0 , u2,0 ) ∈ W0 1 () × W0 2 () of E0 , that is, (0, 0) ∈ ∂E0 (u1,0 , u2,0 ),

(6.68)

such that max{E0 (u1,+ , u2,+ ), E0 (u1,− , u2,− )} < E0 (u1,0 , u2,0 ) = inf max E0 (γ (t)), γ ∈ 0≤t≤1

(6.69) with 1,p1

 = {γ ∈ C([0, 1], W0

1,p2

() × W0

()) : γ (0) = (u1,− , u2,− ), γ (1) = (u1,+ , u2,+ )}.

(6.70)

Exploiting the explicit expression of the generalized gradient ∂E0 (u1,0 , u2,0 ), we infer from (6.68), (6.62), and (6.58) that (u1,0 , u2,0 ) is a solution of (6.46). Moreover, using (6.69), it turns out that (u1,0 , u2,0 ) = (u1,+ , u2,+ )

and

(u1,0 , u2,0 ) = (u1,− , u2,− ).

(6.71)

We can obtain a path γˆ ∈  with the property E0 (γˆ (t)) < 0 for all t ∈ [0, 1]. The construction of such a path γˆ is performed on the pattern of the construction made in the proof of Theorem 5.5 (see the proof of [47, Theorem 4.1]). Then by (6.69) and (6.70) we have E0 (u1,0 , u2,0 ) < 0. 1,p

1,p

Therefore (u1,0 , u2,0 ) ∈ W0 1 () × W0 2 () is a nontrivial solution of problem (6.46). The nonlinear regularity theory guarantees that (u1,0 , u2,0 ) ∈ C01 () × C01 (). Finally, combining with (6.71), the extremal properties of constant-sign solutions (u1,+ , u2,+ ) and (u1,− , u2,− ) ensure that the nontrivial solution (u1,0 , u2,0 ) is neither positive nor negative. This completes the proof.

262 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

6.4 NOTES The general system (6.1) is studied in [184] from where the results in Section 6.1 are taken. The equation version of system (6.1) has been considered in [19]. The case when in (6.1) one takes μ1 = μ2 = 0, i.e., system (6.5), was investigated in [48] through the method of subsolution–supersolution (see also [38] for more particular situations). For the study of equations of type (6.6) and (6.7) exhibiting full gradient dependence, we refer to [68,80–82,212,226]. Equations with (p, q)-Laplacian but without gradient dependence in the right-hand side can be found in [143,191]. For the equations involving the p-Laplacian without gradient dependence in their right-hand side studied through variational and topological methods, we refer to [169]. See also the notes of Chapter 4. A main feature of problem (6.1) is that the parameters μ1 , μ2 enter the leading operators of the elliptic equations and not, as usually, in the right-hand sides of the elliptic equations with lower-order terms. The presence of four operators of type p-Laplacian in the principal parts of the equations, namely pi and qi for i = 1, 2, makes the problem highly nonlinear. Another important aspect of problem (6.1) is that the nonlinearities in the right-hand side depend on the solution and its gradient (which is often expressed as dealing with convection terms), which is rarely handled in the literature, especially in the system setting due to the interaction between different equations. The presence of convection terms is a major difficulty in the study mainly because such problems cannot be handled through variational methods due to the lack of variational structure. Theorem 6.1 establishes the existence of weak solutions to problem (6.1) by using the theory of pseudomonotone operators. The pseudomonotone operators are thoroughly investigated in [38,169,228,238]. Next, a uniqueness result for problem (6.1) is proven in Theorem 6.2. For the case of equations, we refer to [19] for a uniqueness result regarding equations of type (6.6), (6.7), and to [188] in the situation where the functions fi do not depend on the gradient of the solution. In Theorems 6.4, 6.5, and 6.6 we examine the asymptotic behavior of solutions to problem (6.1) as (μ1 , μ2 ) → (0, 0), (μ1 , μ2 ) → (+∞, 0), (μ1 , μ2 ) → (0, +∞), and (μ1 , μ2 ) → (+∞, +∞), respectively. Problems (6.5), (6.6), and (6.7) become limiting cases of problem (6.1), demonstrating a strong unifying effect for (6.1). These asymptotic properties are obtained through a priori estimates. In Section 6.2 we study the Dirichlet boundary value problem (6.1) through the method of subsolution–supersolution. In this respect we follow the results obtained in Motreanu–Vetro–Vetro [185]. In comparison with the existence, uniqueness, and asymptotic results with respect to the parameters (μ1 , μ2 ) given in Section 6.1, here we provide location and enclosure properties by subsolution–supersolution approach for the general system (6.1). We mention that the method of subsolution–supersolution for the case of (6.1) with μ1 = μ2 = 0 was investigated in [48], and without gradient dependence in the right-hand sides in [38]. Here we are able to provide a unifying approach. The

Nonlinear Elliptic Systems Chapter | 6

263

method of subsolution–supersolution for system of equations differs substantially from what happens in the case of single equations. It is worth noticing that for systems one cannot define separately the notions of subsolution and supersolution, but the notion of subsolution–supersolution (or trapping region). It is worth pointing out that assuming condition H(f1 , f2 ) of Section 6.2 we cannot apply the known results as available in [38,48,184], which hold under more restrictive hypotheses. Indeed, in [48] the driving differential operator is more specific, and for the nonlinearities in the reaction terms the following growth is supposed: there exist constants ci ≥ 0 and functions ρi ∈ Lpi () such that p2

|f1 (x, s1 , s2 , ξ1 , ξ2 )| ≤ ρ1 (x) + c1 (|ξ1 |p1 −1 + |ξ2 | p1 ), |f2 (x, s1 , s2 , ξ1 , ξ2 )| ≤ ρ2 (x) + c2 (|ξ1 |

p1 p2

+ |ξ2 |p2 −1 )

for a.a. x ∈  and all s = (s1 , s2 ) ∈ [u1 (x), u1 (x)]×[u2 (x), u2 (x)], ξ1 , ξ2 ∈ RN , which is more restrictive than hypothesis H(f1 , f2 ) because pi − 1 =

pi pi < ∗ pi (pi )

for i = 1, 2. In [38] the setting is even more restrictive. In [184] it is as in Section 6.1 where, due to a completely different approach, assumptions of another type are imposed, namely the generalized sign condition in Section 6.1. Theorem 6.7 establishes under hypothesis H(f1 , f2 ) the existence and location of a solution to problem (6.1) within the ordered rectangle (or trapping region) [u1 , u1 ] × [u2 , u2 ] determined by a subsolution–supersolution. The proof relies on the study of an associated auxiliary problem as demonstrated in Theorem 6.8. In comparison with previous results, the main contribution here consists in the use of a cut-off function adapted to the general growth condition in assumption H(f1 , f2 ). In Theorem 6.9 the existence of a positive solution (u1 , u2 ) of system (6.1) is shown, meaning that both components are positive. The essential point for obtaining a positive solution is the construction of an appropriate subsolution–supersolution of problem (6.1) permitting an application of Theorem 6.8. One can find in [184] criteria to get a negative solution, i.e., having both components negative, as well as hybrid solutions (u1 , u2 ), in the sense that the components u1 and u2 are of opposite constant sign. Section 6.3 is based on Carl–Motreanu [47]. The main result is stated in Theorem 6.21, which postulates the existence of three nontrivial solutions of system (6.46) in variational form. Two of these solutions are of opposite constant sign: one is positive in the sense that both components are positive and one is negative in the sense that both components are negative. They have extremal properties, which means that the former is minimal among the class of positive solutions, while the latter is maximal in the class of negative solutions.

264 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Also there exists a third nontrivial solution of system (6.46) between the extremal opposite constant-sign solutions, which is neither positive nor negative. The complete proof of Theorem 6.21 can be found in [47]. The approach relies on subsolution–supersolution for systems and on variational methods. A significant aspect of Section 6.3 is that the smooth problem (6.46) is handled by means of a nonsmooth problem involving a locally Lipschitz potential, which is not differentiable. The nonsmooth problem is smooth on the relevant domain, which makes it possible to achieve solvability for the original problem (6.46). A crucial point is that a comparison argument permits checking that the critical points of the nonsmooth functional are solutions of system (6.46) within the prescribed trapping regions. Another relevant feature is that we use some associated equations, namely (6.47) and (6.48), in the study of problem (6.46). An open problem for nonlinear elliptic systems is to find solutions for which each component is sign-changing.

Chapter 7

Singular Quasilinear Elliptic Systems 7.1 THE CASE OF SINGULARITIES WITH RESPECT TO THE SOLUTION Consider the system of quasilinear elliptic equations ⎧ −p u = f1 (u, v) in , ⎪ ⎪ ⎪ ⎨ − v = f (u, v) in , q 2 ⎪ u, v > 0 in , ⎪ ⎪ ⎩ u, v = 0 on ∂

(7.1)

on a bounded domain  ⊂ RN with a C 1,α boundary ∂ for some α ∈ (0, 1), for which we allow a singularity at zero. As usual, p and q stand for the 1,p 1,q p-Laplacian and q-Laplacian differential operators on W0 () and W0 (), respectively. For avoiding repetitive discussions, we suppose that 1 < p, q ≤ N . The reaction terms in problem (7.1) are expressed by continuous functions fi : (0, +∞) × (0, +∞) → (0, +∞), i = 1, 2, for which we assume the growth conditions:  β β m1 s1α1 s2 1 ≤ f1 (s1 , s2 ) ≤ M1 s1α1 s2 1 for all s1 , s2 > 0, with M1 , m1 > 0, and α1 ∈ R, β1 < 0 such that |α1 | − β1 < min{1, p − 1}, (7.2)  β β m2 s1α2 s2 2 ≤ f2 (s1 , s2 ) ≤ M2 s1α2 s2 2 for all s1 , s2 > 0, with M2 , m2 > 0, and β2 ∈ R, α2 < 0 such that |β2 | − α2 < min{1, q − 1}. (7.3) It is clear from conditions (7.2) and (7.3) that the system (7.1) possesses singularity at the origin. In the analysis of problem (7.1) we use the Banach spaces Lp () and 1,p W0 () endowed with the usual norms  · p and  ·  := ∇ · p , respectively, 1,q as well as the analogous spaces Lq () and W0 (). Actually, for reasons of nonlinear regularity theory, the natural spaces for solutions are the spaces 1,β C0 () = {u ∈ C 1,β () : u = 0 on ∂} with β ∈ (0, 1). The differential opNonlinear Differential Problems with Smooth and Nonsmooth Constraints. DOI: 10.1016/B978-0-12-813386-6.00007-9 265 Copyright © 2018 Elsevier Inc. All rights reserved.

266 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

erators in problem (7.1) are the p-Laplacian p u = div (|∇u|p−2 ∇u) for all 1,p 1,q u ∈ W0 () and the q-Laplacian q u = div (|∇u|q−2 ∇u) for all u ∈ W0 (). 1,p We denote by λ1,p and λ1,q the first eigenvalue of −p on W0 () and of 1,q −q on W0 (), respectively (see Proposition 1.57). Corresponding to them, we denote by uˆ 1,p the Lp -normalized positive eigenfunction of −p associated p−1 to λ1,p , that is, −p uˆ 1,p = λ1,p uˆ 1,p in , uˆ 1,p = 0 on ∂, with uˆ 1,p p = 1, and by uˆ 1,q the Lq -normalized positive eigenfunction of −q associated to q−1 λ1,q , that is, −q uˆ 1,q = λ1,q uˆ 1,q in , uˆ 1,q = 0 on ∂, with uˆ 1,q q = 1. As known from the strong maximum principle (see Theorem 2.19), there exist positive constants l1 and l2 such that l1 uˆ 1,p (x) ≤ uˆ 1,q (x) ≤ l2 uˆ 1,p (x) for all x ∈ .

(7.4)

r Recall that for any 1 < r < +∞ we denote r = r−1 . For obtaining comparison principles and a priori estimates for problem (7.1), we need some technical tools. Let the functions w1 and w2 be defined as the solutions of the equations ⎧ ⎧ α2 β ⎪ ⎪ ⎨ −q w2 = w2 in , ⎨ −p w1 = w1 1 in , and (7.5) w2 > 0 in , w1 > 0 in , ⎪ ⎪ ⎩ ⎩ w2 = 0 on ∂. w1 = 0 on ∂

From (7.5) and [101] we know that they satisfy c0 uˆ 1,p (x) ≤ w1 (x) ≤ c1 uˆ 1,p (x) and c0 uˆ 1,q (x) ≤ w2 (x) ≤ c1 uˆ 1,q (x) in , (7.6) with positive constants c0 , c1 , c0 , c1 . Consider now the functions z1 , z2 defined by ⎧ ⎨ w β1 in  \ δ , 1 (7.7) z1 = 0 on ∂ −p z1 = ⎩ −w β1 in  , 1

and

 −q z2 =

δ

w2α2

in  \ δ ,

−w2α2

in δ ,

z2 = 0 on ∂,

(7.8)

where δ = {x ∈  : d (x, ∂) < δ} with a fixed δ > 0 sufficiently small. Then (7.6) and the Hardy–Sobolev inequality in [5, Lemma 2.3] ensure that the right-hand side in (7.7) and (7.8)

Singular Quasilinear Elliptic Systems Chapter | 7

267



are elements of W −1,p () and W −1,q (), respectively. According to Proposition 1.56 and Theorem 1.55, this guarantees the existence and uniqueness of z1 and z2 in (7.7) and (7.8). On the basis of (7.5), (7.6), (7.7), (7.8), the monotonicity of the operators −p and −q , and [109, Corollary 3.1], we find the estimates c0 uˆ 1,p (x) ≤ z1 (x) ≤ c1 uˆ 1,p (x), 2 Denote

c0 uˆ 1,q (x) ≤ z2 (x) ≤ c1 uˆ 1,q (x) in . 2 (7.9)

 R := max max uˆ 1,p (x), max uˆ 1,q (x) , 

x∈

x∈

and fix a constant μ = μ(δ) > 0 such that uˆ 1,p (x), uˆ 1,q (x) ≥ μ in  \ δ . Consider the Dirichlet problems ⎧ γ1 ⎪ in , ⎨ −p ξ1 = ξ1 and ξ1 > 0 in , ⎪ ⎩ ξ1 = 0 on ∂

⎧ γ2 ⎪ ⎨ −q ξ2 = ξ2 ξ2 > 0 ⎪ ⎩ ξ2 = 0

in , in , on ∂,

(7.10)

with constants γ1 and γ2 satisfying −1 < γi < min{0, αi + βi }, i = 1, 2.

(7.11)

As known from [101], problems (7.10) possess unique solutions ξ1 and ξ2 , and there exist constants c3 ≥ c2 > 0 and c3 ≥ c2 > 0 such that c2 uˆ 1,p (x) ≤ ξ1 (x) ≤ c3 uˆ 1,p (x) and c2 uˆ 1,q (x) ≤ ξ2 (x) ≤ c3 uˆ 1,q (x).

(7.12)

For a constant C > 0, set (u, v) = C(z1 , z2 ) and (u, v) =

1 (ξ1 , ξ2 ). C

(7.13)

Useful comparison properties can be provided. Proposition 7.1. Assume that conditions (7.2) and (7.3) hold. Then, if C > 0 is small enough in (7.13), we obtain   m2 uα2 v β2 if β2 ≥ 0, m1 uα1 v β1 if α1 ≥ 0, −p u ≤ − q v ≤ m2 uα2 v β2 if β2 < 0, m1 uα1 v β1 if α1 < 0, (7.14)

268 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

and  −p u ≥

M1 uα1 v β1 M1

uα1 v β1

if α1 ≥ 0, if α1 < 0,

 − q v ≥

M2 uα2 v β2 M2

uα2 v β2

if β2 ≥ 0, if β2 < 0. (7.15)

Proof. We provide a proof only for the first inequalities in (7.14) and (7.15) because the others can be established in the same way. By (7.13) and (7.7) we find −β1

w1 1 ≤ 0 < m1 in δ ,

−β1

w1 1 ≤ 0 < m1 in δ .

u−α1 v − β1 (−p u) = −C p−1−α1 +β1 z1−α1 ξ2

u−α1 v − β1 (−p u) = −C p−1+α1 +β1 ξ1−α1 ξ2

and

β

β

Now we turn to the corresponding estimates on  \ δ . If α1 ≥ 0, by (7.13), (7.7), (7.6), (7.9), (7.12), and (7.2), it is seen that u−α1 v −β1 (−p u) −β

= C p−1−α1 +β1 z1−α1 ξ2 1 w1 1 c

−α1 0 ≤ C p−1−α1 +β1 (c3 uˆ 1,q )−β1 (c0 uˆ 1,p )β1 uˆ 1,p 2 β −α ≤ C p−1−α1 +β1 2α1 c0 1 1 (c3 R)−β1 μβ1 −α1 ≤ m1 in  \ δ β

provided the constant C > 0 is sufficiently small. If α1 < 0, we deduce from (7.13), (7.7), (7.6), (7.12), and (7.2) that u

−α1

v

−β1

(−p u) ≤ C p−1+α1 +β1 c3−α1



c0 μ c3

β1

R −α1 −β1 ≤ m1 in  \ δ

provided C > 0 is sufficiently small. Therefore the first part in (7.14) holds true. Now we prove the first inequality in (7.15). If α1 ≥ 0, we obtain from (7.13), (7.10), (7.12), (7.9), (7.4), and (7.11) that u −α1 v −β1 (−p u) ≥ C −(p−1)+α1 −β1 (c3 uˆ 1,p )γ1 −α1 γ −α ≥ C −(p−1)+α1 −β1 c31 1 γ −α1

≥ C −(p−1)+α1 −β1 c31

 

c0 l1 2 c0 l1 2



c0 uˆ 1,q 2

−β1

−β1

γ −(α1 +β1 )

1 uˆ 1,p

−β1

R γ1 −(α1 +β1 ) ≥ M1 in 

Singular Quasilinear Elliptic Systems Chapter | 7

269

provided C > 0 is sufficiently small. If α1 < 0, we note from (7.13), (7.10), (7.12), (7.9), (7.4), (7.11), and (7.2) that u−α1 v −β1 (−p u) ≥ C −(p−1)−α1 −β1

c −α1  c 0

0

2

2

−β1 l1

c31 R γ1 −(α1 +β1 ) γ

≥ M1 in  provided the constant C > 0 is sufficiently small. This completes the proof of the first inequality in (7.15), whence the proof. At this point we are able to prove the existence and regularity of solutions of the singular system (7.1). Theorem 7.2. Under hypotheses (7.2) and (7.3), problem (7.1) has a (positive) 1,β 1,β solution (u, v) ∈ C0 () × C0 () with some β ∈ (0, 1). Proof. By using the functions (u, v) and (u, v) constructed in (7.13), we define the set 

K = (y1 , y2 ) ∈ C() × C() : u ≤ y1 ≤ u and v ≤ y2 ≤ v in  . A direct verification entails that K is a closed, bounded, and convex subset of C() × C(). We define the mapping T : K → C() × C() by T (y1 , y2 ) = (u, v), where (u, v) fulfills ⎧ ⎪ ⎨ −p u = f1 (y1 , y2 ) in , (7.16) −q v = f2 (y1 , y2 ) in , ⎪ ⎩ u, v = 0 on ∂. In order to confirm that the operator T is well defined, we show that problem (7.16) admits a unique solution (u, v). Indeed, given (y1 , y2 ) ∈ K, we infer from (7.2), (7.3), (7.13), (7.9), (7.12), and (7.4) the estimates  M1 uα1 v β1 if α1 ≥ 0 α1 +β1 α1 β1 ≤ C1 uˆ 1,p f1 (y1 , y2 ) ≤ M1 y1 y2 ≤ (7.17) M1 uα1 v β1 if α1 < 0 and  β f2 (y1 , y2 ) ≤ M2 y1α2 y2 2

≤

M2 uα2 v β2

if β2 ≥ 0

M2 uα2 v β2

if β2 < 0

α +β2

2 ≤ C2 uˆ 1,q

,

(7.18)

with positive constants C1 , C2 . The estimates (7.17) and (7.18) combined with Hardy–Sobolev inequality (see [5, Lemma 2.3]) and hypotheses (7.2), (7.3) al low us to identify that f1 (y1 , y2 ) ∈ W −1,p () and f2 (y1 , y2 ) ∈ W −1,q ().

270 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Then it is sufficient to rely on Proposition 1.56 and Theorem 1.55 for obtaining the existence and uniqueness of solution (u, v) in (7.16). The regularity theory up to the boundary as found in [129] if αi + βi ≥ 0 and in [101, Theorem B.1] if αi + βi < 0 enables us to get that (u, v) ∈ 1,β 1,β C0 () × C0 (), with some β ∈ (0, 1). Moreover, (7.17), (7.16), and the a priori estimates for singular problems (see [109, Lemma 3.1] or [101]) provide a constant M > 0 such that uC 1,β () , vC 1,β () ≤ M 0

(7.19)

0

whenever (u, v) = T (y1 , y2 ) with (y1 , y2 ) ∈ K. Let us prove the continuity of the operator T . If (y1,n , y2,n ) ∈ K is a sequence with (y1,n , y2,n ) → (y1 , y2 ) in C() × C(), denote (un , vn ) = T (y1,n , y2,n ), which explicitly means that   p−2 |∇un | ∇un · ∇ϕ dx = f1 (y1,n , y2,n )ϕ dx, (7.20)   |∇vn |q−2 ∇vn · ∇ψ dx = f2 (y1,n , y2,n )ψ dx (7.21) 

 1,p

1,q

for all (ϕ, ψ) ∈ W0 () × W0 (). By (7.19) the sequences {un } and {vn } 1,p 1,q are bounded in W0 () and W0 (), respectively. Hence, along a relabeled subsequence, the weak convergence (un , vn ) (u, v) with some (u, v) ∈ 1,p 1,q W0 () × W0 () holds. Inserting ϕ = un − u in (7.20) and ψ = vn − v in (7.21) leads to lim −p un , un − u = lim −q vn , vn − v = 0.

n→∞

n→∞

This conclusion is achieved via Lebesgue’s dominated convergence theorem, which can be applied taking into account (7.17), (7.18), (7.19), and combining with Hardy–Sobolev inequality when αi + βi < 0. Consequently, invok1,p ing Proposition 1.56, the (S)+ -property of −p on W0 () and of −q 1,q 1,p 1,q on W0 () ensures that un → u in W0 () and vn → v in W0 (). Due to (7.20) and (7.21), we obtain (u, v) = T (y1 , y2 ). Furthermore, the bounded1,β 1,β ness of the sequence {(un , vn )} in C0 () × C0 () as known from (7.19) 1,β yields (un , vn ) → (u, v) in C() × C() because the embedding C0 () ⊂ C() is compact. We conclude that T is continuous. Actually, we can establish that the nonlinear operator T : K → C() × C() is compact. Toward this, we point out that the estimates (7.19) and the 1,β compactness of the embedding C0 () ⊂ C() show that T (K) is a relatively compact subset of C() × C(). The compactness of the nonlinear map T follows.

Singular Quasilinear Elliptic Systems Chapter | 7

271

Next we prove the inclusion T (K) ⊂ K. Let (y1 , y2 ) ∈ K and denote (u, v) = T (y1 , y2 ). Making use of the definitions of K and T as given before, as well as of Proposition 7.1 and (7.17), results in  −p u(x) = f1 (y1 (x), y2 (x)) ≤

M1 u(x)α1 v(x)β1

if α1 ≥ 0

M1 u(x)α1 v(x)β1

if α1 < 0

≤ −p u(x) in . Along the same pattern, we are able to find −q v(x) = f2 (y1 (x), y2 (x)) ≤ −q v(x) in . The same reasoning based on Proposition 7.1 and hypotheses (7.1), (7.2) permits getting  −p u(x) = f1 (y1 (x), y2 (x)) ≥

m1 u(x)α1 v(x)β1

if α1 ≥ 0

u(x)α1 v(x)β1

if α1 < 0

m1

≥ −p u(x) in  and −q v(x) = f2 (y1 (x), y2 (x)) ≥ −q v(x) in . Then the strict monotonicity of the operators −p and −q implies that (u, v) ∈ K, which reads as T (K) ⊂ K. We have thus checked that all the requirements to apply Schauder’s fixed point theorem (see, e.g., [237, p. 57]) with the set K and the map T : K → K are fulfilled. This theorem ensures the existence of (u, v) ∈ K satisfying (u, v) = T (u, v). According to the definition of T , we notice that (u, v) is a (positive) solution of problem (7.1). In addition, the fact that the solution (u, v) satisfies (u, v) ∈ K and that the growth conditions (7.2) and (7.3) hold enable us to apply the regularity theory up to the boundary (see [129] for nonsingular degenerate elliptic equations and [101,109] for singular degenerate elliptic equations) to 1,β 1,β infer that (u, v) ∈ C0 () × C0 () with β ∈ (0, 1). The proof is complete.

7.2 THE CASE OF SINGULARITIES WITH RESPECT TO THE GRADIENT OF THE SOLUTION In this section we focus on a class of singular systems where the singularity can also involve the gradient of the solution, which was not possible in the case of the setting in Section 7.1.

272 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Let  ⊂ RN (N ≥ 2) be a bounded domain with C 2 boundary ∂. Given 1 < p, q < +∞, we consider the quasilinear elliptic problem ⎧ −p u = f (u, v, ∇u, ∇v) in , ⎪ ⎪ ⎪ ⎨ − v = g (u, v, ∇u, ∇v) in , q (7.22) ⎪ u, v > 0 in , ⎪ ⎪ ⎩ u, v = 0 on ∂. Here, as before, p and q stand for the p-Laplacian and q-Laplacian differen1,p 1,q tial operators on W0 () and W0 (), respectively. We stress the presence of the convection in the expression of problem (7.22) meaning the dependence on the solution and its gradient for the terms f (u, v, ∇u, ∇v) and g(u, v, ∇u, ∇v). Moreover, these terms can be singular both in the solution (u, v) and its gradient (∇u, ∇v). We assume that f, g : (0, +∞) × (0, +∞) × (RN \ {0}) × (RN \ {0}) → (0, +∞) are continuous functions satisfying the growth conditions: m1 (s1 + |ξ1 |)α1 (s2 + |ξ2 |)β1 ≤ f (s1 , s2 , ξ1 , ξ2 ) ≤ M1 (s1 + |ξ1 |)α1 (s2 + |ξ2 |)β1 (7.23) and m2 (s1 + |ξ1 |)α2 (s2 + |ξ2 |)β2 ≤ g(s1 , s2 , ξ1 , ξ2 ) ≤ M2 (s1 + |ξ1 |)α2 (s2 + |ξ2 |)β2 (7.24) for all s1 , s2 > 0, (ξ1 , ξ2 ) ∈ (RN \ {0}) × (RN \ {0}), with constants m1 , m2 , M1 , M2 > 0, α2 , β1 < 0 and α1 , β2 ∈ R such that α1 β2 > 0

(7.25)

|α1 | − β1 < p − 1 and |β2 | − α2 < q − 1.

(7.26)

and

According to our assumptions, we notice that all the exponents α1 , α2 , β1 , β2 can be negative. To give just an example, we can choose α1 = β2 = − 12 , α2 = β1 = − 14 and p, q ≥ 2. Another relevant feature of our problem is that the assumptions imposed on the functions f and g make the system (7.22) competitive. This is because α2 , β1 < 0, which generally prevents f and g from being increasing with respect to v and u, respectively. We keep the notation of Section 7.1. In particular, we denote by λ1,p and λ1,q 1,p 1,q the first eigenvalues of −p on W0 () and of −q on W0 (), respectively. p In addition, let uˆ 1,p be the L -normalized positive eigenfunction of −p corp−1 responding to λ1,p , that is, −p uˆ 1,p = λ1,p uˆ 1,p in , uˆ 1,p = 0 on ∂, with

Singular Quasilinear Elliptic Systems Chapter | 7

273

uˆ 1,p p = 1. Similarly, let uˆ 1,q be the Lq -normalized positive eigenfunction of q−1 −q corresponding to λ1,q , that is, −q uˆ 1,q = λ1,q uˆ 1,q in , uˆ 1,q = 0 on r ∂, with uˆ 1,q q = 1. For any r ∈ (1, +∞), we denote r = r−1 (the conjugate exponent). Set, as in Section 7.1,   R := max max uˆ 1,p , max uˆ 1,q . (7.27) 



We recall from Proposition 1.57 the fundamental property of uˆ 1,p and uˆ 1,q related to the normal derivatives on the boundary ∂, namely ∂ uˆ 1,p /∂ν < 0 and ∂ uˆ 1,q /∂ν < 0 on ∂, where ν stands for the outer unit normal. In Proposition 1.57 this property is contained in the relation uˆ 1,p , uˆ 1,q ∈ int (C01 ()+ ). Consequently, we can fix numbers μ1 > 0 and μ2 > 0 such that     (7.28) uˆ 1,p + ∇ uˆ 1,p  ≥ μ1 and uˆ 1,q + ∇ uˆ 1,q  ≥ μ2 in . 1 This is certainly possible because, in view of uˆ 1,p , uˆ 1,q ∈ int (C 0 ()+ ),    the 1    properties uˆ 1,p , uˆ 1,q ∈ C0 (), uˆ 1,p , uˆ 1,q > 0 in  and ∇ uˆ 1,p , ∇ uˆ 1,q  > 0 on ∂ hold. Let us also recall from [33, Lemma 1] that if g ∈ L∞ () and if u ∈ W01,r () (r ∈ {p, q}) is the weak solution of the equation  −r u = g in , (7.29) u=0 on ∂,

then there exist a positive constant K(r) depending only on r, N , and , such that 1

∇uL∞ () ≤ K(r)(g∞ ) r−1 .

(7.30)

It is well known that the functions ξ1 and ξ2 defined as the solutions of the equations   −q ξ2 = 1 in , −p ξ1 = 1 in , and (7.31) ξ1 = 0 ξ2 = 0 on ∂ on ∂ belong to C01,α () for certain α ∈ (0, 1). Furthermore, there exist positive constants c1 and c2 such that ξ1 (x) ≥ c1 uˆ 1,p (x) and ξ2 (x) ≥ c2 uˆ 1,q (x) for all x ∈ .

(7.32)

Denote

    L1 = max ξ1 C 1,α () , K(p) and L2 = max ξ2 C 1,β () , K(q) , 0

0

(7.33)

274 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

 θ (α1 , β2 ) =

−1 if α1 , β2 > 0, 0 if α1 , β2 < 0,

(7.34)

which makes sense in view of (7.25). For any sufficiently large constant C > 1 that will be later selected, we set (u, v) = (C −(q−1)

θ(α1 , β2 )

uˆ 1,p , C −(p−1)

θ(α1 , β2 )

uˆ 1,q )

(7.35)

and (u, v) = (C (p−1)

θ(α1 , β2 )

ξ1 , C (q−1)

θ(α1 , β2 )

ξ2 ).

(7.36)

Here θ (α1 , β2 ) is given in (7.34). From (7.35), (7.36), and (7.32) it turns out that if C is sufficiently large, for all x ∈  one has u(x) = C −(q−1) ≤C v(x) = C ≤C

θ(α1 , β2 )

(p−1)θ(α1 , β2 )

c1 uˆ 1,p (x)

θ(α1 , β2 )

c2 uˆ 1,q (x)

ξ1 (x) = u(x),

−(p−1)θ(α1 , β2 ) (q−1)θ(α1 , β2 )

θ(α1 , β2 )

uˆ 1,p (x) ≤ C (p−1) uˆ 1,q (x) ≤ C (q−1)

ξ2 (x) = v(x).

The functions in (7.35) and (7.36) allow us to introduce the sets to be utilized for developing our fixed point argument. Let us set  θ(α1 , β2 ) L1 in  and K1,C := y ∈ C01 () : u ≤ y ≤ u in , ∇y∞ ≤ C (p−1)  ∂y θ(α1 , β2 ) − |∇ uˆ 1,p | on ∂ , ≥ C −(q−1) ∂ν  θ(α1 , β2 ) K2,C := y ∈ C01 () : v ≤ y ≤ v in , ∇y∞ ≤ C (q−1) L2 in  and  ∂y θ(α1 , β2 ) − |∇ uˆ 1,q | on ∂ . ≥ C −(p−1) ∂ν Combining the definitions of K1,C , K2,C and formula (7.28), for every (y1 , y2 ) ∈ K1,C × K2,C we have the estimates y1 + |∇y1 | ≥ u + |∇y1 | ≥ C −(q−1)

θ(α1 , β2 )

μ0

(7.37)

μ0 ,

(7.38)

and y2 + |∇y2 | ≥ v + |∇y2 | ≥ C −(p−1)

θ(α1 , β2 )

for some μ0 > 0. We need to establish comparison results for problem (7.22). First, we analyze the case α1 , β2 > 0.

Singular Quasilinear Elliptic Systems Chapter | 7

275

Proposition 7.3. Assume that conditions (7.23), (7.24), and (7.26) are satisfied with α1 , β2 > 0. Then, taking C > 1 large enough in (7.35) and (7.36), we have 1

−p u ≤ m1 (u + |∇y1 |)α1 (v + L2 C q−1 )β1 in ,

(7.39)

−q v ≤ m2 (u + L1 C

1 p−1

)α2 (v + |∇y2 |)β2 in ,

(7.40)

−p u ≥ M1 (u + L1 C

1 p−1

) (v + |∇y2 |)

in ,

(7.41)

)β2 in 

(7.42)

α1

β1

−q v ≥ M2 (u + |∇y1 |)α2 (v + L2 C

1 q−1

for all (y1 , y2 ) ∈ K1,C × K2,C . Proof. By (7.35), (7.36), (7.34), (7.33), (7.37), (7.27), and (7.26), we derive for every (y1 , y2 ) ∈ K1,C × K2,C that 1

(u + |∇y1 |)−α1 (v + L2 C q−1 )−β1 (−p u) 1

1

= (u + |∇y1 |)−α1 (C q−1 ξ2 + L2 C q−1 )−β1 C ≤C ≤C

−(p−1)+α1 β1 − q−1 q−1 α1 −β1 −p+1 q−1

−(p−1) q−1

p−1

uˆ 1,p

−β1 1 μ−α uˆ 1,p 0 (2L2 )

p−1

−β1 p−1 1 μ−α R ≤ m1 in  0 (2L2 )

provided that C > 1 is sufficiently large. This implies (7.39). From (7.35), (7.36), (7.34), (7.33), (7.38), (7.27), and (7.26), for every (y1 , y2 ) ∈ K1,C × K2,C we obtain 1

(u + L1 C p−1 )−α2 (v + |∇y2 |)−β2 (−q v) 1

1

= (C p−1 ξ1 + L1 C p−1 )−α2 (v + |∇y2 |)−β2 C ≤C

−α2 +β2 −q+1 p−1

−β2

(2L1 )−α2 μ0

−q+1 p−1

q−1

uˆ 1,q

R q−1 ≤ m2 in 

provided that C > 1 is sufficiently large, which ensures (7.40). By (7.36), (7.34), (7.31), (7.33), (7.38), (7.26), and β1 < 0, for every (y1 , y2 ) ∈ K1,C × K2,C it follows that 1

(u + L1 C p−1 )−α1 (v + |∇y2 |)−β1 (−p u) p−1

1

1

= C p−1 (C p−1 ξ1 + L1 C p−1 )−α1 (v + |∇y2 |)−β1 ≥C

p−1+β1 −α1 p−1

−β1

(2L1 )−α1 μ0

≥ M1 in 

provided that C > 1 is sufficiently large, which proves (7.41). By (7.36), (7.34), (7.31), (7.33), (7.37), (7.26), and α2 < 0, for every (y1 , y2 ) ∈ K1,C × K2,C we derive

276 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints 1

(u + |∇y1 |)−α2 (v + L2 C q−1 )−β2 (−q v) q−1

1

1

= C q−1 (u + |∇y1 |)−α2 (C q−1 ξ2 + L2 C q−1 )−β2 ≥C

q−1+α2 −β2 q−1

−β2 2 μ−α ≥ M2 in  0 (2L2 )

provided that C > 1 is sufficiently large. So, we infer (7.42), which completes the proof. In the next proposition we discuss the case α1 , β2 < 0 that can also occur under condition (7.25). Proposition 7.4. Assume that conditions (7.23), (7.24), and (7.26) are satisfied with α1 , β2 < 0. Then, for C > 1 large enough in (7.35) and (7.36), we have −p u ≤ m1 (u + L1 C)α1 (v + L2 C)β1 in ,

(7.43)

−q v ≤ m2 (u + L1 C) (v + L2 C)

(7.44)

α2

β2

in ,

−p u ≥ M1 (u + |∇y1 |)α1 (v + |∇y2 |)β1 in ,

(7.45)

−q v ≥ M2 (u + |∇y1 |) (v + |∇y2 |)

(7.46)

α2

β2

in 

for all (y1 , y2 ) ∈ K1,C × K2,C . Proof. From (7.35), (7.36), (7.34), (7.33), (7.27), and (7.26), we find that (u + L1 C)−α1 (v + L2 C)−β1 (−p u) = (Cξ1 + L1 C)−α1 (Cξ2 + L2 C)−β1 C −p+1 uˆ 1,p

p−1

≤ C −p+1−α1 −β1 (2L1 )−α1 (2L2 )−β1 uˆ 1,p

p−1

−β1

1 ≤ C −p+1−α1 −β1 2−α1 −β1 L−α 1 L2

R p−1 ≤ m1 in 

provided that C > 1 is sufficiently large, which proves (7.43). Using (7.35), (7.36), (7.34), (7.33), (7.27), and (7.26), leads to (u + L1 C)−α2 (v + L2 C)−β2 (−q v) = (Cξ1 + L1 C)−α2 (Cξ2 + L2 C)−β2 C −q+1 uˆ 1,q

q−1

−β2

2 ≤ C −q+1−α2 −β2 2−α2 −β2 L−α 1 L2

R q−1 ≤ m2 in 

provided that C > 1 is sufficiently large, hence (7.44) ensues. According to (7.36), (7.31), (7.34), (7.37), (7.38), and (7.26), for every (y1 , y2 ) ∈ K1,C × K2,C we obtain (u + |∇y1 |)−α1 (v + |∇y2 |)−β1 (−p u) = C p−1 (u + |∇y1 |)−α1 (v + |∇y2 |)−β1

Singular Quasilinear Elliptic Systems Chapter | 7 −α1 −β1

≥ C p−1+α1 +β1 μ0

277

≥ M1 in 

provided that C > 1 is sufficiently large. This shows that (7.45) is true. By (7.36), (7.31), (7.34), (7.37), (7.38), and (7.26), for every (y1 , y2 ) ∈ K1,C × K2,C we can derive (u + |∇y1 |)−α2 (v + |∇y2 |)−β2 (−q v) = C q−1 (u + |∇y1 |)−α2 (v + |∇y2 |)−β2 −α2 −β2

≥ C q−1+α2 +β2 μ0

≥ M2 in 

provided that C > 1 is sufficiently large, so (7.46) holds. This completes the proof. We are now able to provide the existence of (positive) smooth solutions for the singular system (7.22). Theorem 7.5. Assume that conditions (7.23), (7.24), (7.25), and (7.26) hold. Then problem (7.22) has a positive, smooth solution (u, v) in C01,α () × C01,α () for some α ∈ (0, 1). Proof. We use the sets K1,C and K2,C constructed before. Define the map T : K1,C × K2,C → C01 () × C01 () by T (y1 , y2 ) = (u, v), where (u, v) is required to satisfy ⎧ ⎪ ⎨ −p u = f (y1 , y2 , ∇y1 , ∇y2 ) in , (7.47) −q v = g(y1 , y2 , ∇y1 , ∇y2 ) in , ⎪ ⎩ u = v = 0 on ∂. It is essential to point out that the operator T is well defined for C > 1 large enough because problem (7.47) has a unique smooth solution. Indeed, for (y1 , y2 ) ∈ K1,C × K2,C with C > 1 sufficiently large, we derive from (7.23), (7.25), (7.34), (7.35), (7.36), (7.33), (7.37), (7.38), and (7.26) the estimate f (y1 , y2 , ∇y1 , ∇y2 ) ≤ M1 (y1 + |∇y1 |)α1 (y2 + |∇y2 |)β1 ⎧ 1 ⎨ p−1 )α1 (y + |∇y |)β1 if α1 > 0 2 2 ≤ M1 (u + L1 C ⎩ (y + |∇y |)α1 (y + |∇y |)β1 if α1 < 0 1 1 2 2 ⎧ −β1 ⎨ α1p−1 β (2L1 )α1 μ0 1 if α1 > 0 C ≤ M1 ⎩ −α1 −β1 α1 +β1 μ0 if α1 < 0 C  C if α1 > 0 1+θ(α1 ,β2 ) ≤ in . = C (p−1) p−1 if α1 < 0 C

(7.48)

278 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Similarly, from (7.24), (7.25), (7.34), (7.35), (7.36), (7.33), (7.37), (7.38), and (7.26), for (y1 , y2 ) ∈ K1,C × K2,C and taking the constant C > 1 sufficiently large, we have g(y1 , y2 , ∇y1 , ∇y2 ) ≤ M2 (y1 + |∇y1 |)α2 (y2 + |∇y2 |)β2 ⎧ 1 ⎨ α2 q−1 β2 if β2 > 0 ≤ M2 (y1 + |∇y1 |) (v + L2 C ) ⎩ (y + |∇y |)α2 (y + |∇y |)β2 if β2 < 0 1 1 2 2 ⎧ 2 +β2 ⎨ −αq−1 μα0 2 (2L2 )β2 if β2 > 0 C ≤ M2 ⎩ −α2 −β2 α2 +β2 μ0 if β2 < 0 C  C if β2 > 0 1+θ(α1 ,β2 ) in . = C (q−1) ≤ q−1 if β2 < 0 C

(7.49)

In particular, the estimates obtained in (7.48) and (7.49) imply that f (y1 , y2 , ∇y1 , ∇y2 ) ∈ W −1,p () and g(y1 , y2 , ∇y1 , ∇y2 ) ∈ W −1,q (). Then Proposition 1.56 and Theorem 1.55 ensure the existence and uniqueness of solution (u, v) in (7.47). Moreover, the nonlinear regularity theory up to the boundary in [129] guarantees that (u, v) ∈ C01,α () × C01,α (), with some α ∈ (0, 1). In addition, due to the estimates (7.48) and (7.49), there exists a constant M > 0 such that uC 1,α () , vC 1,α () ≤ M 0

(7.50)

0

whenever (u, v) = T (y1 , y2 ) with (y1 , y2 ) ∈ K1,C × K2,C . The (weak) solutions of problem (7.22) coincide with the fixed points of the map T introduced in (7.47). It is straightforward to check that the set K1,C × K2,C is nonempty, closed, bounded, and convex in C01 () × C01 (). In order to check that the map T : K1,C × K2,C → C01 () × C01 () is continuous and compact, let (y1,n , y2,n ) ∈ K1,C × K2,C with (y1,n , y2,n ) → (y1 , y2 ) in C01 () × C01 () as n → ∞. Set (un , vn ) = T (y1,n , y2,n ), which reads as   p−2 |∇un | ∇un · ∇ϕ dx = f (y1,n , y2,n , ∇y1,n , ∇y2,n )ϕ dx, (7.51)   |∇vn |q−2 ∇vn · ∇ψ dx = g(y1,n , y2,n , ∇y1,n , ∇y2,n )ψ dx (7.52) 

 1,p 1,q W0 () × W0 ().

for all (ϕ, ψ) ∈ Inserting ϕ = un − u in (7.51) and ψ = vn − v in (7.52), we see from (7.48) and (7.49) that lim −p un , un − u = lim −q vn , vn − v = 0.

n→∞

n→∞ 1,p

1,q

Then the (S)+ -property of −p on W0 () and of −q on W0 (), which 1,p was stated in Proposition 1.56, ensures that un → u in W0 () and vn → v

Singular Quasilinear Elliptic Systems Chapter | 7

279

1,q

in W0 (). Hence, we can pass to the limit in (7.51) and (7.52). This results in (u, v) = T (y1 , y2 ). Furthermore, since the embedding C01,α () ⊂ C01 () is compact, the boundedness of the sequence {(un , vn )} in C01,α () × C01,α () as known from (7.50) has the effect that up to a subsequence (un , vn ) → (u, v) in C01 () × C01 (). Therefore the map T is continuous. Again through (7.50) it follows that T (K1,C × K2,C ) is bounded in C01,α () × C01,α (). Consequently, the compactness of the embedding C01,α () ⊂ C01 () ensures that T (K1,C × K2,C ) is a relatively compact subset of C01 () × C01 (), which proves the claim. A fundamental step in the proof is to show the invariance property T (K1,C × K2,C ) ⊂ K1,C × K2,C . For any (y1 , y2 ) ∈ K1,C × K2,C , we write as before (u, v) = T (y1 , y2 ). By the definitions of K1,C , K2,C , T and in view of (7.23), (7.24), (7.47), (7.33), (7.34), Propositions 7.3 and 7.4 (specifically, (7.39)–(7.42) and (7.43)–(7.46)), it turns out that −p u = f (y1 , y2 , ∇y1 , ∇y2 ) ≤ M1 (y1 + |∇y1 |)α1 (y2 + |∇y2 |)β1 ⎧ 1 ⎨ p−1 )α1 (v + |∇y |)β1 if α1 > 0 ≤ − u in , 2 ≤ M1 (u + L1 C p ⎩ (u + |∇y |)α1 (v + |∇y |)β1 if α1 < 0 1 2 − q v = g(y1 , y2 , ∇y1 , ∇y2 ) ≤ M2 (y1 + |∇y1 |)α2 (y2 + |∇y2 |)β2 ⎧ 1 ⎨ α2 q−1 β2 if β2 > 0 ≤ − v in , ≤ M2 (u + |∇y1 |) (v + L2 C ) q ⎩ (u + |∇y |)α2 (v + |∇y |)β2 if β2 < 0 1 2 − p u = f (y1 , y2 , ∇y1 , ∇y2 ) ≥ m1 (y1 + |∇y1 |)α1 (y2 + |∇y2 |)β1 ⎧ 1 ⎨ α1 q−1 β1 if α1 > 0 ≥ − u in , (7.53) ≥ m1 (u + |∇y1 |) (v + L2 C ) p ⎩ (u + L C)α1 (v + L C)β1 if α1 < 0 1 2 − q v = g(y1 , y2 , ∇y1 , ∇y2 ) ≥ m2 (y1 + |∇y1 |)α2 (y2 + |∇y2 |)β2 ⎧ 1 ⎨ p−1 )α2 (v + |∇y |)β2 if β2 > 0 ≥ − v in . 2 ≥ m2 (u + L1 C q ⎩ (u + L C)α2 (v + L C)β2 if β2 < 0 1 2

(7.54)

Thus we get the inequality −p u ≤ −p u in . Acting with (u − u)+ ∈ 1,p W0 () yields  {u>u}

(|∇u|p−2 ∇u − |∇u|p−2 ∇u) · (∇u − ∇u) dx

= −p u + p u, (u − u)+ ≤ 0.

280 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

The strict monotonicity of the map ξ → |ξ |p−2 ξ on RN implies that the set {u > u} = {x ∈  : u(x) > u(x)} has zero Lebesgue measure. Therefore the inequality u ≤ u a.e. in  is valid. Actually, proceeding in the same way, the estimates shown before entail u ≤ u ≤ u and v ≤ v ≤ v in .

(7.55)

On the other hand, using (7.47), (7.48), (7.49), we infer through (7.29) and (7.30) that ∇u∞ ≤ K(p)C (p−1)

θ(α1 , β2 )

and ∇v∞ ≤ K(q)C (q−1)

θ(α1 , β2 )

.

In view of (7.33), we get ∇u∞ ≤ L1 C (p−1)

θ(α1 ,β2 )

and ∇v∞ ≤ L2 C (q−1)

θ(α1 ,β2 )

.

(7.56)

The nonlinear comparison principle (see, e.g., [61]) applied to (7.53) and (7.54) provides the inequalities ∂u ∂u ∂v ∂v ≤ and ≤ on ∂. ∂ν ∂ν ∂ν ∂ν

(7.57)

The strong maximum principle applied to uˆ 1,p and uˆ 1,q yields that their normal derivatives are negative on the boundary ∂ (recall that uˆ 1,p , uˆ 1,q ∈ int (C01 ()+ ) as known from Proposition 1.57). This fact, (7.57) and (7.35) imply  ∂u ∂u   θ(α1 , β2 )  ∇ uˆ 1,p  on ∂, ≥− = ∇u = C −(q−1) ∂ν ∂ν  ∂v ∂v   θ(α1 ,β2 )  ∇ uˆ 1,q  on ∂. − ≥− = ∇v = C −(p−1) ∂ν ∂ν

−

(7.58) (7.59)

Gathering (7.55), (7.56), (7.58), and (7.59) enables us to conclude that (u, v) ∈ K1,C × K2,C . Therefore the inclusion T (K1,C × K2,C ) ⊂ K1,C × K2,C holds true. We have checked all the conditions required to apply Schauder’s fixed point theorem to the map T : K1,C × K2,C → K1,C × K2,C . We conclude that there exists a fixed point (u, v) ∈ K1,C × K2,C , that is, (u, v) = T (u, v). Thanks to the definition of the map T , it turns out that (u, v) is a (positive) smooth solution of problem (7.22).

7.3 NOTES Section 7.1 is based on Motreanu–Moussaoui [172] treating a singular quasilinear elliptic system where the singularities are exhibited by the vector field (f1 , f2 ) and involve the solution (u, v). The singularities in problem (7.1) are located at 0 and arise through a competitive structure of the nonlinearities f1 (u, v)

Singular Quasilinear Elliptic Systems Chapter | 7

281

and f2 (u, v). The singularities occur because the exponents β1 and α2 are negative (see (7.2) and (7.3)). In particular, this prevents f1 and f2 from being increasing with respect to v and u, respectively. Consequently, the subsolution– supersolution method cannot be directly implemented in the case of the system (7.1). Additional assumptions are needed as it is apparent within an approach by the method of subsolution–supersolutions in Giacomoni–Schindler– Takáˇc [101]. A different existence result under another type of hypotheses and through adequate truncations can be found in Motreanu–Moussaoui [171]. We also mention the recent result in Alves–Moussaoui [7] where it is allowed to have certain convection terms. The semilinear case in problem (7.1) (i.e., p = q = 2) was handled in Ghergu [96] and Moussaoui–Khodja–Tas [190] using essentially the linearity of the principal part in the system. The complementary situation for the system (7.1) with respect to our setting is what is called the cooperative structure, which means to have positive exponents β1 and α2 in conditions (7.1) and (7.3). For this case we refer to [77,99,100,170]. Theorem 7.2 establishes the existence and regularity of (positive) solutions for problem (7.1). We emphasize that both components of the solution are positive. The proof relies on comparison arguments, which allow us to get a priori estimates. Using these estimates enables us to obtain the main result by applying the Schauder’s fixed point theorem to a fixed point problem associated to the system (7.1). It is also worth mentioning that a careful inspection of the proof of Proposition 7.1 reveals that the constant C > 0 in the definition of the subsolution–supersolutions (u, v) and (u, v) in (7.13) can be precisely estimated. Section 7.2 follows closely Motreanu–Moussaoui–Zhang [173], where for the first time the system (7.1) is studied with convection terms f (u, v, ∇u, ∇v), g(u, v, ∇u, ∇v) exhibiting singularities in both the solution and its gradient. Without loss of generality the singularity is located at zero. The singularity involving the gradient of the solution is a novelty in the literature. Singular systems with convection terms and singularities only in the solution were examined in the semilinear case, i.e., p = q = 2, by Alves–Carriao–Faria [4] and by Alves–Moussaoui [7]. For singular elliptic systems without gradient dependence in the lower order terms, we refer to [5,6,61,77,96,97,111,158,170–172]. The main result in Section 7.2 is Theorem 7.5 providing a positive solution with regularity information for the singular system (7.22) with singularities for the functions f and g in the right-hand side of the equations possibly occurring not only for u and v but also for their gradients ∇u and ∇v as it is apparent from conditions (7.23)–(7.26). The singularities in the gradient of the solutions produce serious technical difficulties because there are no available estimates for the gradient when it involves singularities. Relevant comparison arguments and a priori estimates for the singular system (7.22) are indicated in Proposition 7.3, which permits us in the proof of Theorem 7.5 to control the gradients ∇u and ∇v despite the fact that they can be singular. This enables us to apply the Schauder’s fixed point theorem. The most challenging point in the fixed point

282 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

approach is to identify the appropriate set where the Schauder’s fixed point theorem should be applied taking into account the behavior of the gradient up to the boundary of the domain. It is worth pointing out a close analogy with the method of subsolutions–supersolutions in a singular framework. An open problem for nonlinear elliptic problems with singularities in the solutions and gradients of the solutions is to establish the existence of multiple positive solutions.

Chapter 8

Evolutionary Variational and Quasivariational Inequalities 8.1 EVOLUTIONARY VARIATIONAL INEQUALITIES First, we present some notation. For a Banach space Z with the topological dual Z ∗ , the duality pairing between Z and Z ∗ will be denoted by ·, ·Z . By  · Z we denote the norm on Z. The domain and the graph of a multivalued ∗ map F : Z → 2Z are given by D(F ) := {x ∈ Z : F (x) = ∅}, and G(F ) := {(x, w) : x ∈ D(F ), w ∈ F (x)}, respectively. We denote the (effective) domain of a functional  : Z → R ∪ {+∞}, which is ≡ +∞, by D() := {x ∈ Z : (x) < +∞}. The strong and weak convergence are specified by → and , respectively. We need multivalued versions of some definitions formulated in the singlevalued setting of Section 1.3. ∗

Definition 8.1. Given a Banach space X, let F : X → 2X be a multivalued map. (i) F is called monotone, if u − v, x − yX ≥ 0 for every (x, u), (y, v) ∈ G(F ). (ii) F is called maximal monotone, if F is monotone and G(F ) is not included in the graph of any other monotone map from X to X∗ . (iii) F is called generalized pseudomonotone if for any sequence {(xn , wn )} ⊂ G(F ) with xn  x and wn  w such that lim supwn , xn − xX ≤ 0, we have w ∈ F (x) and wn , xn X → w, xX .

n→∞

∗

Definition 8.2. A multivalued map F : X → 2X on a Banach space X is called pseudomonotone if it satisfies the conditions: (P M1) For each x ∈ X, the set F (x) is nonempty, closed, and convex in X ∗ . (P M2) If {(xn , wn )} ⊂ G(F ) is a sequence such that xn  x and lim supwn , xn − xX ≤ 0, n→∞

then for each y ∈ X there exists w(y) ∈ F (x) satisfying lim infwn , xn − yX ≥ w(y), x − yX . n→∞

Nonlinear Differential Problems with Smooth and Nonsmooth Constraints. DOI: 10.1016/B978-0-12-813386-6.00008-0 283 Copyright © 2018 Elsevier Inc. All rights reserved.

284 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

(P M3) The map F is upper semicontinuous from each finite-dimensional subspace of X to the space X ∗ endowed with the weak topology. Remark 8.3. It is shown in Kenmochi [117, Lemma 1.2] that, given a reflexive ∗ Banach space X, a map F : X → 2X with D(F ) = X verifying conditions (P M1) and (P M2) of Definition 8.2 is pseudomonotone if it satisfies: (P M4) For each x ∈ X and for each bounded subset B of X, there exists a constant c(B, x) such that whenever (z, u) ∈ G(F ) with z ∈ B, one has u, z − xX ≥ c(B, x). For instance, condition (P M4) is fulfilled by any monotone map F with D(F ) = X, as well as by bounded maps. ∗

Remark 8.4. A pseudomonotone map F : X → 2X on a reflexive Banach space X is generalized pseudomonotone. Conversely, a bounded, generalized ∗ pseudomonotone map F : X → 2X on a reflexive Banach space X satisfying condition (P M1) is pseudomonotone (see [38, Propositions 2.122, 2.123]). We also quote the following result from Kenmochi [117, Proposition 4.1]. ∗

Proposition 8.5. Let F : X → 2X be a set-valued map on a reflexive Banach space X satisfying (P M1), (P M2), and (P M4), let C be a nonempty, closed, convex, and bounded subset of X, let  : X → R ∪ {+∞} be lower semicontinuous, convex, with C ∩ D() = ∅, and let f ∈ X ∗ . Then there exists x ∈ C ∩ D() such that for some w ∈ F (x) we have w − f, z − xX ≥ (x) − (z)

for every z ∈ C.

Now we present general existence results for elliptic and evolutionary variational inequalities. Theorem 8.6. Consider reflexive Banach spaces X and Y with a compact linear map i : X → Y , and a closed convex subset K of X with nonempty interior int K. We assume that (A1 ) L : D(L) ⊂ X → X ∗ is a linear, maximal monotone operator. ∗ (A2 ) A : X → 2X is a (possibly multivalued) bounded, pseudomonotone operator. ∗ (A3 ) B : Y → 2Y is a (possibly multivalued) bounded operator, which has nonempty, convex, closed values, and whose graph is sequentially strongly–weakly closed (i.e., if (yn , bn ) ∈ G(B) with yn → y in Y and bn  b in Y ∗ , then (y, b) ∈ G(B)). (A4 )  : K → R ∪ {+∞} is a convex, lower semicontinuous function, with int K ⊂ D(). (A5 ) f ∈ X ∗ , and when the set K is unbounded in X there exist u0 ∈ D(L) ∩ int K and an r > 0 satisfying a, v − u0 X + b, iv − iu0 Y > f, v − u0 X for all a ∈ A(v), b ∈ B(iv), v ∈ X with vX > r.

Evolutionary Variational and Quasivariational Inequalities Chapter | 8 285

Then there exists a solution of the following evolutionary variational inequality: find u ∈ K ∩ D(L) ∩ D() such that for some a ∈ A(u) and b ∈ B(iu) we have L(u) + a − f, v − uX + b, iv − iuY + (v) − (u) ≥ 0 Proof. Define  : X → R ∪ {+∞} by  (x) (x) = +∞

for all v ∈ K. (8.1)

if x ∈ K, if x ∈ X \ K.

From assumption (A4 ) it follows that  is convex, lower semicontinuous, and ≡ +∞. Problem (8.1) is equivalent to the statement: find u ∈ D(L) ∩ D() such that for some a ∈ A(u) and b ∈ B(iu) one has L(u) + a − f, v − uX + b, iv − iuY + (v) − (u) ≥ 0 for all v ∈ X. (8.2) In turn, (8.2) is equivalent to the inclusion problem: find u ∈ D(L) ∩ D(∂) such that f ∈ L(u) + A(u) + i ∗ B(iu) + ∂(u).

(8.3)

The notation ∂ stands for the subdifferential of  in the sense of convex analysis. Indeed, admitting (8.2) yields −(L(u) + a − f + i ∗ b) ∈ ∂(u), thus (8.3) ensues. Conversely, assuming (8.3), inequality (8.2) follows by definition of the convex subdifferential ∂(u). Therefore, in order to complete the proof, we need to show that (8.3) is solvable. Knowing from hypothesis (A4 ) that int K ⊂ D(), it turns out that int K ⊂ D(∂) ⊂ K.

(8.4)

On the other hand, hypothesis (A1 ) implies that D(L) is dense in X

(8.5)

as shown in [238, Theorem 32.L] (see also Proposition 1.52 (ii)). Then (8.5) gives D(L) ∩ int K = ∅.

(8.6)

Due to (8.4) and (8.6), we are able to apply the sum theorem for maximal monotone operators obtaining L + ∂ : X → 2X

∗

is maximal monotone.

(8.7)

286 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

We claim that i ∗ Bi : X → 2X

∗

is pseudomonotone.

(8.8)

Indeed, in view of assumption (A3 ), the values of the operator i ∗ Bi are nonempty and convex. Let x ∈ X and {bn } ⊂ B(ix) with i ∗ bn → z in X ∗ . By (A3 ), the sequence {bn } is bounded in Y ∗ , so along a relabeled subsequence we have bn  b in Y ∗ with b ∈ B(ix), so z = i ∗ b showing that the multivalued mapping i ∗ Bi is closed valued. Furthermore, we note that if xn  x in X and wn  w in X ∗ , with wn ∈ i ∗ B(ixn ), then ixn → ix in Y (because i : X → Y is compact) and wn = i ∗ bn with bn ∈ B(ixn ). Since, by assumption (A3 ), B is bounded, along a relabeled subsequence we get bn  b in Y ∗ . Using once again (A3 ), this results in b ∈ B(ix). We infer that i ∗ bn → i ∗ b in Y ∗ taking into account that the mapping i ∗ is compact, thus we can conclude that wn → w = i ∗ b ∈ i ∗ B(ix). From this, recalling Definition 8.2 and taking into account (A3 ) and Remark 8.3, we easily derive (8.8). Since the sum of pseudomonotone operators is pseudomonotone, (A2 ) and (8.8) imply A + i ∗ Bi : X → 2X

∗

is pseudomonotone and bounded.

(8.9)

At this point, on the basis of (8.7), (8.9), and assumption (A5 ), the main theorem on pseudomonotone perturbations of maximal monotone mappings (see [238, Theorem 32.A and problem 32.4*]) ensures that the inclusion (8.3) holds true for some u ∈ D(L) ∩ D(∂). The second inclusion in (8.4) provides u ∈ K. The equivalence between problems (8.1) and (8.3) completes the proof. Remark 8.7. The evolutionary character of the variational inequality (8.1), as well as subsequent problems, is determined by the linear, maximal monotone operator L : D(L) ⊂ X → X ∗ , which generally is an unbounded linear operator (see Definition 1.46). The model case for the unbounded linear operator L is the time derivative, so incorporating in our setting evolutionary partial differential equations and inclusions. Next we formulate an existence result for multivalued elliptic variational inequalities where, contrary to Theorem 8.6, the constraint set K can have an empty interior. Theorem 8.8. Given the reflexive Banach spaces X and Y with a compact linear map i : X → Y , a nonempty closed convex subset K of X, and f ∈ X ∗ , assume conditions (A2 ), (A3 ), and (H1 )  : X → R∪{+∞} is convex and lower semicontinuous; x0 ∈ K ∩D(). (H2 ) If K is unbounded then for every sequence {xn } ⊂ K ∩ D() with xn X → +∞ and xxnnX  x in X, for every an ∈ A(xn ) and bn ∈

Evolutionary Variational and Quasivariational Inequalities Chapter | 8 287

B(ixn ) one has lim inf n→∞

 1  an + i ∗ bn , xn − x0 X + (xn ) > f, xX . xn X

(8.10)

Then there exists a solution of the following variational inequality: find x ∈ K ∩ D() such that for some a ∈ A(x) and for some b ∈ B(ix) we have a − f, y − xX + b, iy − ixY + (y) − (x) ≥ 0

for all y ∈ K. (8.11)

Proof. For every integer n, we introduce Kn := K ∩BX (x0 , n), where BX (x0 , n) stands for the closed ball in X of center x0 and of radius n. Here x0 ∈ K ∩ D() is the point postulated in assumption (H1 ). Note that Kn is closed, convex, and bounded in X with x0 ∈ Kn . By virtue of (A2 ) and (A3 ), the proof of Theo∗ rem 8.6 shows that the multivalued map A + i ∗ Bi : X → 2X is bounded and pseudomonotone. By hypothesis (H1 ) we are thus allowed to apply Proposition 8.5, which ensures the existence of xn ∈ Kn ∩ D() such that for some an ∈ A(xn ) and for some bn ∈ B(ixn ) we have an + i ∗ bn − f, z − xn X ≥ (xn ) − (z)

for all z ∈ Kn .

(8.12)

Now we claim that there exists an integer k > 0 with the property xk − x0 X < k.

(8.13)

Arguing by contradiction, admit that for every positive integer n, xn − x0 X = n. Due to the reflexivity of X, along a subsequence we can suppose that in X for some x ∈ X. By inserting z = x0 in (8.12), we obtain

xn xn X

x

an + i ∗ bn , xn − x0 X + (xn ) − (x0 ) ≤ f, xn − x0 . Letting n → ∞ results in   x n − x0 (xn ) ∗ X + lim sup an + i bn , ≤ f, xX , xn X xn X n→∞ contradicting (8.10). Therefore assertion (8.13) is proven. Let an arbitrary y ∈ K. By (8.13), for sufficiently small t > 0 we know that xk + t (y − xk ) − x0 X < k. With a sufficiently small t > 0, we can choose z = xk + t (y − xk ) in (8.12) for n = k. Using the convexity of , we deduce ak + i ∗ bk − f, y − xk X ≥ (xk ) − (y),

288 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

where ak ∈ A(xk ) and bk ∈ B(ixk ) are independent of y. We conclude that xk solves (8.11), so the proof is complete. The following is an existence result for variational inequalities whose proof involves again pseudomonotone multivalued maps. Theorem 8.9. Given a reflexive Banach space X, a nonempty closed convex subset K of X, and f ∈ X ∗ , assume (H1 ), and ∗

(C1 ) A : X → 2X is a multivalued map satisfying conditions (P M1), (P M2), and (P M4). (C2 ) If K is unbounded then for every {xn } ⊂ K ∩ D() with xn X → +∞ and xxnnX  x in X, for every an ∈ A(xn ), we have lim inf n→∞

 1  an , xn − x0 X + (xn ) > f, xX . xn X

Then there exists a solution of the variational inequality: find x ∈ K ∩ D() such that for some a ∈ A(x) we have a − f, y − xX + (y) − (x) ≥ 0

for all y ∈ K.

The proof can be carried out through the arguments used in Theorem 8.8 with B = 0 and by using Remark 8.3. The next result focuses on variational inequalities containing a multivalued operator A which is maximal monotone (see Definition 8.1 (ii)). Generally, we can have D(A) = X, so none of the above theorems is applicable. For the sake of simplicity, we suppose that L = 0 and B = 0. Theorem 8.10. Let K be a nonempty, closed, and convex subset of a reflexive Banach space X, let f ∈ X ∗ , and let  : X → R ∪ {+∞} be convex, lower semicontinuous. Assume the following conditions: ∗

(S1 ) A : D(A) ⊂ X → 2X is a maximal monotone map, with 0 ∈ K ∩ int (D(A)) ∩ int (D(∂)), 0 ∈ A(0), and the weak closure in X of K ∩ D(A) ∩ D(∂) is contained in D(A). (S2 ) For every sequence {xn } ⊂ K ∩ D(A) ∩ D(∂) with xn X → +∞ and xn xn X  z in X, we have lim inf n→∞

(xn ) > f, zX . xn X

(8.14)

(S3 ) For every x ∈ K ∩ D(A) ∩ D() there exists a = a(x) ∈ A(x) such that if z ∈ K ∩ D() there are a sequence tn → 0+ as n → ∞ with x +

Evolutionary Variational and Quasivariational Inequalities Chapter | 8 289

tn (z − x) ∈ D(A) and a sequence ξn ∈ A((1 − tn )x + tn z) satisfying lim infξn , z − xX ≤ a, z − xX . n→∞

Then there exists x ∈ K ∩ D(A) ∩ D() such that for a = a(x) ∈ A(x) given in (S3 ), we have a − f, z − xX ≥ (x) − (z)

for all z ∈ K.

(8.15)

Proof. Since X is reflexive, we can consider a norm on X such that X and ∗ X∗ become strictly convex. Define the multivalued map T : X → 2X by T := A + NK + ∂, where NK denotes the subdifferential of the indicator function of K (equivalently, NK (x) is the cone of exterior normals to K at x ∈ K). From assumption (S1 ) we know that T is a maximal monotone map (see, e.g., [238, Theorem 32.I]). Then a classical surjectivity result ensures that there exists xn ∈ D(T ) = K ∩D(A)∩D(∂) such that f ∈ T (xn )+n J (xn ), where J : X → X ∗ stands for the duality mapping of X and the sequence {n } satisfies n → 0+ . Therefore, for every n, there exist wn ∈ A(xn ), vn ∈ NK (xn ), and un ∈ ∂(xn ) such that wn + vn + un + n J (xn ) − f, y − xn X = 0 for all y ∈ X. Thanks to the inequalities vn , y − xn X ≤ 0 and (y) − (xn ) ≥ un , y − xn X whenever y ∈ K, it turns out that wn + n J (xn ) − f, y − xn X ≥ (xn ) − (y)

for all y ∈ K.

From this and the monotonicity of A we infer that w + n J (y) − f, y − xn X ≥ (xn ) − (y) for all w ∈ A(y) and y ∈ K ∩ D(A).

(8.16)

We claim that {xn } is bounded in X. If not, there would exist a relabeled subsequence such that xn X → +∞ as n → ∞. In view of the reflexivity of X, we can find z ∈ X such that up to a subsequence xn z xn X

in X.

Setting y = 0 and w = 0 in (8.16), which is possible due to assumption (S1 ), then dividing by xn X and letting n → ∞ gives lim sup n→∞

(xn ) ≤ f, zX . xn X

Since this contradicts assumption (S2 ), the claim holds.

290 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

The reflexivity of X enables us to pass to a subsequence still denoted by {xn } converging weakly in X to some x ∈ K. Letting n → ∞ in (8.16) yields w − f, y − xX ≥ (x) − (y)

for all w ∈ A(y) and y ∈ K ∩ D(A). (8.17)

Setting y = 0 and w = 0 in (8.17) shows that x ∈ D(), whereas from the last part of assumption (S1 ) we know that x ∈ D(A). To x ∈ K ∩ D(A) ∩ D() there corresponds an element a ∈ A(x) as stated in hypothesis (S3 ). Let z ∈ K ∩ D(). We can find tn → 0+ as n → ∞ with x + tn (z − x) ∈ D(A) and a sequence ξn ∈ A((1 − tn )x + tn z) that fulfills lim infξn , z − xX ≤ a, z − xX . n→∞

(8.18)

Since (1 − tn )x + tn z ∈ K ∩ D(A), we can use y = (1 − tn )x + tn z in (8.17), rendering through the convexity of  that w − f, z − xX ≥ (x) − (z)

for all w ∈ A((1 − tn )x + tn z).

Hence we obtain ξn − f, z − xX ≥ (x) − (z). Then by means of (8.18) we get (8.15). Observe that (8.15) is valid for any z ∈ K with (z) = +∞, which completes the proof. In order to get more insight on Theorem 8.10, we present a result in the spirit of Minty’s formulation for evolutionary variational inequalities. Theorem 8.11. Let K be a nonempty, closed, and convex subset of a reflexive Banach space X, let L : D(L) ⊂ X → X ∗ be linear, maximal monotone, let ∗ A : D(A) ⊂ X → 2X be a maximal monotone map, let  : X → R ∪ {+∞} be a convex, lower semicontinuous functional ≡ +∞, let K ⊂ int (D(A)) ∩ int (D(∂)), and let f ∈ X ∗ . Assume that there exists an element x ∈ K ∩ D(L) such that for every z ∈ K ∩ D(L) and for every az ∈ A(z), we have L(z) + az − f, z − xX ≥ (x) − (z).

(8.19)

Then there exists a ∈ A(x) such that L(x) + a − f, z − xX ≥ (x) − (z)

for all z ∈ K ∩ D(L).

Proof. Since for every z ∈ K ⊂ D(∂), z∗ ∈ ∂(z), and y ∈ X, we have (y) − (z) ≥ z∗ , y − zX ,

(8.20)

Evolutionary Variational and Quasivariational Inequalities Chapter | 8 291

it follows from (8.19) that for any x ∈ K ∩ D(L), z ∈ K ∩ D(L), z∗ ∈ (z), and az ∈ A(z), L(z) + az + z∗ − f, z − xX ≥ 0.

(8.21)

∗

We define the multivalued map T : X → 2X by T := L + A + NK + ∂, where NK denotes the subdifferential in the sense of convex analysis of the indicator function of K. It is known that T is a maximal monotone map with D(T ) = K ∩ D(L) (see, e.g., [238, Theorem 32.1]). For any uz ∈ NK (z) and z ∈ K ∩ D(L), we note from (8.21) that L(z) + az + z∗ + uz − f, z − xX = L(z) + az + z∗ − f, z − xX + uz , z − xX ≥ 0, implying by the maximal monotonicity of T that f ∈ (L + A + ∂ + NK )(x). Consequently, there exist a ∈ A(x), u ∈ NK (x), and x ∗ ∈ ∂(x) satisfying L(x) + a + u + x ∗ − f, z − xX = 0 for all z ∈ X, which, by using the inequalities u, z −xX ≤ 0 and (z)−(x) ≥ x ∗ , z −xX whenever z ∈ K ∩ D(L), confirms that (8.20) is true, and the proof is complete. Remark 8.12. Condition (S3 ) in Theorem 8.8 represents a multivalued version of Minty’s technique for variational inequalities driven by single-valued monotone, hemicontinuous operators. Complementing this, Theorem 8.11 provides a Minty formulation for evolutionary variational inequalities where it is shown that, by imposing the assumption K ⊂ int (D(A)) ∩ int (D(∂)), condition (S3 ) can be dropped. We also note that the asymptotic recessive condition (8.14) in Theorem 8.9 is a requirement that does not involve A, in particular, it is not necessary to ask for any coercivity of A.

8.2 EVOLUTIONARY QUASIVARIATIONAL INEQUALITIES For the sake of clarity we state the fixed point theorem of Kluge [123] that will be used in the sequel. Theorem 8.13. Let Z be a real reflexive Banach space and let D ⊂ Z be nonempty, convex, bounded, and closed. Assume that P : D → 2D is a setvalued map such that for every u ∈ D, the set P (u) is nonempty, closed, and convex, and whose graph G(P ) is weakly closed. Then P has a fixed point, that is, z ∈ P (z). Remark 8.14. The request in Theorem 8.13 on the set D to be bounded in Z can be replaced by requiring that the image P (D) be bounded. To see this it is sufficient to apply Theorem 8.13 to the closed convex hull co(P (D)) of P (D) in place of D.

292 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Let X and Y be reflexive Banach spaces with a compact linear map i : X → Y , and let L : D(L) ⊂ X → X ∗ be a linear, maximal monotone map. ∗ ∗ Let A : X → 2X and B : Y → 2Y be multivalued maps, let K be a nonempty, closed, and convex subset of X, let  : X → R ∪ {+∞} be a functional ≡ +∞, and let C : K → 2K be a multivalued map such that for any z ∈ K, C(z) is a closed, convex subset of K with nonempty interior in X, and let f ∈ X ∗ . With these data, we formulate the following quasivariational inequality: find x ∈ C(x) ∩ D(L) ∩ D() such that for some a ∈ A(x) and b ∈ B(ix) we have L(x) + a − f, z − xX + b, iz − ixY + (z) − (x) ≥ 0 for all z ∈ C(x) ∩ D(L).

(8.22)

The evolutionary feature of the inequality problem (8.22) comes from the unbounded linear operator L, which plays the role of time derivative in evolution equations and inclusions. In order to focus on the evolutionary quasivariational inequality (8.22), we proceed through a parametric approach by means of evolutionary variational inequalities of type (8.1). Namely, for every w ∈ K, we consider the following parametric evolutionary variational inequality: find x ∈ C(w) ∩ D(L) ∩ D() such that for some a ∈ A(x) and b ∈ B(ix) we have L(x) + a − f, z − xX + b, iz − ixY + (z) − (x) ≥ 0 for all z ∈ C(w) ∩ D(L).

(8.23)

We define a multivalued map, called variational selection, S : K → 2K

(8.24)

by assigning to each y ∈ K the set S(y) of all solutions of (8.23) with w = y. It is essential to note that if x is a fixed point of the multivalued map S in (8.24), that is, x ∈ S(x), then x solves the evolutionary quasivariational inequality (8.22). With the data in problem (8.22) we formulate the conditions: (A5 ) f ∈ X ∗ , and if the set K is unbounded in X, then there exist r > 0 and a point  C(w) (8.25) u0 ∈ D(L) ∩ int w∈K

satisfying a, v − u0 X + b, iv − iu0 Y > max{f, v − u0 X , f − L(u0 ), v − u0 X } for all a ∈ A(v), b ∈ B(iv), v ∈ X with vX > r.

Evolutionary Variational and Quasivariational Inequalities Chapter | 8 293

(A6 ) The operator A + i ∗ Bi is monotone on C(w), that is, av − au , v − uX + bv − bu , iv − iuY ≥ 0 for all u, v ∈ C(w), au ∈ A(u), av ∈ A(v), bu ∈ B(iu), bv ∈ B(iv), w ∈ K. (A7 ) For every x ∈ C(w) ∩ D(L) ∩ D() there exist Ax ∈ A(x) and Bx ∈ B(ix) such that   lim sup at , z − xX + bt , iz − ixY ≤ Ax , z − xX + Bx , iz − ixY t→0+

whenever z ∈ C(w)∩D(L), at ∈ A(x +t (z −x)), bt ∈ B(ix +t (iz −ix)). (A8 ) The map C : K → 2K has the following Mosco-type continuity properties (see [159,160]): (i) If {wn } ⊂ K and un ∈ C(wn ) ∩ D(L) ∩ D() satisfy wn  w in X and un  u in X, with u ∈ D(L) and L(un )  L(u) in X ∗ , then u ∈ C(w). (ii) For every sequence {wn } ⊂ K ∩ D(L) with wn  w in X and for every v ∈ C(w) ∩ D(L), there exist a subsequence {wnk } of {wn } and a sequence vk ∈ C(wnk ) ∩ D(L) with vk → v in X and (vk ) → (v). Theorem 8.15. Assume that conditions (A1 )–(A4 ), (A5 ), (A6 )–(A8 ) hold. Then there exists a solution of the quasivariational inequality (8.22). Proof. Conditions (A1 )–(A5 ) are verified with C(w) in place of K for each w ∈ K, so Theorem 8.6 with K substituted by C(w) can be applied. This ensures that the parametric problem (8.23) is solvable for every w ∈ K, so the variational selection S : K → 2K introduced in (8.24) has nonempty values. We split the proof in several parts. Step 1. For every w ∈ K, problem (8.23) is equivalent to: find x ∈ C(w) ∩ D(L) ∩ D() such that for every z ∈ C(w) ∩ D(L), a ∈ A(z) and b ∈ B(iz), we have L(z) + a − f, z − xX + b, iz − ixY + (z) − (x) ≥ 0.

(8.26)

Indeed, if x ∈ C(w) ∩ D(L) ∩ D() is a solution of (8.23), then it solves (8.5) on account of (A1 ) and (A6 ). Conversely, suppose that x ∈ C(w) ∩ D(L) ∩ D() is a solution of (8.26) and let z ∈ C(w) ∩ D(L). Since C(w) ∩ D(L) is a convex set, we can set zt := (1 − t)x + tz ∈ C(w) ∩ D(L) for all t ∈ (0, 1] in (8.26) obtaining 0 ≤ L(zt ) + at − f, zt − xX + bt , izt − ixY + (zt ) − (x) whenever at ∈ A(zt ) and bt ∈ B(izt ). The linearity of L and the convexity of  as postulated in condition (A4 ) imply 0 ≤ L(z) + (1 − t)L(x) + at − f, z − xX + bt , iz − ixY + (z) − (x). (8.27)

294 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Letting t → 0 in (8.27), by hypothesis (A7 ) we find that x solves problem (8.23). Step 2. The set S(w) is convex and closed in X whenever w ∈ K. Using Step 1, we first check that the solution set of (8.23) is convex in X. To this end, fix w ∈ K. Let u and v solve (8.26), so u, v ∈ C(w) ∩ D(L) ∩ D() such that for every z ∈ C(w) ∩ D(L) and for every a ∈ A(z) and b ∈ B(iz) we have L(z) + a − f, z − uX + b, iz − iuY + (z) − (u) ≥ 0, L(z) + a − f, z − vX + b, iz − ivY + (z) − (v) ≥ 0. Let t ∈ [0, 1]. From the convexity of , it is seen that L(z) + a − f, z − (1 − t)u − tvX + (z) − ((1 − t)u + tv) ≥ 0, which in view of Step 1 entails the convexity assertion. Now let {un } ⊂ S(w) with un → u in X. We infer that un ∈ C(w) ∩ D(L) ∩ D(), and for some an ∈ A(un ) and bn ∈ B(iun ) we have L(un ) + an − f, v − un X + bn , iv − iun Y + (v) − (un ) ≥ 0

(8.28)

for all v ∈ C(w) ∩ D(L). Since C(w) is a closed subset of K, we get u ∈ C(w). By hypotheses (A2 ) and (A3 ), the sequences {an } and {bn } are bounded in X ∗ and Y ∗ , respectively. Consequently, along relabeled subsequences an  a in X ∗ and bn  b in Y ∗ . Observing that lim an , un − uX = 0, we deduce from (A2 ) n→∞

that a ∈ A(u) because then A is generalized pseudomonotone (see Remark 8.4). Similarly, in view of (8.8), we derive that b ∈ B(iu). Recalling that int (C(w)) is nonempty, we can choose a ball B in X with B ⊂ C(w). Then (8.28), in conjunction with the density of D(L) in X, guarantees that the sequence {L(un )} is uniformly bounded below on the ball B, which ensures that {L(un )} is bounded in X ∗ . Since L is linear, maximal monotone, so weakly graph closed, we have that u ∈ D(L) and L(un )  L(u) in X ∗ . We also note from (8.28) that u ∈ D() thanks to the lower semicontinuity of  as required in (A4 ). Altogether, we can pass to the limit in (8.28) as n → ∞, proving that u ∈ S(w). Step 3. The image S(K) of S is bounded in X. Arguing by contradiction, we assume that there exists a sequence {un } ⊂ S(K) satisfying un X → +∞ as n → ∞.

(8.29)

Necessarily, by (8.29), K is unbounded. Let wn ∈ K verify un ∈ S(wn ). Consequently, un ∈ C(wn ) ∩ D(L) ∩ D() and there exist an ∈ A(un ) and bn ∈

Evolutionary Variational and Quasivariational Inequalities Chapter | 8 295

B(iun ) fulfilling (8.28). Due to (A5 ) we can insert v = u0 into (8.28), which by the monotonicity of L reads as an , un − u0 X + bn , iun − iu0 Y + (un ) − (u0 ) ≤ f − L(u0 ), un − u0 X . Taking into account (8.29), we come to a contradiction with the inequality in hypothesis (A5 ). In view of this contradiction, Step 3 is valid. Step 4. The graph G(S) of S is sequentially weakly closed in X × X. Let {(wn , un )} ⊂ G(S) converge weakly to (w, u) in X × X. The fact that un ∈ S(wn ) means that wn ∈ K, un ∈ C(wn ) ∩ D(L) ∩ D() and the inequality (8.28) holds for some an ∈ A(un ) and bn ∈ B(iun ). Note that w ∈ K. Owing to assertion (8.25) in (A5 ), we can choose a ball B in X such that B ⊂ C(wn ) for all n. Then (8.28) and the density of D(L) in X ensure that the sequence {L(un )} is uniformly bounded below on the ball B, which guarantees that {L(un )} is bounded in X ∗ . Since L, being linear and maximal monotone, is weakly graph closed, we obtain that u ∈ D(L) and L(un )  L(u) in X ∗ . Then according to assumption (A8 ) (i) we have that u ∈ C(w). Moreover, by (A4 ) and (8.28) we derive that u ∈ D(). Since u ∈ C(w) ∩ D(L) ∩ D(), we can invoke assumption (A8 ) (ii), obtaining a subsequence of {wn }, denoted again by {wn }, corresponding to which there exists a sequence {zn } with zn ∈ C(wn ) ∩ D(L), zn → u in X and (zn ) → (u). We can thus insert v = zn into (8.28), resulting in L(un ) + an − f, zn − un X + bn , izn − iun Y + (zn ) − (un ) ≥ 0, which reads as an , un − uX ≤ an , zn − uX + L(un ) − f, zn − un X + bn , izn − iun Y + (zn ) − (un ).

(8.30)

On the other hand, the monotonicity of L, in conjunction with the facts that unk  u in X and L(unk )  L(u) in X ∗ , implies lim inf L(un ), un X ≥ L(u), uX . n→∞

(8.31)

Because the map i : X → Y is compact, we have that iun → iu in Y . We also notice from hypothesis (A3 ) and (8.8) that along a relabeled sequence bn  b in Y ∗ with b ∈ B(iu). Thanks to the boundedness of the map A, up to a subsequence, we have an  a in X ∗ . So (8.30), (8.31), and (A4 ) yield lim sup an , un − uX ≤ 0. n→∞

296 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

This property, assumption (A2 ) and Remark 8.4 entail a ∈ A(u) and an , un X → a, uX .

(8.32)

Now let an arbitrary v ∈ C(w) ∩ D(L). On the basis of assumption (A8 ) (ii), there exists vk ∈ C(wnk ) ∩ D(L) such that vk → v in X and (vk ) → (v). By testing (8.28) with v = vk when n = nk , we get L(unk ) + ank − f, vk − unk X + bnk , ivk − iunk Y + (vk ) − (unk ) ≥ 0. (8.33) Letting k → ∞ in (8.33) and taking into account (8.31) and (8.32) lead to L(u) + a − f, v − uX + b, iv − iuY + (v) − (u) ≥ 0, which proves that (w, u) ∈ G(S). This confirms that G(S) is sequentially weakly closed in X × X. Finally, we apply Theorem 8.13 by taking D as the closed convex hull of S(K) in X and P as the restriction of S to this set D (see Remark 8.14). Steps 2–4 show that all the hypotheses in Theorem 8.13 are verified. Consequently, through Theorem 8.13 and Remark 8.14, we are able to obtain a fixed point of the multivalued map S, which represents a solution of the quasivariational inequality (8.22). This completes the proof. The following estimate from Alber–Notik [3] will be useful. Lemma 8.16. Let (Z,  · ) be a reflexive Banach space with its dual Z ∗ . Let ∗ A : Z → 2Z be a monotone map and let x¯ ∈ int (D(A)). Then there exists a constant r = r(x) ¯ > 0 such that for every (x, w) ∈ G(A) we have w, x − x ¯ ≥ rw − (x − x ¯ + r)c,

(8.34)

¯ ≤ r and w  ∈ A(x  )} < +∞. where c := sup{w  : x  − x We now resolve an evolutionary quasivariational inequality with a maximal monotone map A: find x ∈ C(x) ∩ D(L) ∩ D() such that for some a ∈ A(x) we have L(x) + a − f, z − xX + (z) − (x) ≥ 0 for all z ∈ C(x) ∩ D(L). (8.35) As before we proceed through a parametric evolutionary variational inequality: find x ∈ C(w) ∩ D(L) ∩ D() such that for some a ∈ A(x) we have L(x) + a − f, z − xX + (z) − (x) ≥ 0 for all z ∈ C(w) ∩ D(L). (8.36)

Evolutionary Variational and Quasivariational Inequalities Chapter | 8 297

Theorem 8.17. Let K be a nonempty, closed, and convex subset of a reflexive Banach space X, let f ∈ X ∗ , and let  : X → R ∪ {+∞} be convex, lower semicontinuous. Assume (A1 ), (A8 ), and the conditions: ∗

(T1 ) A : D(A) ⊂ X → 2X is a maximal monotone map, with  0 ∈ int C(w) ⊂ K ⊂ int (D(A)) ∩ int (D()) w∈K

and 0 ∈ A(0). (T2 ) For every sequence {xn } ⊂ K with xn X → +∞, and we have lim inf n→∞

(xn ) > f, zX . xn X

xn xn X

 z in X,

(8.37)

Then the evolutionary quasivariational inequality (8.35) has a nonempty solution set. Proof. The assumptions of Theorem 8.10 are fulfilled for every C(w) in place of K. Specifically, condition (S3 ) is true because A is hemicontinuous on K due to (T1 ), whereas (T1 ) and (T2 ) imply (S1 ) and (S2 ), respectively. Then Theorem 8.10 ensures that the parametric evolutionary variational inequality (8.36) is solvable for every w ∈ K. Therefore, the associated variational selection S : K → 2K (see (8.24)) has nonempty values for every w ∈ K. We are going to show the existence of a fixed point of S. Using Theorem 8.11, it is seen that (8.36) is equivalent to the following evolutionary variational inequality: find x ∈ C(w) ∩ D(L) such that for every a ∈ A(z), we have L(z) + a − f, z − xX + (z) − (x) ≥ 0

for all z ∈ C(w) ∩ D(L). (8.38)

From (8.38) it is obvious that the solution set S(w) is convex. It is also clear that it is closed. Here the essential fact is that A is locally bounded on K as guaranteed by assumption (T1 ). Let us show that the image S(K) is bounded in X. By contradiction, suppose that there exists a sequence {un } ⊂ S(K) such that un X → +∞ as n → ∞. Let wn ∈ K be such that un ∈ S(wn ), that is un ∈ C(wn ) ∩ D(L) and there exists an ∈ A(un ) such that L(un ) + an − f, z − un X ≥ (un ) − (z)

for all z ∈ C(wn ) ∩ D(L). (8.39)

In view of assumption (T1 ) we can set z = 0 in (8.39). Using the monotonicity of L and A, we arrive at (un ) − (0) ≤ f, un X .

298 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

From here a contradiction to (8.37) in assumption (T2 ) arises. Therefore S(K) is bounded in X. Next we prove that the graph G(S) of S is sequentially weakly closed in X × X. Let {(wn , un )} ⊂ G(S) converge weakly to (w, u) in X × X. We know that wn ∈ K and un ∈ C(wn ) ∩ D(L) satisfies for some an ∈ A(un ) the inequality L(un ) + an − f, v − un X + (v) − (un ) ≥ 0

for all v ∈ C(wn ) ∩ D(L). (8.40)

By assumption (T1 ), we can choose a ball B in X such that B ⊂ C(wn ) for all n. Then (8.40), the local boundedness of the monotone map A on K (see (T1 )) and the density of D(L) in X ensure that the sequence {L(un )} is bounded in X ∗ . Since L is weakly graph closed, it turns out that u ∈ D(L) and L(un )  L(u) in X ∗ . Then assumption (A8 ) (i) yields that u ∈ C(w). We claim that the sequence {an } is bounded. Since u ∈ C(w) ∩ D(L), we find by assumption (A8 ) (ii) a relabeled subsequence of {wn } and a sequence {zn } with zn ∈ C(wn ) ∩ D(L), zn → u in X and (zn ) → (u). Thus we can insert v = zn into (8.40), getting L(un ) + an − f, zn − un X + (zn ) − (un ) ≥ 0. Lemma 8.16 applied to A − f implies that there are constants c > 0 and r > 0 as in the estimate (8.34) such that ran − f X ≤ an − f, un − uX + c(r + un − uX ) = an − f, un − zn X + an − f, zn − uX + c(r + un − uX ) ≤ an − f X∗ zn − uX + c(r + un − uX ) + L(un ), zn − un X + (zn ) − (un ), thereby (r − zn − uX ) an − f X∗ ≤ c(r + un − uX ) + L(un ), zn − un X + (zn ) − (un ).

(8.41)

Since zn → u and the right-hand side of the inequality (8.41) stays bounded, we deduce the boundedness of {an } as claimed. Let v ∈ C(w) ∩ D(L). By assumption (A8 ) (ii), there exists vk ∈ C(wnk ) ∩ D(L) such that vk → v in X and (vk ) → (v) as k → ∞. From (8.40) with v = vk and ank ∈ A(unk ) we infer that L(unk ) + ank − f, vk − unk X + (vk ) − (unk ) ≥ 0. Then, for any av ∈ A(v), (8.42) enables us to find

(8.42)

Evolutionary Variational and Quasivariational Inequalities Chapter | 8 299

L(v) + av , unk − vX ≤ L(v) + av , unk − vX + L(unk ) + ank − f, vk − unk X + (vk ) − (unk ) = L(unk ) + ank , vk − vX + L(unk ) + ank − L(v) − av , v − unk X + f, unk − vk X + (vk ) − (unk ) ≤ L(unk ) + ank , vk − vX + f, unk − vk X + (vk ) − (unk ), where we used the monotonicity of L and A, too. In the limit we obtain L(v) + av , u − vX ≤ (v) − (u) + f, u − v. We can now apply Theorem 8.13, choosing for D the closed convex hull of S(K) in X and for P the restriction of S to this set D. As shown before all the requirements in Theorem 8.13 are fulfilled, which provides a fixed point of S, thus a solution of the quasivariational inequality (8.35). This completes the proof. The next part of this section concerns the evolutionary quasivariational inequality (8.22) with compact constraints C(w), which will permit relaxing other data, in particular dropping the requirement on the linear map i to be compact. Suppose that X and Y are reflexive Banach spaces and a continuous linear map i : X → Y is given. Consider the maps L : W := D(L) ⊂ X → X ∗ , ∗ ∗ A : X → 2X , and B : Y → 2Y such that conditions (A1 ), (A2 ), and (A3 ) are satisfied. We notice that W endowed with the graph norm xW = xX + L(x)X∗

for x ∈ W

is a reflexive Banach space, and further assume that W is separable. Let K be a nonempty compact convex subset of W , let  : X → R ∪ {+∞} be a convex and lower semicontinuous function with K ⊂ D(∂), let C : K → 2K be a multivalued map, and let f ∈ X ∗ . With these data, the quasivariational inequality (8.22) reads as: find x ∈ C(x) ∩ D(L) such that for some a ∈ A(x) and b ∈ B(ix) we have L(x) + a − f, z − xX + b, iz − ixY + (z) − (x) ≥ 0 for all z ∈ C(x). (8.43) An auxiliary result from Lunsford [140, Theorem 3.1] is necessary in the following. Theorem 8.18. Let X be a separable Banach space with its topological dual X∗ and let K be a nonempty, compact, and convex subset of X. Let the multivalued ∗ maps F : K → 2X and G : K → 2K satisfy the conditions:

300 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

(i) The map M : K → 2K defined by 

M(x) := z ∈ K : inf w, x − zX ≤ 0 w∈F (x)

has a closed graph. (ii) G is lower semicontinuous and has closed graph and nonempty convex values. Then there exists a fixed point x ∈ G(x) such that inf w, x − zX ≤ 0

w∈F (x)

for all z ∈ G(x).

If, in addition, F (x) is weakly compact, then there exists w ∈ F (x) such that w, x − zX ≤ 0

for all z ∈ G(x).

The statement of our result on problem (8.43) is formulated below. Theorem 8.19. Under assumptions (A1 ), (A2 ), (A3 ), if ∂ is bounded on K and the multivalued map C : K → 2K is lower semicontinuous with closed graph and nonempty convex values, then the evolutionary quasivariational inequality (8.43) has a nonempty solution set. Proof. By the definition of the convex subdifferential ∂, it is sufficient to find x ∈ C(x) such that for some a ∈ A(x), b ∈ B(ix), and c ∈ ∂(x) one has L(x) + a + c − f, z − xX + b, iz − ixY ≥ 0 for all z ∈ C(x). Since K ⊂ D(∂), we can introduce for every x ∈ K the set 

  L(x) + a + c − f, x − zX + b, ix − izY ≤ 0 . M(x) := z ∈ K : inf a∈A(x) b∈B(ix) c∈∂(x)

(8.44) Notice that x ∈ M(x) for every x ∈ K. We claim that the multivalued map M : K → 2K defined in (8.44) has a closed graph in the topology induced by W . Indeed, let (xn , zn ) ∈ G(M) be a sequence such that xn → x and zn → z in W . It is seen from (8.44) that there exist an ∈ A(xn ), bn ∈ B(ixn ), cn ∈ ∂(xn ) such that 1 L(xn ) + an + cn − f, xn − zn X + bn , ixn − izn Y ≤ . n

(8.45)

The fact that xn → x in X, zn → z in X, and L(xn ) → L(x) in X ∗ ensures L(xn ), xn − zn X → L(x), x − zX

as n → ∞.

(8.46)

Evolutionary Variational and Quasivariational Inequalities Chapter | 8 301

It turns out from assumptions (A2 ) and (A3 ) that along relabeled subsequences one has an  a in X ∗ and bn  b in Y ∗ for some a ∈ A(x) and b ∈ B(ix). Moreover, the boundedness of ∂ on K renders that along a relabeled subsequence one has cn  c in X ∗ , with c ∈ ∂(x) because ∂ is maximal monotone. Then, taking into account (8.45) and (8.46), we infer that L(x) + a + c − f, x − zX + b, ix − izY   ≤ lim sup L(xn ) + an + cn − f, xn − zn X + bn , xn − zn X ≤ 0, n→∞

which entails z ∈ M(x), so our claim holds true. Hence hypothesis (i) in Theorem 8.18 is verified. Since we assumed that C : K → 2K is lower semicontinuous with closed graph and nonempty convex values, hypothesis (ii) in Theorem 8.18 holds also true. Therefore we are able to apply Theorem 8.18 to X = W , F = (L + A + i ∗ Bi + ∂)|K − f , G = C, and our set K. Our hypotheses ensure that the set F (x) is bounded, convex, and closed in X ∗ , so weakly compact in W ∗ . The final assertion of Theorem 8.18 enables us to complete the proof. Remark 8.20. Sufficient conditions to meet the properties required for the multivalued map C : K → 2K in Theorem 8.19 can be found in [140]. We end the section with two applications related to Theorem 8.15. In order to avoid technical details and emphasize the main ideas, we take the set K = X. The first application deals with an elliptic quasivariational inequality in the form of a hemivariational inequality. Given a bounded domain  in RN and 1,p a number p ∈ [2, +∞), we consider the reflexive Banach spaces W0 () p and L () endowed with their usual norms. By Theorem 1.20 we know that 1,p the inclusion W0 () ⊂ Lp () is a dense compact embedding. The negative  1,p (Dirichlet) p-Laplacian − p : W0 () → W −1,p () (see Proposition 1.56) satisfies 1,p

p

− p u + p v, u − vW 1,p () ≥ c(p)u − v 0

1,p

W0 ()

for all u ∈ W0 (), (8.47)

with some constant c(p) > 0. It has a first eigenvalue λ1 > 0 (see Proposi tion 1.57). Let f ∈ W −1,p () and consider a convex, continuous function 1,p  : W0 () → R. Next we fix a function J : Lp () → R which is Lipschitz continuous on the bounded sets in Lp () and whose generalized gradient p ∂J : Lp () → 2L () fulfills 0 ∈ ∂J (0) and p

ξ − η, u − vLp () ≥ −c0 u − vLp () for all u, v ∈ Lp (), ξ ∈ ∂J (u), η ∈ ∂J (v), with a constant c0 < λ1 c(p).

(8.48)

302 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Gathering all these data, we formulate the following quasivariational in1,p equality: given ρ > 0, find u ∈ W0 () satisfying uW 1,p () ≤ ρ + uLp ()

(8.49)

0

such that for some b ∈ ∂J (u) we have − p u − f, v − uW 1,p () + b, v − uLp () + (v) − (u) ≥ 0

(8.50)

0

1,p

for all v ∈ W0 () with vW 1,p () ≤ ρ + uLp () . 0

Theorem 8.21. If the locally Lipschitz function J : Lp () → R verifies (8.48), then for any ρ > 0, problem (8.49)–(8.50) possesses at least one solution. 1,p

Proof. Fix a number ρ > 0 and set X = W0 () and Y = Lp (). Our goal 1,p is to apply Theorem 8.15 to K = W0 (), L = 0, A = − p , B = ∂J , and a 1,p convex, lower semicontinuous function  : W0 () → R. It is clear that conditions (A1 ) and (A4 ) are satisfied. It is known from (8.47) that − p is uniformly monotone, so taking into account Proposition 1.56 it is pseudomonotone. Therefore assumption (A2 ) is verified. In order to check assumption (A3 ), let bn ∈ ∂J (yn ) with yn → y in Lp ()  and bn  b in Lp (). Then Proposition 1.39 (ii) ensures that b ∈ ∂J (y). Furthermore, ∂J has nonempty, convex, closed values, and it is a bounded operator because J is supposed to be Lipschitz continuous on the bounded sets. This follows from Proposition 1.40. Altogether, we can conclude that assumption (A3 ) is fulfilled. 1,p 1,p We define the multivalued map C : W0 () → 2W0 () by

 1,p C(w) := v ∈ W0 () : vW 1,p () ≤ ρ + wLp () 0

1,p

for all w ∈ W0 (). By (8.47) and (8.48), the following estimate is valid − p u − f, uW 1,p () + b, uLp () 0

p ≥ (c(p) − λ−1 1 c0 )uW 1,p () 0

− f W −1,p () uW 1,p () 0

1,p

for every u ∈ W0 () and b ∈ ∂J (u). Since c0 < λ1 c(p), we derive that hypothesis (A5 ) is satisfied with u0 = 0. Through an estimate of the same type based again on (8.47) and (8.48) we find that hypothesis (A6 ) holds true. 1,p Given u, z ∈ C(w) with w ∈ W0 (), from Proposition 1.39 it turns out that lim supbt , z − uLp () ∈ {b, z − uLp () : b ∈ ∂J (u)} t→0+

Evolutionary Variational and Quasivariational Inequalities Chapter | 8 303

whenever bt ∈ ∂J (ix + t (z − x)) for t > 0 small. This enables us to obtain lim supbt , z − uLp () ≤ max{b, z − uLp () : b ∈ ∂J (u)} t→0+

= Bu , z − uLp () , for some Bu ∈ ∂J (u), because the generalized gradient ∂J (u) is weakly∗ com pact in Lp (). Combining with the continuity of − p , this proves condition (A7 ). 1,p For checking condition (A8 ), let {wn } ⊂ W0 () and un ∈ C(wn ) satisfy 1,p wn  w and un  u in W0 (). It is thus known that un W 1,p () ≤ ρ + 0

wn Lp () and along a relabeled subsequence wn → w in Lp () by applying Theorem 1.20. In the limit we get uW 1,p () ≤ ρ + wLp () , so u ∈ C(w), 0

which yields condition (A8 ) (i). 1,p 1,p Let {wn } ⊂ W0 () with wn  w in W0 () and v ∈ C(w). We set vn :=

ρ + wn Lp () v. ρ + wLp () 1,p

Along a relabeled subsequence we have vn → v in W0 () and vn ∈ C(wn ) because v ∈ C(w) and thus vn W 1,p () = 0

ρ + wn Lp () vW 1,p () ≤ ρ + wn Lp () , 0 ρ + wLp ()

which shows that condition (A8 ) (ii) holds, too. Since all the hypotheses of Theorem 8.15 are verified, we are in a position to apply it to problem (8.49)–(8.50), which leads to the desired conclusion. We pass to our second application which concerns an evolutionary quasivariational inequality. Let a bounded domain  in RN and numbers p ∈ [2, +∞) 1,p and τ > 0. We set X := Lp ((0, τ ); W0 ()), which is a reflexive Banach space   under the usual norm and has the dual X ∗ = Lp ((0, τ ); W −1,p ()), where  ∗ 1 < p < +∞ and 1/p + 1/p = 1. Let f ∈ X , and let us introduce the map ∗ A : X → 2X by A(u)(t) = − p (u(t))

for all u ∈ X, t ∈ [0, τ ].

(8.51)

Fix a (possibly nonlinear) compact operator T : X → X. We state the evolutionary quasivariational inequality: given a number ρ > 0, find u ∈ X with ∗ u = du dt ∈ X and u(0) = 0 such that uX ≤ ρ + T uX

(8.52)

304 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

and u + A(u) − f, v − uX + (v) − (u) ≥ 0

(8.53)

for all v ∈ X with vX ≤ ρ + T uX . Theorem 8.22. For any ρ > 0, problem (8.52)–(8.53) possesses at least one solution. Proof. Fix ρ > 0. The time derivative is an unbounded linear operator L : D(L) ⊂ X → X ∗ defined by L(u) = u for all u ∈ D(L), with the domain D(L) = {u ∈ X : u ∈ X ∗ , u(0) = 0}. It is a maximal monotone operator (see, e.g., [38, Lemma 2.149]), so assump∗ tion (A1 ) in Theorem 8.6 is satisfied. The map A : X → 2X introduced in (8.51) is monotone, hemicontinuous, and bounded (see, e.g., [238, p. 878]), so pseudomonotone, which implies assumption (A2 ) in Theorem 8.6. Hypothesis (A3 ) is verified with B = 0. Choose K = X, which makes hypothesis (A4 ) be satisfied. Let us define the multivalued map C : X → 2X by C(w) = {v ∈ X : vX ≤ ρ + T wX } for all w ∈ X. On account of (8.47), the operator A introduced in (8.51) is coercive, which renders assumption (A5 ) true, for instance, with u0 = 0. Since the operator A is monotone, whereas B = 0, condition (A6 ) is valid, while condition (A7 ) holds because the operator A is hemicontinuous. In order to check condition (A8 ) (i), we consider sequences {wn } ⊂ X and un ∈ C(wn ) satisfying wn  w and un  u in X. We have that un X ≤ ρ + T wn X and, thanks to the compactness of the mapping T , we can pass to a relabeled subsequence such that T wn → T w in X. In the limit it results in uX ≤ ρ + T wX , that is, u ∈ C(w), thus (A8 ) (i) holds. For proving condition (A8 ) (ii), we consider {wn } ⊂ X with wn  w in X and v ∈ C(w). Setting vn :=

ρ + T wn X v ρ + T wX

and using the compactness of the map T , as well as that v ∈ C(w), it is straightforward to show that along a relabeled subsequence vn → v in X and vn ∈ C(wn ). Since the function  is continuous, condition (A8 ) (ii) is satisfied. Therefore all the hypotheses of Theorem 8.15 hold true, which enables us to infer through Theorem 8.15 the solvability of problem (8.52)–(8.53).

Evolutionary Variational and Quasivariational Inequalities Chapter | 8 305

8.3 NOTES This chapter is conducted following Khan–Motreanu [118] and is devoted to evolutionary variational and quasivariational inequalities. The evolutionary feature of these problems is introduced through an unbounded, linear, maximal operator L, which is interpreted as abstract time derivative. Quasivariational inequalities not only subsume variational inequalities and nonlinear partial differential inequations, but also provide a unified framework for general boundary value problems with nonstandard unilateral boundary conditions. For different developments and applications we refer to [20,102,104,105,119,124,189,193]. Roughly speaking, for existence and approximation regarding quasivariational inequalities it is required that a variational inequality and a fixed point problem be solved simultaneously. Section 8.1 sets forth several general results on elliptic and evolutionary variational inequalities incorporating numerous boundary value problems. We mention Theorem 8.6, extending different existence results for evolutionary variational inequalities, as well as variational–hemivariational inequalities as, for instance, [117, Theorem 4.1] and [132, Theorem 3.1]. Specifically, for variational–hemivariational inequalities, if g : Y → R is a locally Lipschitz ∗ function, denoting its generalized gradient by ∂g : Y → 2Y , we can choose B = ∂g. Recall that if the function g is convex, the generalized gradient ∂g becomes the subdifferential of g in the sense of convex analysis. Notice that Theorem 8.6 can involve different types of pseudomonotone operators. In Theorems 8.8 and 8.9, under different hypotheses, the assumption on the constraint set K to have a nonempty interior is dropped. Theorem 8.9 gives a new existence result for elliptic variational inequalities with generalized pseudomonotone maps. The recessive assumption (H2 ) extends all the classical coercivity conditions. Theorem 8.10 manages elliptic variational inequalities driven by maximal monotone operators covering new situations. Theorem 8.11 is based on a multivalued version of Minty’s technique for variational inequalities governed by single-valued monotone operators. Another trait of our results is that the imposed assumptions exploit the interplay between the properties of the operators involved in the problem and the geometry of the sets of constraints, for instance, their exterior normal cones. Section 8.2 deals with quasivariational inequalities which incorporate evolutionary inequality problems thanks to the presence of a possibly unbounded linear operator L whose prototype is the time derivative L(x) = x  on a space such as X = Lp ((0, τ ), V ), with 1 < p < +∞, τ > 0, and a reflexive Banach space V . For this evolutionary setting we refer to Zeidler [238] (see also [38]). The specific feature of the quasivariational inequalities treated here is that the constraint set C(x) depends on the solution x. This dependence poses serious challenges in extending the properties from variational inequalities to quasivariational inequalities. The quasivariational inequality (8.22) represents a general statement covering significant elliptic and evolutionary problems. In this direction we mention

306 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

just a simple but relevant example, which in fact lies at the beginning of the theory. Namely, if the map A is single-valued with D(A) = X,  = 0, B = 0, and L = 0, then (8.22) reduces to the quasivariational inequality given by Bensoussan–Lions [26]: find x ∈ C(x) such that A(x) − f, z − xX ≥ 0

for every z ∈ C(x).

If, additionally, C(x) = K for every x ∈ K, then it recovers the standard statement of variational inequality (see [122]): find x ∈ K such that A(x) − f, z − xX ≥ 0

for every z ∈ K.

A powerful technique for solving quasivariational inequalities is by finding fixed points of the associated variational selection which is defined in (8.24). This procedure is implemented for resolving the general quasivariational inequality stated in problem (8.22) thus treating in a unifying way elliptic and evolutionary problems in Theorem 8.15. The proof is based on Mosco-type continuity properties and Kluge’s fixed point theorem for multivalued maps. In Theorem 8.17 we give another existence result for quasivariational inequalities where this time the focus is on maximal monotone maps. The main tool in the proof is an extension of Minty’s technique for evolutionary variational inequalities provided in Theorem 8.11. The object of Theorem 8.19 is the study of evolutionary quasivariational inequalities with compact constraints and taking as underlying space the domain of the linear, maximal monotone operator L endowed with the graph norm. Applications of the abstract results, specifically of Theorem 8.15, are given in Theorems 8.21 and 8.22. An open study in the field of variational and quasivariational inequalities is the qualitative theory of solutions.

Chapter 9

Control Problems for Evolutionary Differential Inclusions 9.1 SEMILINEAR EVOLUTIONARY INCLUSIONS WITH CONTROL PARAMETER We start by mentioning certain needed preliminaries about the semilinear evolutionary inclusion on a complex separable Hilbert space H with the scalar product ·, ·H and the associated norm  · H :  x  (t) ∈ A(t)x(t) + Bu(t) + G(t, x(t)), t ∈ J = [0, b], (9.1) x(0) = x0 , where {A(t) : t ∈ J } is a family of linear closed densely defined operators A(t) on H (see Definition 1.48) such that the domain of A(t) does not depend on t , B is a bounded linear operator from a Hilbert space U (the admissible controls set) into H , G : J × H → 2H is a multivalued map, and x0 ∈ H . The control function u belongs to L2 (J, U ). For notation convenience the resolvent of A(t) whenever it exists is denoted by R(λ, A(t)) (see Section 1.3 in Chapter 1). We can regard (9.1) as a control problem where the control function u is in L2 (J, U ). We may consider the control problem on a Banach space, but for simplicity we deal with a Hilbert space. We assume the following conditions taken from [198, p. 150]: (A1) the operators A(t) (t ∈ J ) have a common domain D which is dense in H ; (A2) for each t ∈ J , the resolvent R(λ, A(t)) of A(t) exists for all λ ∈ C with Re λ ≤ 0 and there exists a constant C1 > 0 independent of t and λ such that R(λ, A(t)) ≤

C1 ; |λ| + 1

(A3) there exist constants β ∈ (0, 1] and C2 > 0 such that whenever t, s, τ ∈ J ,    A(t) − A(s) A−1 (τ ) ≤ C2 |t − s|β ; (A4) for each t ∈ J there exists λ ∈ ρ(A(t)) (the resolvent set of A(t)) such that the resolvent R(λ, A(t)) is a compact operator. Nonlinear Differential Problems with Smooth and Nonsmooth Constraints. DOI: 10.1016/B978-0-12-813386-6.00009-2 307 Copyright © 2018 Elsevier Inc. All rights reserved.

308 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Definition 9.1. ([198, p. 129]) A two-parameter family of bounded linear operators U (t, s), 0 ≤ s ≤ t ≤ b, on a Banach space E is called an evolution system if the following conditions are satisfied: (i) U (s, s) = I , U (t, r)U (r, s) = U (t, s) for all 0 ≤ s ≤ r ≤ t ≤ b; (ii) (t, s) → U (t, s) is strongly continuous on the set 0 ≤ s ≤ t ≤ b. Here I denotes the identity map of H . Lemma 9.2 ([198, p. 150]). Under assumptions (A1)–(A3), there exists a unique evolution system U (t, s) on the set 0 ≤ s ≤ t ≤ b in the sense of Definition 9.1 satisfying: (i) There is a constant M > 0 such that U (t, s) ≤ M for 0 ≤ s ≤ t ≤ b; (ii) For 0 ≤ s < t ≤ b, one has U (t, s) : H → D, and t → U (t, s) is strongly differentiable with the derivative ∂t∂ U (t, s) ∈ L(H, H ) strongly continuous on the set 0 ≤ s < t ≤ b. Moreover, for 0 ≤ s < t ≤ b, ∂  ∂ M   U (t, s) + A(t)U (t, s) = 0,  U (t, s) = A(t)U (t, s) ≤ , ∂t ∂t t −s A(t)U (t, s)A−1 (s) ≤ M; (iii) For every y ∈ D and t ∈ (0, b], U (t, s)y is differentiable with respect to s on 0 ≤ s ≤ t ≤ b and ∂ U (t, s)y = U (t, s)A(s)y. ∂s Lemma 9.3 ([86]). Let {A(t) : t ∈ J } satisfy conditions (A1)–(A4). If {U (t, s) : 0 ≤ s ≤ t ≤ b} is the evolution system generated by {A(t) : t ∈ J } as given in Lemma 9.2, then {U (t, s) : 0 ≤ s ≤ t ≤ b} is a compact operator whenever t − s > 0. Taking advantage of what was said above, we introduce the following concept of solution for the differential system (9.1). Definition 9.4. Given u ∈ L2 (J, U ), a function x ∈ C(J, H ) is called a mild solution of system (9.1) if there exists f ∈ L2 (J, H ) such that f (t) ∈ G(t, x(t)) for a.e. t ∈ J and  t  t x(t) = U (t, 0)x0 + U (t, s)Bu(s) ds + U (t, s)f (s) ds for all t ∈ J. 0

0

We also consider the following semilinear evolution hemivariational inequality: ⎧ ⎪ ⎨ −x  (t) + A(t)x(t) + Bu(t), vH + F 0 (t, x(t); v) ≥ 0 for all t ∈ J, all v ∈ H, ⎪ ⎩ x(0) = x0 ,

(9.2)

Control Problems for Evolutionary Differential Inclusions Chapter | 9

309

where F 0 (t, ·; ·) stands for the generalized directional derivative of a locally Lipschitz function F (t, ·) : H → R (see Section 1.2 for the definition of F 0 (t, ·; ·)). Let us note that Definition 9.4 provides the notion of mild solution for problem (9.2), too. Indeed, according to the definition of the generalized gradient in Section 1.2, problem (9.2) is equivalent to the differential inclusion  x  (t) − A(t)x(t) − Bu(t) ∈ ∂F (t, x(t)), t ∈ J, x(0) = x0 , where we denote ∂F (t, x) = ∂F (t, ·)(x). Hence Definition 9.4 becomes as follows. Definition 9.5. Given u ∈ L2 (J, U ), a function x ∈ C(J, H ) is called a mild solution of the hemivariational inequality (9.2) if there exists f ∈ L2 (J, H ) such that f (t) ∈ ∂F (t, x(t)) for a.e. t ∈ J and  t  t x(t) = U (t, 0)x0 + U (t, s)Bu(s) ds + U (t, s)f (s) ds for all t ∈ J. 0

0

The set Kb (G) = {x(b) ∈ H : x(·) is a mild solution of the system (9.1) corresponding to a control u ∈ L2 (J, U ) with initial value x0 ∈ H } represents the reachable set of the system (9.1). For the general theory of linear control systems we refer to [63]. Definition 9.6. The system (9.1) is said to be approximately controllable on the interval J = [0, b] if Kb (G) = H , where Kb (G) denotes the closure of Kb (G) in H . In particular, the corresponding linear system is approximately controllable on J if Kb (0) = H . We impose the following hypotheses on the multivalued map G : J × H → 2H in problem (9.1): H(G) (i) G is jointly measurable on J × H and has nonempty, convex, and weakly compact values; (ii) for a.e. t ∈ J , G(t, ·) has a strongly-weakly closed graph, i.e., if xn → x in H and hn  h in H with hn ∈ G(t, xn ), then h ∈ G(t, x); (iii) there exist a function a ∈ L2 (J ) and a constant c > 0 such that G(t, x)H := sup{ξ H : ξ ∈ G(t, x)} ≤ a(t) + cxH for a.a. t ∈ J and all x ∈ H . In order to establish the existence of mild solutions for problem (9.1), we 2 introduce the multivalued map N : L2 (J, H ) → 2L (J,H ) by N (x) = {w ∈ L2 (J, H ) : w(t) ∈ G(t, x(t))

for a.e. t ∈ J,

whenever x ∈ L (J, H )}. 2

310 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

As in [155, Lemma 5.3] we can prove the result. Lemma 9.7. If hypotheses H(G) hold, then for each x ∈ L2 (J, H ) the set N (x) is nonempty, convex, and weakly compact. Applying [154, Lemma 11], we also get a useful closure statement. Lemma 9.8. If conditions H(G) hold, then the multivalued operator N defined above fulfills the property: if xn → x in L2 (J, H ), wn  w in L2 (J, H ), and wn ∈ N (xn ), then w ∈ N (x). A key tool in the sequel is a fixed point theorem for multivalued maps that we quote from [107]. Theorem 9.9. If E is a Banach space, ⊂ E is nonempty, closed, and convex with 0 ∈ , and : → 2 is an upper semicontinuous multifunction that maps bounded sets into relatively compact sets and has nonempty, compact, convex values, then one of the following statements is true: (a) the set V = {x ∈ : x ∈ λ (x), λ ∈ (0, 1)} is unbounded; (b) has a fixed point, i.e., there exists x ∈ such that x ∈ (x). Now we are able to provide the existence of mild solutions to problem (9.1). Theorem 9.10. Assume that the hypotheses (A1)–(A4) and H(G) hold. Then, for each u ∈ L2 (J, U ), problem (9.1) has a mild solution on J in the sense of Definition 9.4. Proof. By Lemma 9.7, the multivalued map : C(J, H ) → 2C(J,H ) defined by 

(x) = h ∈ C(J, H ) :  t   h(t) = U (t, 0)x0 + U (t, s) f (s) + Bu(s) ds, f ∈ N (x) 0

has nonempty values. In order to show that has a fixed point, we check that

fulfills the conditions of Theorem 9.9. For any x ∈ C(J, H ), (x) is convex due to the convexity of N (x). We subdivide the rest of the proof into five steps. Step 1. is bounded, that is it maps bounded sets into bounded sets in C(J, H ). Given r > 0, we set Br := {x ∈ C(J, H ) : xC(J,H ) ≤ r}. For every ϕ ∈

(x) with x ∈ Br , there exists f ∈ N (x) such that  t  t ϕ(t) = U (t, 0)x0 + U (t, s)Bu(s) ds + U (t, s)f (s) ds, t ∈ J. (9.3) 0

0

Using Lemma 9.2 (i), H(G) (iii), and Hölder’s inequality, we obtain for all t ∈ J that  t  t ϕ(t)H ≤ U (t, 0)x0 H + U (t, s)Bu(s)H ds + U (t, s)f (s)H ds 0

0

Control Problems for Evolutionary Differential Inclusions Chapter | 9

311

 t

   B u(s)U + a(s) + cx(s)H ds ≤ M x0 H + 0 

√ √ ≤ M x0 H + B uL2 (J,U ) b + aL2 (J ) b + crb , which proves the claim.  Step 2. (Br ) := { (x) : x ∈ Br } is equicontinuous for every r > 0. For each x ∈ Br and ϕ ∈ (x), there exists f ∈ N (x) such that (9.3) holds. By assumption H(G) (iii), we note for 0 < τ1 < τ2 ≤ b and δ > 0 small enough that ϕ(τ2 ) − ϕ(τ1 )H ≤ U (τ2 , 0)x0 − U (τ1 , 0)x0 H  τ1    +  U (τ2 , s) − U (τ1 , s) Bu(s) + f (s) H ds 0 τ2   U (τ2 , s) Bu(s) + f (s) H ds + τ   1 ≤  U (τ2 , 0) − U (τ1 , 0) x0 H √   + sup U (τ2 , s) − U (τ1 , s) (B uL2 (J,U ) + aL2 (J ) ) b + crb s∈[0,τ1 −δ]

√   √ + M (B uL2 (J,U ) + aL2 (J ) )(2 δ + τ2 − τ1 ) + cr(2δ + τ2 − τ1 ) . Lemma 9.3 ensures the compactness of U (t, s) for t − s > 0. It is seen that ϕ(τ2 ) − ϕ(τ1 )H → 0 uniformly with respect to x ∈ Br as τ2 → τ1 and δ → 0. In addition, if τ1 = 0 and 0 < τ2 ≤ b, it is easy to prove that ϕ(τ2 ) − x0 H → 0 uniformly with respect to x ∈ Br as τ2 → 0. Altogether, we find that Step 2 holds true. Step 3. is completely continuous. Fix any r > 0. We need to show that (Br ) is relatively compact in C(J, H ). Taking into account Steps 1 and 2 and making use of Ascoli–Arzelà theorem, we must prove that for every t ∈ (0, b] the set (t) := {ϕ(t) : ϕ ∈ (Br )} is relatively compact in H . Given x ∈ Br , ϕ ∈ (x) and t ∈ (0, b], choose δ ∈ (0, t) and define  ϕ δ (t) := U (t, 0)x0 +

t−δ

  U (t, s) Bu(s) + f (s) ds

0



t−δ

= U (t, 0)x0 + U (t, t − δ)

  U (t − δ, s) Bu(s) + f (s) ds.

0

Lemma 9.3 ensures the compactness of U (t, s) for t − s > 0. This, in conjunc t−δ tion with the boundedness of 0 U (t − δ, s)(Bu(s) + f (s)) ds, implies that the set δ (t) := {ϕ δ (t) : ϕ ∈ (Br )} is relatively compact in H .

312 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Moreover, for every ϕ ∈ (Br ), it turns out that √ √   ϕ(t) − ϕ δ (t)H ≤ M B uL2 (J,U ) δ + aL2 (J ) δ + crδ → 0 as δ → 0, with a constant M > 0. Therefore the set (t) is totally bounded, so in view of Step 2, it is relatively compact in H . Step 4. has a closed graph. Let xn → x∗ in C(J, H ) and ϕn → ϕ∗ in C(J, H ), with ϕn ∈ (xn ). The fact that ϕn ∈ (xn ) means that there exists fn ∈ N (xn ) such that 

t

ϕn (t) = U (t, 0)x0 +



t

U (t, s)Bu(s) ds +

0

U (t, s)fn (s) ds.

(9.4)

0

By assumption H(G) (iii), we can show that the sequence {fn } is bounded in L2 (J, H ). Hence, passing to a subsequence if necessary, we may suppose that fn  f∗ in L2 (J, H ).

(9.5)

From (9.4), (9.5) and the compactness of the operator U (t, s) for t − s > 0 as stated in Lemma 9.3, we get 

t

ϕn (t) → U (t, 0)x0 +

 U (t, s)Bu(s) ds +

0

0

t

U (t, s)f∗ (s) ds.

(9.6)

Since ϕn → ϕ∗ in C(J, H ) and fn ∈ N (xn ), from Lemma 9.8 and (9.6), we obtain f∗ ∈ N (x∗ ), hence ϕ∗ ∈ (x∗ ). Step 5. A priori estimate and conclusion. Step 4 guarantees that the graph of is closed. Then it follows from [155, Proposition 3.3.12(2)] that is upper semicontinuous. Thus from Steps 1–4 we know that the multivalued map satisfies all the hypotheses of Theorem 9.9. It remains to decide on the alternative of Theorem 9.9. To this end we claim that the set V := {x ∈ C(J, H ) : x ∈ λ (x), 0 < λ < 1} is bounded. Let x ∈ V and λ ∈ (0, 1) with x ∈ λ (x). Then there exists f ∈ N (x) such that  t  t x(t) = λU (t, 0)x0 + λ U (t, s)Bu(s) ds + λ U (t, s)f (s) ds. 0

0

Control Problems for Evolutionary Differential Inclusions Chapter | 9

313

By assumption H(G) (iii) we derive the estimate  t  t   a(s) + cx(s)H ds u(s)U ds + M x(t)H ≤ Mx0 H + MB 0 0  t x(s)H ds for all t ∈ J , ≤ ρ + Mc 0

 √  where ρ = M x0 H + B uL2 (J,U ) + aL2 (J ) b , and with a constant M > 0. Then Gronwall’s inequality yields x(t)H ≤ ρeMct , showing that the set V is bounded. Now Theorem 9.9 provides a fixed point for , thus a mild solution of problem (9.1). On the basis of Theorem 9.10 we can conduct the study of the approximate controllability for the control problem (9.1). For this we rely on the associated linear control system  x  (t) = A(t)x(t) + Bu(t), t ∈ J = [0, b], (9.7) x(0) = x0 . The controllability map corresponding to (9.7) is the nonnegative bounded linear operator on H defined by  b b U (b, s)BB ∗ U ∗ (b, s) ds.

0 = 0

Here B ∗

and U ∗ (t, s) denote the adjoint operators of B

and U (t, s), respectively. Therefore, the inverse of εI + 0b exists for any ε > 0, so the resolvent R(ε, − 0b ) = (εI + 0b )−1 is well defined; see also Section 1.3. Lemma 9.11. Given x1 ∈ H and ε > 0, there exists a unique (optimal) control uε (·) ∈ L2 (J, U ) at which the functional S : L2 (J, U ) → R defined by  b S(u) = x u (b) − x1 2H + ε u(t)2U dt, (9.8) 0

where

x u (·)

is the unique mild solution of (9.7), attains its minimum value and uε (t) = B ∗ U ∗ (b, t)R(ε, − 0b )(x1 − U (b, 0)x0 ), x

uε

(b) − x1 = εR(ε, − 0b )(U (b, 0)x0

− x1 ).

(9.9) (9.10)

Proof. The existence and uniqueness of an optimal control is the consequence of a general theorem on linear regulator problems (see [62]). We will establish Eqs. (9.9) and (9.10). The functional S in (9.8) is convex and differentiable. It

314 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

has a minimizer uε which is characterized by S  (uε ) = 0, that is, for all v ∈ L2 (J, U ), one has  b S(uε + hv) − S(uε ) S  (uε ), vU dt = lim 0= h→0 h 0  1 ε ε ε ε x u +hv (b) − x1 , x u +hv (b) − x1 H − x u (b) − x1 , x u (b) − x1 H = lim h→0 h   b  ε  ε ε ε u (t) + hv(t), u (t) + hv(t)U − u (t), u (t)U dt +ε 0 b



 ε x u (b) − x1 , U (b, t)Bv(t)H + εuε (t), v(t)U dt

=2 

0 b

 ε B ∗ U ∗ (b, t)(x u (b) − x1 ) + εuε (t), v(t)U dt.

=2 0

Since v ∈ L2 (J, U ) is arbitrary, we get 1 ε uε (t) = − B ∗ U ∗ (b, t)(x u (b) − x1 ) a.e. on J. ε

(9.11)

Then, in view of (9.11), we have  b uε x (b) = U (b, 0)x0 + U (b, s)Buε (s) ds 0

 1 b ε = U (b, 0)x0 − U (b, s)BB ∗ U ∗ (b, s)(x u (b) − x1 ) ds ε 0 1 b uε = U (b, 0)x0 − 0 (x (b) − x1 ), ε implying ε

(εI + 0b )x u (b) = εU (b, 0)x0 + 0b x1 = εU (b, 0)x0 + (εI + 0b )x1 − εx1 . We obtain (9.10) because x u (b) − x1 = ε(εI + 0b )−1 (U (b, 0)x0 − x1 ) = εR(ε, − 0b )(U (b, 0)x0 − x1 ). ε

From (9.11) and (9.10) we derive 1 ε uε (t) = − B ∗ U ∗ (b, t)(x u (b) − x1 ) ε 1 ∗ ∗ = − B U (b, t)ε(εI + 0b )−1 (U (b, 0)x0 − x1 ) ε = B ∗ U ∗ (b, t)R(ε, − 0b )(x1 − U (b, 0)x0 ). Therefore (9.9) holds, which completes the proof.

Control Problems for Evolutionary Differential Inclusions Chapter | 9

315

Through Lemma 9.11, [24, Theorem 2] can be adapted from the timeinvariant case to our time depending system (9.7). Lemma 9.12. The linear control system (9.7) is approximately controllable on J in the sense of Definition 9.6 if and only if εR(ε, − 0b ) → 0 as ε → 0+ in L(H, H ). At this point, we develop a fixed point approach. From Lemma 9.7 we know that for any x ∈ C(J, H ) ⊂ L2 (J, H ) there holds N (x) = ∅. Hence, for every ε > 0, we can define the multivalued map ε : C(J, H ) → 2C(J,H ) with nonempty values as follows: 

ε (x) = h ∈ C(J, H ) :  t   h(t) = U (t, 0)x0 + U (t, s) Buε (s) + f (s) ds, f ∈ N (x) , 0

where



uε (t) = B ∗ U ∗ (b, t)R(ε, − 0b ) x1 − U (b, 0)x0 −

b

 U (b, s)f (s) ds .

0

It is worth noting that for G = 0, we get (9.9). We seek fixed points for ε . Theorem 9.13. Suppose that conditions (A1)–(A4) and H(G) hold. Then the multivalued map ε : C(J, H ) → 2C(J,H ) possesses a fixed point. Proof. The proof is carried out by applying Theorem 9.9 to the multivalued map ε . First, we note that the assumptions guarantee the applicability of Theorem 9.10. Using that N (x) is convex, it is clear that ε (x) is convex for each x ∈ C(J, H ). As for Theorem 9.10, we subdivide the proof into steps. Step 1. ε is bounded. We must show that ε maps BR = {x ∈ C(J, H ) : xC(J,H ) ≤ R} into a bounded set in C(J, H ) for every R > 0. Let x ∈ BR and ϕ ∈ ε (x). Then there exists f ∈ N (x) such that  t  t ϕ(t) = U (t, 0)x0 + U (t, s)Buε (s) ds + U (t, s)f (s) ds, t ∈ J. (9.12) 0

0

Notice from H(G) (iii) that √  MB  uε (t)U ≤ x1 H + Mx0 H + MφL2 (J ) b =: , ε with a constant M > 0 and φ ∈ L2 (J ) given by φ(t) = sup{ξ H : ξ ∈ G(t, x(t)), x ∈ BR } for a.e. t ∈ J. In view of (9.12), this entails that the assertion of Step 1 is valid.

316 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

 Step 2. ε (BR ) := { ε (x) : x ∈ BR } is equicontinuous for every R > 0. As in the proof of Theorem 9.10, for x ∈ BR and ϕ ∈ ε (x), there exists f ∈ N (x) satisfying (9.12). Using the estimate of uε (t)U in Step 1, we obtain the equicontinuity property on the pattern of Step 2 in the proof of Theorem 9.13. Step 3. ε is completely continuous. Let δ > 0. For each x ∈ BR and t > δ, we introduce 

t−δ

ϕ δ (t) := U (t, 0)x0 +

  U (t, s) Buε (s) + f (s) ds

0



= U (t, 0)x0 + U (t, t − δ)

t−δ

  U (t − δ, s) Buε (s) + f (s) ds,

0

where f is related to x through the definition of ε . The estimate for uε (t)U in Step 1 and the compactness of U (t, s) for t −s > 0 imply that the set δ (t) := {ϕ δ (t) : ϕ ∈ ε (BR )} is relatively compact in H for every t > δ. We also observe that for every ϕ ∈ ε (BR ) we have 

t

ϕ(t) − ϕ δ (t)H ≤ M

t−δ

√    B + φ(s) ds ≤ M Bδ + φL2 (J ) δ ,



with a constant M > 0 and φ ∈ L2 (J ) in Step 1. We infer that the set (t) := {ϕ(t) : ϕ ∈ ε (BR )} is totally bounded, so relatively compact in H . By Ascoli– Arzelà theorem and Step 2, we conclude that ε is completely continuous. Step 4. ε is upper semicontinuous. We claim that ε has a closed graph. Let xn → x∗ and ϕn → ϕ∗ in C(J, H ), with ϕn ∈ ε (xn ). Thus there exists fn ∈ N (xn ) such that   t

U (t, s) BB ∗ U ∗ (b, s)R(ε, − 0b ) x1 − U (b, 0)x0 ϕn (t) = U (t, 0)x0 +  −

0

b

 U (b, τ )fn (τ ) dτ + fn (s) ds. 

(9.13)

0

Assumption H(G) (iii) ensures that {fn }n≥1 ⊂ L2 (J, H ) is bounded, hence passing to a subsequence if necessary, fn  f∗ in L2 (J, H ).

(9.14)

By (9.13), (9.14), and the compactness of the operator U (t, s) for t − s > 0, we get   t

U (t, s) BB ∗ U ∗ (b, s)R(ε, − 0b ) x1 − U (b, 0)x0 ϕn (t) → U (t, 0)x0 + 0

Control Problems for Evolutionary Differential Inclusions Chapter | 9

 − 0

b

  U (b, τ )f∗ (τ ) dτ + f∗ (s) ds as n → ∞.

317

(9.15)

Since ϕn → ϕ∗ in C(J, H ) and fn ∈ N (xn ), Lemma 9.8 and (9.15) yield f∗ ∈ N (x∗ ), so ϕ∗ ∈ ε (x∗ ), which expresses that ε has a closed graph. Finally, [155, Proposition 3.3.12(2)] entails that ε is upper semicontinuous. Step 5. A priori estimate. We show that the set := {x ∈ C(J, H ) : x ∈ λ ε (x), 0 < λ < 1} is bounded in C(J, H ). Given x ∈ and λ ∈ (0, 1) with x ∈ λ ε (x), there exists f ∈ N (x) such that  x(t) = λU (t, 0)x0 + λ

t

  U (t, s) Buε (s) + f (s) ds.

0

From H(G) (iii), we deduce the estimate  t U (t, s)Buε (s) dsH x(t)H ≤ U (t, 0)x0 H +  0  t + U (t, s)f (s) dsH 0 √ ≤ Mx0 H + MφL2 (J ) b √  M 2 B2 b x1 H + Mx0 H + MφL2 (J ) b , + ε with a constant M > 0, which proves Step 5. By Steps 1–4 we know that the multivalued map ε fulfills all the conditions of Theorem 9.9, whereas Step 5 decides on the alternative in the conclusion of Theorem 9.10. Therefore ε has a fixed point, which completes the proof. Now we are in a position to provide the main result of approximate controllability. Theorem 9.14. Assume that the hypotheses of Theorem 9.14 are fulfilled and there exists a function φ ∈ L2 (J ) such that G(t, x)H := sup{ξ H : ξ ∈ G(t, x)} ≤ φ(t) for a.a. t ∈ J and all x ∈ H. If the associated linear control system (9.7) is approximately controllable on J , then the system (9.1) is approximately controllable on J in the sense of Definition 9.6. Proof. For every ε > 0, Theorem 9.13 provides a fixed point x ε ∈ C(J, H ) of the multivalued operator ε , so x ε is a mild solution of (9.1) corresponding to

318 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

the control uε assigned with ε . In other words, there exists f ε ∈ N (x ε ) such that  t

x ε (t) = U (t, 0)x0 + U (t, s) BB ∗ U ∗ (b, s)R(ε, − 0b )(x1 − U (b, 0)x0  −

0

b

 U (b, τ )f ε (τ ) dτ ) + f ε (s) ds.

(9.16)

0

Since I − 0b R(ε, − 0b ) = εR(ε, − 0b ), we get from (9.16) and the definitions of the operators 0b and R(ε, − 0b ) that x ε (b) = x1 − εR(ε, − 0b )E(f ε ), with 

b

E(f ε ) := x1 − U (b, 0)x0 −

U (b, τ )f ε (τ ) dτ.

0

By virtue of our assumption that G(t, x)H ≤ φ(t) it follows that 

b

0

√ f ε (s)H ds ≤ φL2 (J ) b.

Consequently, the net {f ε } is bounded in L2 (J, H ), thus there is a subsequence, still denoted by {f ε }, converging weakly to f in L2 (J, H ). Set  b U (b, τ )f (τ ) dτ. z := x1 − U (b, 0)x0 − 0

Due to the compactness in Lemma 9.3, it is readily seen that 

b

E(f ) − zH ≤ ε

  U (b, τ ) f ε (τ ) − f (τ ) H dτ → 0

(9.17)

0

as ε → 0. From Lemma 9.12 that can be applied because the system (9.7) is approximately controllable and on the account of (9.17), it turns out that x ε (b) − x1 H = εR(ε, 0b )E(f ε )H

  ≤ εR(ε, 0b )(z)H + εR(ε, 0b ) E(f ε ) − z H ≤ εR(ε, 0b )(z)H + E(f ε ) − zH → 0 as ε → 0.

Thanks to the fact that x1 ∈ H is arbitrary, we conclude that the multivalued system (9.1) is approximately controllable on J .

Control Problems for Evolutionary Differential Inclusions Chapter | 9

319

A relevant special case is the control hemivariational inequality (9.2). Here the mild solutions are understood in the sense of Definition 9.5. The hypotheses on the function F : J × H → R in (9.2) are as follows: H(F ) (i) the function t → F (t, x) is measurable for all x ∈ H ; (ii) the function x → F (t, x) is locally Lipschitz for a.e. t ∈ J ; (iii) there exist a function a ∈ L2 (J ) and a constant c > 0 such that ∂F (t, x)H := sup{f H : f ∈ ∂F (t, x)} ≤ a(t) + cxH for a.a. t ∈ J and all x ∈ H . Corollary 9.15. Assume that hypotheses (A1)–(A4) and H(F ) are satisfied. If the associated linear control system (9.7) is approximately controllable on J , then the control hemivariational inequality (9.2) is approximately controllable on J as described in Definition 9.6 for the particular situation of (9.2). Proof. Using Proposition 1.39 it is easy to see that hypotheses H(F ) (i)–(iii) imply conditions H(G) for G(t, x) = ∂F (t, x), so Theorem 9.14 can be applied. In view of Definitions 9.5 and 9.6, the stated conclusion is achieved. We end this section with an application to a heat equation. For J = [0, b] and = (0, 1) ⊂ R, consider the following initial–boundary value problem: ⎧ ⎪ ∂2 ∂ ⎪ ⎪ x(t, y) = 2 x(t, y) + a(t)x(t, y) + Bu(t, y) + ϕ(t, y), t ∈ J, y ∈ , ⎨ ∂t ∂y ⎪ x(t, 0) = x(t, 1) = 0, t ∈ J, ⎪ ⎪ ⎩ x(0, y) = x0 (y), y ∈ . (9.18) The physical interpretation is that x(t, y) stands for the temperature at the point y ∈ and time t ∈ J . On the data in (9.18) we suppose that a : J → (0, +∞) is continuous, x0 ∈ L2 ((0, 1)), ϕ = ψ + χ , with −ψ(t, y) ∈ ∂F (t, x(t, y)), (t, y) ∈ J × , for a measurable function F provided F (t, ·) is locally Lipschitz on R, so its generalized gradient denoted ∂F is well defined. A simple example of a function F satisfying hypotheses H(F ) which is relevant for our purposes is F (t, η) = F (η) = max{h1 (η), h2 (η)}, where hi : R → R (i = 1, 2) are convex quadratic functions. In order to match the setting of Corollary 9.15, take H = L2 ((0, 1)) and, for every t ∈ [0, b], the operator A(t) defined by A(t)x = x  + a(t)x with the common domain D := {x ∈ H : x, x  are absolutely continuous, x  ∈ H, x(0) = x(1) = 0}.

320 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

The family of operators {A(t) : t ∈ [0, b]} verifies conditions (A1)–(A4) and in view of Lemma 9.2 it generates a compact evolution system U (t, s) given by U (t, s) = T (t − s)e−

t s

a(τ ) dτ

,

where T (t) for t > 0 is the compact semigroup generated by the operator 2 2 Ax = x  for x ∈ D (see [87]). The √ eigenvalues of A are −n π with the corresponding eigenvectors en (y) = 2 sin(nπy), n ∈ N. Thus, for x ∈ D, we have ∞  (−n2 + a(t))x, en H en ,

A(t)x =

n=1

while for each x ∈ H , it holds U (t, s)x = U ∗ (t, s)x =

∞ 

 2 π 2 (t−s)− t s

e−n

a(τ ) dτ

x, en H en .

n=1

The control space in problem (9.18) is chosen to be the Hilbert space U := {u : u =

∞ 

un en with

n=2 ∞ 

n=2

L(U, H ) in (9.18) is given by ∞ 

u2n < +∞},

n=2

endowed with the norm uU = (

Bu = 2u2 e1 +

∞ 

u2n )1/2 , whereas the mapping B ∈

un en for all u =

n=2

∞ 

un en ∈ U .

n=2

It follows that B ∗ u = (2u1 + u2 )e2 +

∞ 

un en for all u =

n=3

∞ 

un en ∈ H ,

n=1

which gives B ∗ U ∗ (t, s)x = e− +

t s

a(τ ) dτ

∞ 

e−n

(2x1 e−π

2 π 2 (t−s)

2 (t−s)

+ x2 e−4π

2 (t−s)

)e2

 xn en

n=3

for all x =

∞  n=1

xn en ∈ H . We see that, if B ∗ U ∗ (t, s)x = 0 for some t ∈ J ,

then x = 0. This results in the fact that the linear control system corresponding to (9.18) is approximately controllable on J (see [63, Theorem 4.1.7] and [213] for results of this type). Therefore the hypotheses of Corollary 9.15 are satisfied, so problem (9.18) is approximately controllable.

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9.2 EVOLUTIONARY PROBLEMS DRIVEN BY VARIATIONAL INEQUALITIES The problem considered in this section consists of an evolution equation driven by a variational inequality ⎧ ⎪ ˙ = Ax(t) + f (t, x(t), u(t)), t ∈ [0, T ], ⎨ x(t) (9.19) u(t) ∈ S(K, g(t, x(t), ·), φ), t ∈ [0, T ], ⎪ ⎩ x(0) = x0 , where S(K, g(t, x(t), ·), φ) stands for the solution set of the variational inequality: find u : [0, T ] → K such that g(t, x(t), u(t)), v − u(t) + φ(v) − φ(u(t)) ≥ 0 for all v ∈ K.

(9.20)

Here, with real Banach spaces E and E1 , K is a convex subset of E1 , A : D(A) ⊂ E → E is the infinitesimal generator of a C0 -semigroup eAt in E, φ : E1 → (−∞, +∞] is a convex lower semicontinuous ≡ +∞ functional, and f : [0, T ] × E × E1 → E and g : [0, T ] × E × K → E1∗ are given mappings. We introduce a notion of solution for problem (9.19) in a suitable mild sense. Definition 9.16. A pair of functions (x, u), with x ∈ C([0, T ], E) and u : [0, T ] → K measurable, is said to be a mild solution of problem (9.19) if  t x(t) = eAt x0 + eA(t−s) f (s, x(s), u(s)) ds, t ∈ [0, T ], 0

and u(s) ∈ S(K, g(s, x(s), ·), φ), s ∈ [0, T ]. If (x, u) is a mild solution of (9.19), then x is called the mild trajectory and u the variational control. We prepare our results on problem (9.19) by providing needed information on the variational inequality (9.20). We recall a useful classical result. Lemma 9.17 (Ky Fan [79]). Let K be a nonempty subset of a Hausdorff topological vector space E1 and let G : K → 2E1 be a set-valued mapping with the properties: (a) G is a KKM mapping, that is, for any {v1 , v2 , . . . , vn } ⊂ K, one has that n  G(vi ); its convex hull co{v1 , v2 , . . . , vn } is contained in (b) G(v) is closed in E1 for every v ∈ K; (c) G(v0 ) is compact in E1 for some v0 ∈ K.  Then G(v) = ∅.

i=1

v∈K

Given a reflexive Banach space E1 , a nonempty subset K ⊆ E1 , a mapping Q : K → E1∗ , and a function φ : E1 → R ∪ {+∞} which is ≡ +∞, we consider the following variational inequality: find u ∈ K such that Q(u), v − u + φ(v) − φ(u) ≥ 0 for all v ∈ K.

(9.21)

322 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Theorem 9.18. Assume that: (i) Q : K → E1∗ is monotone on K, that is, Q(v) − Q(u), v − u ≥ 0

for all u, v ∈ K,

and satisfies lim infQ(λu + (1 − λ)v), v − u ≤ Q(v), v − u λ→0+

for all u, v ∈ K;

(ii) φ : E1 → R ∪ {+∞} is convex, lower semicontinuous, and ≡ +∞; (iii) if the set K is unbounded in E1 , there exist u0 ∈ K and r > 0 such that Q(v), v − u0  + φ(v) − φ(u0 ) > 0 for all v ∈ K with vE1 > r. Then u ∈ K is a solution of (9.21) if and only if it fulfills Q(v), v − u + φ(v) − φ(u) ≥ 0

for all v ∈ K.

(9.22)

Moreover, the solution set of (9.21) is nonempty, convex and closed in E1 . Proof. Clearly, if u ∈ K is a solution of (9.21), it is also a solution of (9.22) by the monotonicity of Q in assumption (i). Conversely, assume that u ∈ K is a solution of (9.22). Using the convexity of the set K, one has uλ := λv + (1 − λ)u ∈ K for all λ ∈ (0, 1) and all v ∈ K, so Q(uλ ), uλ − u + φ(uλ ) − φ(u) ≥ 0. Then by assumption (ii), there holds Q(uλ ), v − u + φ(v) − φ(u) ≥ 0.

(9.23)

Letting λ → 0 in (9.22) and making use of hypotheses (i) and (ii) imply that u ∈ K solves problem (9.21), proving the required equivalence. In order to establish the properties of the solution set of (9.21), we first suppose that the set K is bounded in E1 . We introduce the multivalued map G : K → 2K by   G(v) := u ∈ K : Q(v), v − u + φ(v) − φ(u) ≥ 0 for all v ∈ K. We notice that G(v) is convex and v ∈ G(v) whenever v ∈ K. The set G(v) is weakly closed in E1 for all v ∈ K. Indeed, taking a sequence {un } ⊂ G(v) with un  u in E1 , we have Q(v), v − un  + φ(v) − φ(un ) ≥ 0.

(9.24)

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Letting n → ∞ in (9.24), on the basis of assumption (ii), results in Q(v), v − u + φ(v) − φ(u) ≥ 0, which means that u ∈ G(v). Now we show that G is a KKM mapping. Arguing by contradiction, assume n  that there exist a finite subset {v1 , v2 , . . . , vn } ⊂ K and u0 = λi vi with λi ∈ [0, 1] and

n 

λi = 1 satisfying u0 ∈ /

i=1

n 

i=1

G(vi ), or equivalently

i=1

Q(vi ), vi − u0  + φ(vi ) − φ(u0 ) < 0 for all i ∈ {1, 2, . . . , n}.

(9.25)

By hypothesis (i), for each i ∈ {1, . . . , n} there holds Q(u0 ) − Q(vi ), vi − u0  ≤ 0.

(9.26)

Then (9.25) and (9.26) lead to Q(u0 ), vi − u0  + φ(vi ) − φ(u0 ) < 0 for all i ∈ {1, . . . , n}, from which the following contradiction arises: 0 = Q(u0 ), u0 − u0  + φ(u0 ) − φ(u0 ) ≤

n 

  λi Q(u0 ), vi − u0  + φ(vi ) − φ(u0 ) < 0.

i=1

Therefore G is a KKM mapping. Since K is bounded, closed, and convex in the reflexive Banach space E1 , it follows that G(v) is weakly compact inE1 for each v ∈ K. We are in a position to apply Lemma 9.17 ensuring that G(v) = ∅. Hence the solution set of v∈K

problem (9.21) is nonempty. Next we suppose that the set K is unbounded in E1 . For every integer n ≥ 1, using the element u0 ∈ K given in assumption (iii) we set Kn := {x ∈ K : x − u0 E1 ≤ n}, which is a bounded, closed and convex subset of E1 . According to the previous case, we can find un ∈ Kn such that Q(un ), v − un  + φ(v) − φ(un ) ≥ 0 for all v ∈ Kn .

(9.27)

We claim that there exists an integer k ≥ 1 for which one has uk − u0 E1 < k.

(9.28)

Arguing by contradiction, assume that un − u0 E1 = n for every n ≥ 1. Inserting v = u0 in (9.27), we observe that Q(un ), un − u0  + φ(un ) − φ(u0 ) ≤ 0,

324 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

which contradicts assumption (iii) provided n is sufficiently large. Hence (9.28) holds. Let x ∈ K. By (9.28), for sufficiently small t > 0 we have that uk + t (x − uk ) − x0 E1 < k. This enables us to set v = uk + t (x − uk ) in (9.27) for n = k, which through the convexity of  yields Q(uk ), x − uk  + φ(x) − φ(uk ) ≥ 0. Consequently, u = uk is a solution of problem (9.21). The equivalence between problems (9.21) and (9.22), in conjunction with assumption (ii), shows that the solution set of (9.21) is closed and convex in E1 , which completes the proof. Corollary 9.19. Let K be a nonempty, compact, and convex subset of a real Banach space E1 . Assume that Q : K → E1∗ and φ : E1 → R verify conditions (i), (ii) in Theorem 9.18. Then the conclusion of Theorem 9.18 is valid. Before continuing, some prerequisites are necessary concerning the measurability of multivalued mappings. Definition 9.20. Let E and E1 be Banach spaces and let an interval I ⊂ R. We say that F : I × E → 2E1 with nonempty values is superpositionally measurable if, for every measurable multivalued mapping Q : I → 2E with nonempty compact values, the composition  : I → 2E1 given by (t) = F (t, Q(t)) is measurable. Theorem 9.21 (see [238]). If F : I × E → 2E1 with nonempty compact values satisfies the Carathéodory condition or F is upper or lower semicontinuous, then F is superpositionally measurable in the sense of Definition 9.20. Theorem 9.22 ([115, Theorem 1.3.5]). Let E and E1 be Banach spaces. Assume that, for some 0 < T < +∞, the multivalued mapping G : [0, T ] × E → 2E1 with nonempty compact values satisfies: (a) for every x ∈ E, G(·, x) : [0, T ] → 2E1 has a strongly measurable selection; (b) for a.e. t ∈ [0, T ], G(t, ·) : E → 2E1 is upper semicontinuous. Then, for every strongly measurable function q : [0, T ] → E, there exists a strongly measurable selection g : [0, T ] → E1 of the composition M : [0, T ] → 2E1 , M(t) = G(t, q(t)) for a.e. t ∈ [0, T ]. We focus on certain properties of the solution set for a class of variational inequalities related to problem (9.19). Theorem 9.23. Assume that K is a nonempty, compact, and convex subset of a real Banach space E1 . Assume that E is a real separable Banach space and, for some T > 0, the mapping g : [0, T ] × E × K → E1∗ is such that g(·, ·, u) is

Control Problems for Evolutionary Differential Inclusions Chapter | 9

325

continuous from [0, T ] × E to E1∗ endowed with the weak∗ topology whenever u ∈ K. In addition, we assume that for every (t, x) ∈ [0, T ] × E the mappings Q := g(t, x, ·) and φ : E1 → R ∪ {+∞} satisfy hypotheses (i), (ii) in Theorem 9.18. Then the multivalued mapping U : [0, T ] × E → 2K defined by U (t, x) := {u ∈ K : g(t, x, u), v − u + φ(v) − φ(u) ≥ 0

for all v ∈ K} (9.29)

for all (t, x) ∈ [0, T ] × E has nonempty values and fulfills: (U1 ) U is upper semicontinuous; (U2 ) U is superpositionally measurable. Proof. Corollary 9.19 guarantees that for every t ∈ [0, T ] and x ∈ E the set U (t, x) is nonempty, so the multivalued mapping U : [0, T ] × E → 2K has nonempty values. In order to prove assertion (U1 ), we need to show that for each closed subset C ⊂ K, the set U − (C) := {(t, x) ∈ [0, T ] × E : U (t, x) ∩ C = ∅} is closed in R × E. To this end, let a sequence {(tn , xn )} ⊂ U − (C) satisfy (tn , xn ) → (t, x) in R × E. Thus we can choose un ∈ U (tn , xn ) ∩ C, and (9.29) and Corollary 9.19 render g(tn , xn , v), v − un  + φ(v) − φ(un ) ≥ 0 for all v ∈ K.

(9.30)

By the compactness of K, passing to a relabeled subsequence, we may assume that un → u in E1 for some u ∈ C. Letting n → ∞ in (9.30), from our assumptions we obtain g(t, x, v), v − u + φ(v) − φ(u) ≥ 0 for all v ∈ K. This reads as u ∈ U (t, x) ∩ C, whence (t, x) ∈ U − (C), thereby (U1 ). Finally, assertion (U1 ) allows us to apply Theorem 9.21, which ensures that (U2 ) holds true. This completes the proof. In order to study problem (9.19) with the set of constraints K ⊂ E1 we impose the following assumptions on the mapping f : [0, T ] × E × E1 → E in (9.19): H(f ) (i) for every (t, x) ∈ [0, T ] × E and every convex set D ⊂ K, the set f (t, x, D) is convex in E; (ii) there exists ψ ∈ L1 ([0, T ]) such that f (t, x, u)E ≤ ψ(t)(1 + xE ) for all (t, x, u) ∈ [0, T ] × E × K; (iii) f (·, x, u) : [0, T ] → E is measurable for every (x, u) ∈ E × E1 ; (iv) f (t, ·, ·) : E × E1 → E is continuous for a.e. t ∈ [0, T ]; (v) there exists k ∈ L1 ([0, T ]) such that, for a.e. t ∈ [0, T ], every x0 , x1 ∈ E and u ∈ K, f (t, x0 , u) − f (t, x1 , u)E ≤ k(t)x0 − x1 E .

326 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

Remark 9.24. A special case of condition H(f ) (i) is when f (t, x, ·) : K → E1 is affine for all (t, x) ∈ [0, T ] × E. In Pang–Stewart [196] and Liu–Loi– Obukhovskii [136] a function f of the form f (t, x, u) = g(t, x) + B(t, x)u is utilized. Next we briefly discuss the important notion of measure of noncompactness that is needed in the sequel. Definition 9.25. Let E be a Banach space and let (A, ≤) be a partially ordered set. A mapping β : 2E → A is called a measure of noncompactness (MNC, for short) in E if β(co ) = β( ) for every ∈ 2E . An important example of MNC in the sense of Definition 9.25 is the Hausdorff MNC χ defined by diameters of sets:  χ( ) := inf  > 0 :∃ i ∈ 2E with diam( i ) ≤ , i = 1, . . . , n, and ⊆

n 

 i .

i=1

We also use the monotone nonsingular MNC in the space C([0, T ], E) introduced in [115, Example 2.1.4]. Fixing a constant L > 0, for each nonempty bounded set ⊂ C([0, T ], E) it is equal to ν( ) := max {γ (ω), modC (ω)}, ω∈( )

(9.31)

where ( ) denotes the collection of all countable subsets of , while γ (ω) := sup e−Lt χ(ω(t)), modC (ω) := lim sup max x(t1 ) − x(t2 )E . t∈[0,T ]

δ→0 x∈ω |t1 −t2 |≤δ

In (9.31) the maximum is viewed in the sense of the order on A = R2 . Let β1 and β2 be real MNCs (that is, A = [0, +∞]) in E1 and E2 , respectively. For k ≥ 0, the multivalued map F : X ⊂ E1 → K(E2 ) is said to be (k, β1 , β2 )-bounded if β2 (F ( )) ≤ kβ1 ( ) for all ⊆ X. The next criterion of (k, β1 , β2 )-boundedness is quoted from [115, Proposition 2.2.2]. Theorem 9.26. Let χ1 and χ2 be the Hausdorff MNCs in E1 and E2 , respectively. Suppose that X ⊂ E1 and F : X × E1 → 2E2 with nonempty compact values satisfy the conditions: (i) there exists k ∈ R+ such that for every x ∈ X, F (x, ·) : E1 → K(E2 ) is k-Lipschitz with respect to the Hausdorff metric H , that is, H (F (x, y1 ), F (x, y2 )) ≤ ky1 − y2  for all y1 , y2 ∈ E1 ;

Control Problems for Evolutionary Differential Inclusions Chapter | 9

327

(ii) the set F ( × {y}) is relatively compact in E2 for every bounded subset ⊂ X and y ∈ E1 . Then G : X → K(E2 ) defined as G(x) = F (x, x) is (k, χ1 , χ2 )-bounded. The following definition will be useful in the forthcoming multivalued fixed point approach. Definition 9.27. Let X be a closed subset of a Banach space E and let β be a real MNC in E. A multivalued map F : X → 2E with nonempty compact values is said to be β-condensing if there is k ∈ [0, 1) such that β(F ( )) ≤ kβ( ) for every ⊂ X. Relevant properties of the multivalued mapping F : [0, T ] × E → 2E given by F (t, x) = f (t, x, U (t, x)), with U introduced in (9.29), are listed below. Lemma 9.28. Let E and E1 be real Banach spaces, with E separable, and let K be a nonempty compact and convex subset of E1 . Assume that the hypotheses of Theorem 9.23, as well as conditions H(f ), are satisfied. Then: (i) (ii) (iii) (iv)

for all (t, x) ∈ [0, T ] × E, F (t, x) is nonempty, compact, and convex in E; for every x ∈ E, F (·, x) has a strongly measurable selection; for a.e. t ∈ [0, T ], F (t, ·) is upper semicontinuous; for every bounded subset D ⊂ E, there exists l ∈ L1 ([0, T ]) such that χ(F (t, D)) ≤ l(t)χ(D) for a.e. t ∈ [0, T ], where χ is the Hausdorff MNC in E.

Proof. (i) Corollary 9.19 and Theorem 9.23 ensure that the multivalued mapping U takes nonempty, compact and convex values in K. Then hypotheses H(f ) (i), (iv) imply the validity of assertion (i). (ii) By virtue of H(f ) (iii)–(iv), we have that f (·, x, ·) : [0, T ] × E1 → E satisfies the Carathéodory condition. Notice that U (t, x) is nonempty, compact in E1 for all (t, x) ∈ [0, T ] × E. Invoking assertion (U1 ) of Theorem 9.23, together with Theorem 9.21, we infer that F (·, x) = f (·, x, U (·, x)) is measurable for every x ∈ E, so it is strongly measurable because E is separable (see Definition 1.1). Therefore, taking into account (i), F (·, x) has a strongly measurable selection whenever x ∈ E (see [115]). (iii) For any t ∈ [0, T ], as known from property (U2 ) in Theorem 9.23, the multivalued mapping F1 = (·, U (t, ·)) : E → E × K is upper semicontinuous, whereas hypothesis H(f ) (iv) guarantees that the mapping F2 = f (t, ·, ·) : E × E1 → E is continuous. Then, since F (t, ·) = F2 ◦ F1 , we can infer that F (t, ·) is upper semicontinuous.

328 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

(iv) Fixing t ∈ [0, T ], in view of Theorem 9.23, the multivalued mapping G : E × E → 2E given by G(x, y) = f (t, y, U (t, x)) for all (x, y) ∈ E × E has nonempty, compact, convex values. For each bounded subset D ⊂ E, G(D, y) = f (t, y, U (t, D)) is relatively compact in E whenever y ∈ E due to the facts that U (t, D) ⊂ K and K is compact. Hence the requirement (ii) in Theorem 9.26 is verified. Let x, y  , y  ∈ E. If z ∈ G(x, y  ), there is u ∈ U (t, x) such that z = f (t, y  , u), and so z = f (t, y  , u) ∈ G(x, y  ). Assumption H(f ) (v) enables us to get z − z E = f (t, y  , u) − f (t, y  , u)E ≤ k(t)y  − y  E . In this way, the requirement (i) in Theorem 9.26 is fulfilled. Consequently, we can apply Theorem 9.26, obtaining the desired conclusion with l = k ∈ L1 ((0, T )). From parts (ii) and (iii) of Lemma 9.28 we see that the assumptions of Theorem 9.22 with F in place of G are fulfilled. Hence F (·, q(·)) has strongly measurable selections for every q ∈ C([0, T ], E). So, it is meaningful to define 1 the set-valued mapping PF : C([0, T ], E) → 2L ([0,T ],E) by PF (q) := {g : g is strongly measurable and g(t) ∈ F (t, q(t)) for a.e. t ∈ [0, T ]} for all q ∈ C([0, T ], E). In view of part of Lemma 9.28 (i), we can introduce the multivalued map : C([0, T ], E) → 2C([0,T ],E) with nonempty, closed, convex values by 



x := y ∈ C([0, T ], E) : y(t) = e x0 + At

t

e

A(t−s)

 h(s) ds, h ∈ PF (x) .

0

(9.32) Here A : D(A) ⊂ E → E is the infinitesimal generator of the C0 -semigroup eAt in E as given in problem (9.19). Theorem 9.29. Under the hypotheses of Lemma 9.28, the multivalued map

in (9.32) is upper semicontinuous and ν-condensing in the sense of Definition 9.27 on every closed bounded subset of C([0, T ], E), with ν constructed in (9.31). Proof. The conclusion is achieved by combining Lemma 4.2.1, Corollary 5.1.2, Theorem 5.1.3, and Theorem 5.1.4 of [115] with Lemma 9.28. The proof of our main result relies on the following fixed point theorems that we cite from [115].

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Theorem 9.30. If S is a nonempty, closed, convex subset of a Banach space E and F : S → 2S is a closed, β-condensing multivalued map in the sense of Definition 9.27, with nonempty compact convex values and a nonsingular MNC β in E, then the set of fixed points of F is nonempty. Theorem 9.31. Let C be a nonempty closed subset of a Banach space E and let F : C → 2E be a closed multivalued mapping, with nonempty compact values, which is β-condensing on every bounded subset of C, with a monotone MNC β in E. If the set of fixed points of F is bounded, then it is compact in E. We are now able to present our main result regarding problem (9.19). Theorem 9.32. Under the hypotheses of Lemma 9.28, the solution set of problem (9.19) in the sense of Definition 9.16 is nonempty and the set of all corresponding mild trajectories of (9.19) is compact in C([0, T ], E). Proof. Consider the evolutionary differential inclusion associated to problem (9.19):  x(t) ˙ ∈ Ax(t) + F (t, x(t)), t ∈ [0, T ], (9.33) x(0) = x0 , where, as before, F (t, x) = f (t, x, U (t, x)) with U (t, x) defined in (9.29). Step 1. The solution set of (9.33) is nonempty. By (9.32) it is equivalent to show that the set of fixed points of is nonempty. Fix a positive constant L sufficiently large such that 

t

M

e−L(t−s) ψ(s)ds < 1 for all t ∈ [0, T ],

(9.34)

0

with ψ ∈ L1 ([0, T ]) in assumption H(f ) (ii) and M := max eAt . According t∈[0,T ]

to (9.34), we may choose r > 0 such that   M x0 E + ψL1 ([0,T ]) + Mr



t

e−L(t−s) ψ(s)ds ≤ r for all t ∈ [0, T ].

0

(9.35) For the equivalent norm on the space C([0, T ], E) given by x∗ := max e−Lt x(t)E , t∈[0,T ]

we consider the closed ball

  Br (0) := x ∈ C([0, T ], E) : x∗ ≤ r .

330 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

We prove that maps Br (0) into Br (0). Indeed, let y ∈ x with x ∈ Br (0). From (9.32), there is h ∈ PF (x) such that  t eA(t−s) h(s)ds y(t) = eAt x0 + 0

whenever t ∈ [0, T ]. Then H(f ) (ii) yields the estimate  t e−Lt y(t)E = e−Lt  eAt x0 + eA(t−s) h(s)ds E 0



≤ M(x0 E + ψL1 ([0,T ]) ) + Mx∗

t

e−L(t−s) ψ(s)ds.

0

Taking into account that x ∈ Br (0) and (9.35), we find  t −Lt e y(t)E ≤ M(x0 E + ψL1 ([0,T ]) ) + Mr e−L(t−s) ψ(s)ds ≤ r, 0

which means that y∗ ≤ r. Through Theorem 9.29, we apply Theorem 9.30 with S = Br (0) and F =

obtaining that the set of fixed points of is nonempty, so Step 1 holds true. Step 2. The solution set of (9.33) is compact in C([0, T ], E). If x ∈ C([0, T ], E) is solution of (9.33), then with certain h ∈ PF (x), from the growth condition H(f ) (ii) we have for any t ∈ [0, T ] that  t At x(t)E ≤ e  x0 E + eA(t−s)  h(s)E ds, 0    t ψ(s)x(s)E ds ≤ M x0 E + ψL1 ([0,T ]) + 0

with M > 0 introduced in Step 1. Through Gronwall’s inequality, we get the estimate x(t)E ≤ M(x0 E + ψL1 ([0,T ]) )e

MψL1 ([0,T ])

showing that the set of fixed points of is bounded in C([0, T ], E). Due to Theorem 9.29, we can invoke Theorem 9.31 with F = in its statement. We deduce that the solution set of problem (9.33), which coincides with the set of fixed points of , is compact in C([0, T ], E). Consequently, the set of all mild trajectories of problem (9.19) is compact in C([0, T ], E). Step 3. The solution set of (9.19) is nonempty. Assertion (U2 ) of Theorem 9.23 guarantees that multivalued map U : [0, T ] × E → 2K is superpositionally measurable as formulated in Definition 9.20. This allows us to apply Filippov implicit function lemma (see

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331

[85,115]) to conclude that for every trajectory x ∈ C([0, T ], E) of (9.33) there exists a measurable selection u(t) ∈ U (t, x(t)) such that x(t) ˙ = Ax(t) + f (t, x(t), u(t)) for t ∈ [0, T ], which ensures that (x, u) is a mild solution of problem (9.19) in the sense of Definition 9.25. The proof is complete.

9.3 NOTES Section 9.1 focuses on the approximate controllability for control systems and inclusions, which is a property frequently studied (see [24,63,137,142,213]). Our basic reference is Liu–Li–Motreanu [135] from which the main results are extracted. Specifically, we investigate the approximate controllability of the control inclusion problem (9.1), which contains as a particular case the control hemivariational inequality (9.2). Hemivariational inequalities have been recognized as an interesting subject in mathematics, constituting also one of the best tools to describe the mechanical problems with nonsmooth and nonconvex energy potentials. For existence of solutions for hemivariational inequalities under different hypotheses, we refer to [29,37,39,58,133,167,174,177,192,194,199,208]. Motivated by the wide applications in engineering, physics, and finance, some works focused on the optimal control problems for hemivariational inequalities as [110,153,197]. A powerful method to study the controllability properties of nonlinear systems is to use the fixed point theory (see, e.g., [24,137,142,213]). The first paper in this direction for hemivariational inequalities is Liu–Li [134]. For investigating the control problems (9.1) and (9.2) we also develop a fixed point approach. The first main result of Section 9.1 is stated as Theorem 9.10, providing the existence of solutions in a mild sense of the control evolutionary inclusion problem (9.1), which is more general than the inequality problem (9.2). Next, to (9.1) and (9.2) we associate a linear control system (9.7). The main controllability result presented in Theorem 9.13 establishes that the approximate controllability of the associated linear system (9.7) implies the approximate controllability of the control evolutionary inclusion problem (9.1), and in particular of the control hemivariational inequality (9.2). It extends the criterion of approximate controllability in [213, Theorem 4.5] and [134, Theorem 4.4] from the time-invariant case to a control inclusion problem with time-depending operators A(t). Explicit expressions for the involved approximate controls and corresponding states are indicated. Theorem 9.14 sets forth an application of the approximate controllability result to a heat equation. The object of Section 9.2 is a nonstandard problem stated in (9.19) which consists in mixing an evolutionary differential equation with a variational inequality depending mutually on their solutions. The exposition of this section is carried based on the results of Liu–Zeng–Motreanu [139]. Aspects related to problem (9.19) have been examined until now only in a finite-dimensional context or with A = 0 and other particularities (see [108,128, 136,196]). In those works an ordinary differential equation is parameterized by

332 Nonlinear Differential Problems with Smooth and Nonsmooth Constraints

a variable required to solve a variational inequality in the state variable of the differential equation. Such problems taking into account simultaneously both dynamics and constraints in the form of inequalities arise in electrical circuits with ideal diodes, Coulomb friction problems for contacting bodies, economical dynamics, dynamic traffic networks, control systems (see [52,108,128,135,136, 195,196]). Mixing partial differential equations and inequality problems with constraints is an open subject. The crucial point in problem (9.19) is that it is described by a partial differential equation instead of an ordinary differential equation. Hence we can take, for example, the operator A to be the Laplacian (see (1.34)), subject to an infinite-dimensional variational inequality (9.20). This incorporates large classes of problems and models. In order to investigate problem (9.19), we first prove Theorem 9.18, which includes many important situations and extends the results in Costea–R˘adulescu [57] and Liu–Zeng [138]. Assertion (9.22) is an extension of Minty’s technique, whereas assumption (iii) in Theorem 9.18 expresses a generalized coercivity condition. Related results can be found in [118]. Theorem 9.23 shows essential properties of measurability and upper semicontinuity for the solution set of the general variational inequality associated to (9.19). Then, based on these properties, the main result of Section 9.2 is established in Theorem 9.32, stating that the solution set of problem (9.19) is nonempty and compact. Various analytical and topological tools such as KKM theorem, monotonicity, semigroup theory, measure of noncompactness, Filippov implicit function lemma, fixed points for set-valued mappings are used in the proof.

Bibliography

[1] Adams R. Sobolev spaces. Pure Appl Math. New York–London: Academic Press; 1975. [2] Aizicovici S, Papageorgiou NS, Staicu V. The spectrum and an index formula for the Neumann p-Laplacian and multiple solutions for problems with a crossing nonlinearity. Discrete Contin Dyn Syst Ser A 2009;25(2):431–56. [3] Alber YI, Notik AI. Perturbed unstable variational inequalities with unbounded operators on approximately given sets. Set-Valued Anal 1993;1(4):393–402. [4] Alves CO, Carrião PC, Faria LFO. Existence of solutions to singular elliptic equations with convection terms via the Galerkin method. Electron J Differential Equations 2010;(12). 12 pp. [5] Alves CO, Corrêa FJSA. On the existence of positive solution for a class of singular systems involving quasilinear operators. Appl Math Comput 2007;185(1):727–36. [6] Alves CO, Corrêa FJSA, Gonçalves JVA. Existence of solutions for some classes of singular Hamiltonian systems. Adv Nonlinear Stud 2005;5(2):265–78. [7] Alves CO, Moussaoui A. Existence of solutions for a class of singular elliptic systems with convection term. Asymptot Anal 2014;90(3–4):237–48. [8] Amann H, Crandall MG. On some existence theorems for semi-linear elliptic equations. Indiana Univ Math J 1978;27(5):779–90. [9] Ambrosetti A, Brezis H, Cerami G. Combined effects of concave and convex nonlinearities in some elliptic problems. J Funct Anal 1994;122(2):519–43. [10] Ambrosetti A, García Azorero J, Peral I. Multiplicity results for some nonlinear elliptic equations. J Funct Anal 1996;137(1):219–42. [11] Ambrosetti A, Lupo D. On a class of nonlinear Dirichlet problems with multiple solutions. Nonlinear Anal 1984;8(10):1145–50. [12] Ambrosetti A, Mancini G. Sharp nonuniqueness results for some nonlinear problems. Nonlinear Anal 1979;3(5):635–45. [13] Ambrosetti A, Prodi G. A primer of nonlinear analysis. Cambridge Stud Adv Math, vol. 34. Cambridge: Cambridge University Press; 1993. [14] Anane A. Simplicité et isolation de la première valeur propre du p-laplacien avec poids. C R Acad Sci Paris Sér I Math 1987;305(16):725–8. [15] Arias M, Campos J. Radial Fuˇcik spectrum of the Laplace operator. J Math Anal Appl 1995;190(3):654–66. [16] Arias M, Campos J, Gossez J-P. On the antimaximum principle and the Fuˇcik spectrum for the Neumann p-Laplacian. Differential Integral Equations 2000;13(1–3):217–26. [17] Arias M, Campos J, Cuesta M, Gossez J-P. An asymmetric Neumann problem with weights. Ann Inst H Poincaré Anal Non Linéaire 2008;25(2):267–80. [18] Averna D, Marano SA, Motreanu D. Multiple solutions for a Dirichlet problem with p-Laplacian and set-valued nonlinearity. Bull Aust Math Soc 2008;77(2):285–303. [19] Averna D, Motreanu D, Tornatore E. Existence and asymptotic properties for quasilinear elliptic equations with gradient dependence. Appl Math Lett 2016;61:102–7. [20] Baiocchi C, Capelo A. Variational and quasivariational inequalities. Applications to free boundary problems. New York: John Wiley & Sons, Inc.; 1984.

333

334 Bibliography

[21] Barbu V. Nonlinear semigroups and differential equations in Banach spaces. Leiden: Noordhoff International Publishing; 1976. [22] Barletta G, Candito P, Motreanu D. Constant sign and sign-changing solutions for quasilinear elliptic equations with Neumann boundary condition. J Convex Anal 2014;21(1):53–66. [23] Barletta G, Papageorgiou NS. A multiplicity theorem for the Neumann p-Laplacian with an asymmetric nonsmooth potential. J Global Optim 2007;39(3):365–92. [24] Bashirov AE, Mahmudov NI. On concepts of controllability for deterministic and stochastic systems. SIAM J Control Optim 1999;37(6):1808–21. [25] Benouhiba N, Belyacine Z. On the solutions of the (p, q)-Laplacian problem at resonance. Nonlinear Anal 2013;77:74–81. [26] Bensoussan A, Lions J-L. Nouvelles méthodes en contrôle impulsionnel. Appl Math Optim 1974/1975;1(4):289–312. [27] Boccardo L, Gallouët T, Orsina L. Existence and nonexistence of solutions for some nonlinear elliptic equations. J Anal Math 1997;73:203–23. [28] Bonanno G, Candito P. Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian. Arch Math (Basel) 2003;80(4):424–9. [29] Bonanno G, Motreanu D, Winkert P. Boundary value problems with nonsmooth potential, constraints and parameters. Dynam Systems Appl 2013;22(2–3):385–96. [30] Brezis H. Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holl Math Stud, vol. 5. Amsterdam–London: North-Holland Publishing Co.; 1973. [31] Brezis H. Functional analysis, Sobolev spaces and partial differential equations. Universitext. New York: Springer; 2011. [32] Brezis H, Cazenave T. Course notes. [33] Bueno H, Ercole G. A quasilinear problem with fast growing gradient. Appl Math Lett 2013;26(4):520–3. [34] Các NP. On nontrivial solutions of a Dirichlet problem whose jumping nonlinearity crosses a multiple eigenvalue. J Differential Equations 1989;80(2):379–404. [35] Caldiroli P, Montecchiari P. Homoclinic orbits for second order Hamiltonian systems with potential changing sign. Comm Appl Nonlinear Anal 1994;1(1):97–129. [36] Carl S, Heikkilä S. Fixed point theory in ordered sets and applications. From differential and integral equations to game theory. New York: Springer; 2011. [37] Carl S, Le VK. Multi-valued parabolic variational inequalities and related variational– hemivariational inequalities. Adv Nonlinear Stud 2014;14(3):631–59. [38] Carl S, Le VK, Motreanu D. Nonsmooth variational problems and their inequalities. Comparison principles and applications. Springer Monogr Math. New York: Springer; 2007. [39] Carl S, Motreanu D. Extremal solutions of quasilinear parabolic inclusions with generalized Clarke’s gradient. J Differential Equations 2003;191(1):206–33. [40] Carl S, Motreanu D. Constant-sign and sign-changing solutions for nonlinear eigenvalue problems. Nonlinear Anal 2008;68(9):2668–76. [41] Carl S, Motreanu D. General comparison principle for quasilinear elliptic inclusions. Nonlinear Anal 2009;70(2):1105–12. [42] Carl S, Motreanu D. Multiple and sign-changing solutions for the multivalued p-Laplacian equation. Math Nachr 2010;283(7):965–81. [43] Carl S, Motreanu D. Constant-sign and sign-changing solutions of a nonlinear eigenvalue problem involving the p-Laplacian. Differential Integral Equations 2007;20(3):309–24. [44] Carl S, Motreanu D. Sign-changing and extremal constant-sign solutions of nonlinear elliptic problems with supercritical nonlinearities. Comm Appl Nonlinear Anal 2007;14(4):85–100. [45] Carl S, Motreanu D. Sign-changing solutions for nonlinear elliptic problems depending on parameters. Int J Differ Equ 2010:536236. [46] Carl S, Motreanu D. Multiple solutions of nonlinear elliptic variational problems. In: Nonlinear analysis research trends. New York: Nova Sci. Publ.; 2009. p. 235–55.

Bibliography 335

[47] Carl S, Motreanu D. Multiple solutions for elliptic systems via trapping regions and related nonsmooth potentials. Appl Anal 2015;94(8):1594–613. [48] Carl S, Motreanu D. Extremal solutions for nonvariational quasilinear elliptic systems via expanding trapping regions. Monatsh Math 2017;182(4):801–21. [49] Carl S, Perera K. Sign-changing and multiple solutions for the p-Laplacian. Abstr Appl Anal 2002;7(12):613–25. [50] Carriao PC, Miyagaki OH. Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems. J Math Anal Appl 1999;230(1):157–72. [51] Chang KC. Variational methods for nondifferentiable functionals and their applications to partial differential equations. J Math Anal Appl 1981;80(1):102–29. [52] Chen X, Wang Z. Convergence of regularized time-stepping methods for differential variational inequalities. SIAM J Optim 2013;23(3):1647–71. [53] Cianchi A, Maz’ya V. Global gradient estimates in elliptic problems under minimal data and domain regularity. Commun Pure Appl Anal 2015;14(1):285–311. [54] Cingolani S, Degiovanni M. Nontrivial solutions for p-Laplace equations with righthand side having p-linear growth at infinity. Comm Partial Differential Equations 2015;30(7–9):1191–203. [55] Clarke FH. Optimization and nonsmooth analysis. New York: John Wiley & Sons Inc.; 1983. [56] Costea N, Csirik M, Varga C. Linking-type results in nonsmooth critical point theory and applications. Set-Valued Var Anal 2017;25(2):333–56. [57] Costea N, R˘adulescu V. Inequality problems of quasi-hemivariational type involving setvalued operators and a nonlinear term. J Global Optim 2017;52(4):743–56. [58] Costea N, Varga C. Systems of nonlinear hemivariational inequalities and applications. Topol Methods Nonlinear Anal 2013;41(1):39–65. [59] Coti Zelati V, Rabinowitz PH. Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J Amer Math Soc 1991;4(4):693–727. [60] Cuesta M, de Figueiredo D, Gossez JP. The beginning of the Fuˇcik spectrum for the p-Laplacian. J Differential Equations 1999;159(1):212–38. [61] Cuesta M, Takáˇc P. Nonlinear eigenvalue problems for degenerate elliptic systems. Differential Integral Equations 2010;23(11–12):1117–38. [62] Curtain RF, Pritchard AJ. Infinite dimensional linear systems theory. Lecture Notes in Control and Inform Sci, vol. 8. New York: Springer-Verlag; 1978. [63] Curtain RF, Zwart H. An introduction to infinite dimensional linear systems theory. Texts Appl Math, vol. 21. New York: Springer-Verlag; 1995. [64] Damascelli L. Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. Ann Inst H Poincaré Anal Non Linéaire 1998;15(4):493–516. [65] Dancer EN. On the Dirichlet problem for weakly non-linear elliptic partial differential equations. Proc Roy Soc Edinburgh Sect A 1976/1977;76(4):283–300. [66] Dancer EN. Generic domain dependence for nonsmooth equations and the open set problem for jumping nonlinearities. Topol Methods Nonlinear Anal 1993;1(1):139–50. [67] Dancer EN, Perera K. Some remarks on the Fucík spectrum of the p-Laplacian and critical groups. J Math Anal Appl 2001;254(1):164–77. [68] De Figueiredo D, Girardi M, Matzeu M. Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques. Differential Integral Equations 2004;17(1–2):119–26. [69] De Figueiredo D, Gossez J-P. On the first curve of the Fuˇcik spectrum of an elliptic operator. Differential Integral Equations 1994;7(5–6):1285–302. [70] Deng S-G. Positive solutions for Robin problem involving the p(x)-Laplacian. J Math Anal Appl 2009;360(2):548–60. [71] Deimling K. Nonlinear functional analysis. Berlin: Springer-Verlag; 1985. [72] DiBenedetto E. C 1+a local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal 1983;7(8):827–50.

336 Bibliography

[73] Diestel J. Sequences and series in Banach spaces. Grad Texts in Math, vol. 92. New York: Springer-Verlag; 1984. [74] Drábek P. Solvability and bifurcations of nonlinear equations. Harlow: Longman Scientific & Technical; 1982. [75] Drábek P. The least eigenvalue of nonhomogeneous degenerated quasilinear eigenvalue problems. Math Bohem 1995;120(2):169–95. [76] Dunford N, Schwartz JT. Linear operators. I. General theory. London: Interscience Publishers, Ltd.; 1958. [77] El Manouni S, Perera K, Shivaji R. On singular quasi-monotone (p, q)-Laplacian systems. Proc Roy Soc Edinburgh Sect A 2012;142(3):585–94. [78] Evans LC, Gariepy RF. Measure theory and fine properties of functions. Stud Adv Math. Boca Raton (FL): CRC Press; 1992. [79] Fan K. Some properties of convex sets related to fixed point theorems. Math Ann 1984;266(4):519–37. [80] Faraci F, Motreanu D, Puglisi D. Positive solutions of quasi-linear elliptic equations with dependence on the gradient. Calc Var Partial Differential Equations 2015;54(1):525–38. [81] Faria LFO, Miyagaki O, Motreanu D. Comparison and positive solutions for problems with the (p, q)-Laplacian and a convection term. Proc Edinb Math Soc (2) 2014;57(3):687–98. [82] Faria LFO, Miyagaki O, Motreanu D, Tanaka M. Existence results for nonlinear elliptic equations with Leray–Lions operator and dependence on the gradient. Nonlinear Anal 2014;96:154–66. [83] Felmer P, Montenegro M, Quass A. A note on the strong maximum principle and the compact support principle. J Differential Equations 2009;246(1):39–49. [84] Fernández Bonder J, Rossi JD. Existence results for the p-Laplacian with nonlinear boundary conditions. J Math Anal Appl 2001;263(1):195–223. [85] Filippov AF. On some problems of the theory of optimal control. Vestn Mosk Univ 1959;2:25–32. [86] Fitzgibbon WE. Semilinear functional differential equations in Banach spaces. J Differential Equations 1978;29(1):1–14. [87] Fu XL, Zhang WE. Exact null controllability of non-autonomous functional evolution system with nonlocal conditions. Acta Math Sci Ser B 2013;3(3):747–57. ˇ [88] Fuˇcík S. Boundary value problems with jumping nonlinearities. Cas Pˇest Mat 1976;101(1):69–87. [89] Fuˇcík S. Solvability of nonlinear equations and boundary value problems. Dordrecht: D. Reidel Publishing Co.; 1980. [90] Fukagai N, Narukawa K. On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems. Ann Mat Pura Appl (4) 2007;186(3):539–64. [91] García Azorero JP, Manfredi J, Peral Alonso I. Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations. Commun Contemp Math 2000;2(3):385–404. [92] García-Huidobro M, Manásevich R, Ubilla P. Existence of positive solutions for some Dirichlet problems with an asymptotically homogeneous operator. Electron J Differential Equations 1995;1995(10). 22 pp. [93] Gasi´nski L, Papageorgiou NS. Positive solutions for nonlinear elliptic problems with dependence on the gradient. J Differential Equations 2017;263(2):1451–76. [94] Gasi´nski L, Papageorgiou NS. Nonsmooth critical point theory and nonlinear boundary value problems. Ser Math Anal Appl, vol. 8. Boca Raton (FL): Chapman & Hall; 2005. [95] Gasi´nski L, Papageorgiou NS. Nonlinear analysis. Ser Math Anal Appl, vol. 9. Boca Raton (FL): Chapman & Hall; 2006. [96] Ghergu M. Lane–Emden systems with negative exponents. J Funct Anal 2010;258(10):3295–318. [97] Ghergu M. Lane–Emden systems with singular data. Proc Roy Soc Edinburgh Sect A 2011;141(6):1279–94.

Bibliography 337

[98] Ghoussoub N. Duality and perturbation methods in critical point theory. Cambridge Tracts in Math, vol. 107. Cambridge: Cambridge University Press; 1993. [99] Giacomoni J, Hernández J, Moussaoui A. Quasilinear and singular systems: the cooperative case. In: Nonlinear elliptic partial differential equations. Contemp Math, vol. 540. Providence (RI): Amer. Math. Soc.; 2011. p. 79–94. [100] Giacomoni J, Hernández J, Sauvy P. Quasilinear and singular elliptic systems. Adv Nonlinear Anal 2013;2(1):1–41. [101] Giacomoni J, Schindler I, Takáˇc P. Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation. Ann Sc Norm Super Pisa Cl Sci (5) 2007;6(1):117–58. [102] Giannessi F, Khan A. Regularization of non-coercive quasi variational inequalities. Control Cybernet 2000;29(1):91–110. [103] Gilbarg D, Trudinger N. Elliptic partial differential equations of second order. Classics Math. Berlin: Springer-Verlag; 2001. [104] Goeleven D, Motreanu D, Dumont Y, Rochdi M. Variational and hemivariational inequalities: theory, methods and applications, vol. I. Unilateral analysis and unilateral mechanics. Nonconvex Optim Appl, vol. 69. Boston: Kluwer Academic Publishers; 2003. [105] Goeleven D, Motreanu D. Variational and hemivariational inequalities: theory, methods and applications, vol. II. Unilateral problems. Nonconvex Optim Appl, vol. 70. Boston: Kluwer Academic Publishers; 2003. [106] Gol’dshtein V, Motreanu D, Motreanu VV. Non-homogeneous Dirichlet boundary value problems in weighted Sobolev spaces. Complex Var Elliptic Equ 2015;60(3):372–91. [107] Granas A, Dugundji J. Fixed point theory. Springer Monogr Math. New York: SpringerVerlag; 2003. [108] Gwinner J. On a new class of differential variational inequalities and a stability result. Math Program Ser B 2013;139(1–2):205–21. [109] Hai DD. On a class of singular p-Laplacian boundary value problems. J Math Anal Appl 2011;383(2):619–26. [110] Haslinger J, Panagiotopoulos PD. Optimal control of systems governed by hemivariational inequalities. Existence and approximation results. Nonlinear Anal 1995;24(1):105–19. [111] Hernández J, Mancebo FJ, Vega JM. Positive solutions for singular semilinear elliptic systems. Adv Differential Equations 2008;13(9–10):857–80. [112] Hu S, Papageorgiou NS. Handbook of multivalued analysis, vol. I. Theory. Math Appl. Dordrecht: Kluwer Academic Publishers; 1997. [113] Iannizzotto A, Papageorgiou NS. Existence of three nontrivial solutions for nonlinear Neumann hemivariational inequalities. Nonlinear Anal 2009;70(9):3285–97. [114] Jin Z. Multiple solutions for a class of semilinear elliptic equations. Proc Amer Math Soc 1997;125(12):3659–67. [115] Kamenskii M, Obukhovskii V, Zecca P. Condensing multivalued maps and semilinear differential inclusions in Banach spaces. De Gruyter Ser Nonlinear Anal Appl, vol. 7. Berlin: Walter de Gruyter & Co.; 2001. [116] Kavian O. Introduction à la théorie des points critiques et applications aux problèmes elliptiques. Math Appl. Paris: Springer-Verlag; 1993. [117] Kenmochi N. Nonlinear operators of monotone type in reflexive Banach spaces and nonlinear perturbations. Hiroshima Math J 1974;4:229–63. [118] Khan AA, Motreanu D. Existence theorems for elliptic and evolutionary variational and quasivariational inequalities. J Optim Theory Appl 2015;67(3):1136–61. [119] Khan AA, Tammer C, Z˘alinescu C. Regularization of quasi variational inequalities. Optimization 2015;64(8):1703–24. [120] Kim Y-H. A global bifurcation for nonlinear equations with nonhomogeneous part. Nonlinear Anal 2009;71(12):e738–43. [121] Kim I-S, Kim Y-H. Global bifurcation for equations involving nonhomogeneous operators in an unbounded domain. Nonlinear Anal 2010;73(4):1057–64.

338 Bibliography

[122] Kinderlehrer D, Stampacchia G. An introduction to variational inequalities and their applications. Philadelphia (PA): SIAM; 1987. [123] Kluge R. On some parameter determination problems and quasi-variational inequalities. In: Theory of nonlinear operators, Proc. fifth internat. summer school. Berlin: Akademie-Verlag; 1978. p. 129–39. [124] Kravchuk AS, Neittaanmäki PJ. Variational and quasi-variational inequalities in mechanics. Dordrecht: Springer; 2007. [125] Kyritsi ST, O’Regan D, Papageorgiou NS. Existence of multiple solutions for nonlinear Dirichlet problems with a nonhomogeneous differential operator. Adv Nonlinear Stud 2010;10(3):631–57. [126] Ladyzhenskaya OA, Uraltseva NN. Linear and quasilinear elliptic equations. New York: Academic Press; 1968. [127] Lê A. Eigenvalue problems for the p-Laplacian. Nonlinear Anal 2006;64(5):1057–99. [128] Li X, Huang N, O’Regan D. Differential mixed variational inequalities in finite dimensional spaces. Nonlinear Anal 2010;72(9–10):3875–86. [129] Lieberman GM. Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal 1988;12(11):1203–19. [130] Lieberman GM. The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations. Comm Partial Differential Equations 1991;16(2–3):311–61. [131] Lindqvist P. On the equation div (|∇u|p−2 ∇u) + λ|u|p−2 u = 0. Proc Amer Math Soc 1990;109(1):157–64. [132] Liu Z. Existence results for evolution noncoercive hemivariational inequalities. J Math Anal Appl 2004;120(2):417–27. [133] Liu Z. Existence results for quasilinear parabolic hemivariational inequalities. J Differential Equations 2008;244(6):1395–409. [134] Liu Z, Li X. Approximate controllability for a class of hemivariational inequalities. Nonlinear Anal Real World Appl 2015;22:581–91. [135] Liu Z, Li X, Motreanu D. Approximate controllability for nonlinear evolution hemivariational inequalities in Hilbert spaces. SIAM J Control Optim 2015;531(5):3228–44. [136] Liu Z, Loi NV, Obukhovskii V. Existence and global bifurcation of periodic solutions to a class of differential variational inequalities. Internat J Bifur Chaos Appl Sci Engrg 2013;23:1350125. [137] Liu Z, Lv J, Sakthivel R. Approximate controllability of fractional functional evolution inclusions with delay in Hilbert spaces. IMA J Math Control Inform 2014;31(3):363–83. [138] Liu Z, Zeng S. Existence results for a class of hemivariational inequalities involving the stable (g, f, a)-quasimonotonicity. Topol Methods Nonlinear Anal 2016;47(1):195–217. [139] Liu Z, Zeng S, Motreanu D. Evolutionary problems driven by variational inequalities. J Differential Equations 2016;260(9):6787–99. [140] Lunsford ML. Generalized variational and quasi-variational inequalities with discontinuous operators. J Optim Theory Appl 1997;214(1):245–63. [141] Lloyd NG. Degree theory. Cambridge Tracts in Math, vol. 73. Paris: Cambridge University Press; 1978. [142] Mahmudov NI. Approximate controllability of evolution systems with nonlocal conditions. Nonlinear Anal 2008;68(3):536–46. [143] Marano SA, Mosconi SJN, Papageorgiou NS. Multiple solutions to (p, q)-Laplacian problems with resonant concave nonlinearity. Adv Nonlinear Stud 2016;16(1):51–65. [144] Marano SA, Motreanu D, Puglisi D. Multiple solutions to a Dirichlet eigenvalue problem with p-Laplacian. Topol Methods Nonlinear Anal 2013;42(2):277–91. [145] Marano SA, Papageorgiou NS. Constant-sign and nodal solutions of coercive (p, q)-Laplacian problems. Nonlinear Anal 2013;77:118–29. [146] Marcus M, Mizel VJ. Nemitsky operators on Sobolev spaces. Arch Ration Mech Anal 1973;15:347–70.

Bibliography 339

[147] Marino A, Micheletti AM, Pistoia A. A nonsymmetric asymptotically linear elliptic problem. Topol Methods Nonlinear Anal 1994;4(2):289–339. [148] Marino A, Prodi G. Metodi perturbativi nella teoria di Morse. Boll Unione Mat Ital (4) 1975;11:1–32. [149] Martínez SR, Rossi JD. Isolation and simplicity for the first eigenvalue of the p-Laplacian with a nonlinear boundary condition. Abstr Appl Anal 2002;7(5):287–93. [150] Martínez SR, Rossi JD. On the Fuˇcik spectrum and a resonance problem for the p-Laplacian with a nonlinear boundary condition. Nonlinear Anal 2004;59(6):813–48. [151] Mawhin J. Generic properties of nonlinear boundary value problems. In: Differential equations and mathematical physics. Boston (MA): Academic Press; 1992. p. 217–34. [152] Mawhin J, Willem M. Critical point theory and Hamiltonian systems. Appl Math Sci, vol. 74. New York: Springer-Verlag; 1989. [153] Migórski S, Ochal A. Optimal control of parabolic hemivariational inequalities. J Global Optim 2000;17(1–4):285–300. [154] Migórski S, Ochal A. Quasi-static hemivariational inequality via vanishing acceleration approach. SIAM J Math Anal 2009;41(4):1415–35. [155] Migórski S, Ochal A, Sofonea M. Nonlinear inclusions and hemivariational inequalities. Models and analysis of contact problems. Adv Mech Math, vol. 26. New York: Springer; 2013. [156] Miyajima S, Motreanu D, Tanaka M. Multiple existence results of solutions for the Neumann problems via super- and sub-solutions. J Funct Anal 2012;262(5):1921–53. [157] Mih˘ailescu M. An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue. Commun Pure Appl Anal 2011;10(2):701–8. [158] Montenegro M, Suarez A. Existence of a positive solution for a singular system. Proc Roy Soc Edinburgh Sect A 2010;140(2):435–47. [159] Mosco U. Convergence of convex sets and of solutions of variational inequalities. Adv Math 1969;3:512–85. [160] Mosco U. Implicit variational problems and quasi variational inequalities. In: Nonlinear operators and the calculus of variations. Lecture Notes in Math. Berlin: Springer; 1976. p. 83–156. [161] Motreanu D. Generic existence of Morse functions on infinite-dimensional Riemannian manifolds and applications. In: Global differential geometry and global analysis. Lecture Notes in Math, vol. 1481. Berlin: Springer; 1991. p. 175–84. [162] Motreanu D, Motreanu VV. Transversality theory with applications to differential equations. In: Essays in mathematics and its applications. Cham: Springer; 2016. p. 133–57. [163] Motreanu D, Motreanu VV. Generic existence of nondegenerate homoclinic solutions. Lobachevskii J Math 2017;38(2):322–9. [164] Motreanu D, Motreanu VV, Moussaoui A. Location of nodal solutions for quasilinear elliptic equations with gradient dependence. Discrete Contin Dyn Syst Ser S 2018;11(2):293–307. [165] Motreanu D, Motreanu VV, Papageorgiou NS. A unified approach for multiple constant sign and nodal solutions. Adv Differential Equations 2007;12(12):1363–92. [166] Motreanu D, Motreanu VV, Papageorgiou NS. Multiple nontrivial solutions for nonlinear eigenvalue problems. Proc Amer Math Soc 2007;135(1):3649–58. [167] Motreanu D, Motreanu VV, Papageorgiou NS. Positive solutions and multiple solutions at non-resonance, resonance and near resonance for hemivariational inequalities with p-Laplacian. Trans Amer Math Soc 2008;360(5):2527–45. [168] Motreanu D, Motreanu VV, Papageorgiou NS. Multiple constant sign and nodal solutions for nonlinear Neumann eigenvalue problems. Ann Sc Norm Super Pisa Cl Sci (5) 2011;10(2):729–55. [169] Motreanu D, Motreanu VV, Papageorgiou NS. Topological and variational methods with applications to nonlinear boundary value problems. New York: Springer; 2014. [170] Motreanu D, Moussaoui A. Existence and boundedness of solutions for a singular cooperative quasilinear elliptic system. Complex Var Elliptic Equ 2014;59(2):285–96.

340 Bibliography

[171] Motreanu D, Moussaoui A. A quasilinear singular elliptic system without cooperative structure. Acta Math Sci Ser B Engl Ed 2014;34(3):905–16. [172] Motreanu D, Moussaoui A. An existence result for a class of quasilinear singular competitive elliptic systems. Appl Math Lett 2014;38:33–7. [173] Motreanu D, Moussaoui A, Zhang Z. Positive solutions for singular elliptic systems with convection term. J Fixed Point Theory Appl 2014;19(3):2165–75. [174] Motreanu D, Panagiotopoulos PD. Minimax theorems and qualitative properties of the solutions of hemivariational inequalities. Optim Appl, vol. 29. Dordrecht: Kluwer Academic Publishers; 1999. [175] Motreanu D, Papageorgiou NS. Multiple solutions for nonlinear elliptic equations at resonance with a nonsmooth potential. Nonlinear Anal 2004;56(8):1211–34. [176] Motreanu D, Papageorgiou NS. Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator. Nonlinear Anal 2011;139(10):3527–35. [177] Motreanu D, R˘adulescu V. Variational and non-variational methods in nonlinear analysis and boundary value problems. Optim Appl, vol. 67. Dordrecht: Kluwer Academic Publishers; 2003. [178] Motreanu D, Tanaka M. Sign-changing and constant-sign solutions for p-Laplacian problems with jumping nonlinearities. J Differential Equations 2010;249(12):3352–76. [179] Motreanu D, Tanaka M. Existence of solutions for quasilinear elliptic equations with jumping nonlinearities under the Neumann boundary condition. Calc Var Partial Differential Equations 2012;43(1–2):231–64. [180] Motreanu D, Tanaka M. Generalized eigenvalue problems of nonhomogeneous elliptic operators and their application. Pacific J Math 2013;265(1):151–84. [181] Motreanu D, Tanaka M. Multiple existence results of solutions for quasilinear elliptic equations with a nonlinearity depending on a parameter. Ann Mat Pura Appl 2014;193(4):1255–82. [182] Motreanu D, Tanaka M. On a positive solution for (p, q)-Laplace equation with indefinite weight. Minimax Theory Appl 2016;1(1):1–20. [183] Motreanu D, Tanaka M. Existence of positive solutions for nonlinear elliptic equations with convection terms. Nonlinear Anal 2017;152:38–60. [184] Motreanu D, Vetro C, Vetro F. A parametric Dirichlet problem for systems of quasilinear elliptic equations with gradient dependence. Numer Funct Anal Optim 2016;37(12):1551–61. [185] Motreanu D, Vetro C, Vetro F. Systems of quasilinear elliptic equations with dependence on the gradient via subsolution–supersolution method. Discrete Contin Dyn Syst Ser S 2018;11(2):309–21. [186] Motreanu D, Winkert P. On the Fu˘cik spectrum for the p-Laplacian with a Robin boundary condition. In: Nonlinear analysis. Springer Optim Appl, vol. 68. New York: Springer; 2012. p. 471–85. [187] Motreanu D, Winkert P. On the Fu˘cik spectrum for the p-Laplacian with a Robin boundary condition. Nonlinear Anal 2011;74(14):4671–81. [188] Motreanu VV. Uniqueness results for a Dirichlet problem with variable exponent. Commun Pure Appl Anal 2010;9(5):1399–410. [189] Motreanu VV. Existence results for constrained quasivariational inequalities. Abstr Appl Anal 2013:427908. [190] Moussaoui A, Khodja B, Tas S. A singular Gierer–Meinhardt system of elliptic equations in RN . Nonlinear Anal 2009;71(3–4):708–16. [191] Mugnai D, Papageorgiou NS. Wang’s multiplicity result for superlinear (p, q)-equations without the Ambrosetti–Rabinowitz condition. Trans Amer Math Soc 2014;366(9):4919–37. [192] Naniewicz Z, Panagiotopoulos PD. Mathematical theory of hemivariational inequalities and applications. Monogr Textb Pure Appl Math, vol. 188. New York: Marcel Dekker; 1995. [193] Noor MA, Rassias TM. A class of projection methods for general variational inequalities. J Math Anal Appl 2002;268(1):334–43.

Bibliography 341

[194] Panagiotopoulos PD. Hemivariational inequalities. Applications in mechanics and engineering. Appl Math Sci, vol. 44. Berlin: Springer; 1993. [195] Pang J. Frictional contact models with local compliance: semismooth formulation. Z Angew Math Mech 2008;88(6):454–71. [196] Pang J, Stewart DE. Differential variational inequalities. Math Program Ser A 2007;113(2):345–424. [197] Park JY, Park SH. Optimal control problems for anti-periodic quasi-linear hemivariational inequalities. Optimal Control Appl Methods 2007;28(4):275–87. [198] Pazy A. Semigroups of linear operators and applications to partial differential equations. Appl Math Sci, vol. 44. New York: Springer-Verlag; 1983. [199] Peng ZJ, Liu ZH, Liu XY. Boundary hemivariational inequality problems with doubly nonlinear operators. Math Ann 2013;356(4):1339–58. [200] Perera K, Agarwal RP, O’Regan D. Morse theoretic aspects of p-Laplacian type operators. Math Surveys Monogr, vol. 161. Providence (RI): American Mathematical Society; 2010. [201] Pohožaev SI. Equations of the type u = f (x, u, Du). Mat Sb (NS) 1980;113(155)(2(10)):324–38. [202] Ponstein J. Approaches to the theory of optimization. Cambridge Tracts in Math, vol. 77. Cambridge: Cambridge University Press; 1980. [203] Prado H, Ubilla P. Existence of nonnegative solutions for generalized p-Laplacians. In: Reaction diffusion systems. Lect Notes Pure Appl Math, vol. 194. New York: Dekker; 1998. p. 289–98. [204] Precup R. Linear and semilinear partial differential equations. An introduction. de Gruyter Textbook. Basel: Walter de Gruyter & Co.; 2013. [205] Pucci P, Serrin J. The maximum principle. Progr Nonlinear Differential Equations Appl, vol. 73. Basel: Birkhäuser Verlag; 2007. [206] Pucci P, Serrin J. The strong maximum principle revisited. J Differential Equations 2004;196(1):1–66. [207] Rabinowitz PH. Minimax methods in critical point theory with applications to differential equations. Providence (RI): American Mathematical Society; 1986. Published for the Conference Board of the Mathematical Sciences, Washington, DC. [208] R˘adulescu V, Repovš D. Existence results for variational–hemivariational problems with lack of convexity. Nonlinear Anal 2010;73(1):99–104. [209] Ricceri B. Infinitely many solutions of the Neumann problem for elliptic equations involving the p-Laplacian. Bull Lond Math Soc 2001;33(3):331–40. [210] Ricceri B. Nonlinear eigenvalue problems. In: Handbook of nonconvex analysis and applications. Somerville (MA): Int. Press; 2010. p. 543–95. [211] Robinson SB. On the second eigenvalue for nonhomogeneous quasi-linear operators. SIAM J Math Anal 2004;35(5):1241–9. [212] Ruiz D. A priori estimates and existence of positive solutions for strongly nonlinear problems. J Differential Equations 2004;199(1):96–114. [213] Rykaczewski K. Approximate controllability of differential inclusions in Hilbert spaces. Nonlinear Anal 2012;75(5):2701–12. [214] Saut J-C. Generic properties of nonlinear boundary value problems. In: Partial differential equations. Warsaw: Banach Center Publ., PWN; 1983. p. 331–51. [215] Saut J-C, Temam R. Generic properties of nonlinear boundary value problems. Comm Partial Differential Equations 1979;4:293–319. [216] Schechter M. The Fuˇcík spectrum. Indiana Univ Math J 1994;43(4):1139–57. [217] Showalter RE. Monotone operators in Banach space and nonlinear partial differential equations. Math Surveys Monogr, vol. 49. Providence (RI): American Mathematical Society; 1997. [218] Sidiropoulos NE. Existence of solutions to indefinite quasilinear elliptic problems of p–q-Laplacian type. Electron J Differential Equations 2010;(162). 23 pp. [219] Smale S. An infinite dimensional version of Sard’s theorem. Amer J Math 1965;87:861–6.

342 Bibliography

[220] Struwe M. A note on a result of Ambrosetti and Mancini. Ann Mat Pura Appl 1982;131(4):107–15. [221] Szulkin A. Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems. Ann Inst H Poincaré Anal Non Linéaire 2009;3(2):77–109. [222] Tanaka M. Existence of constant sign solutions for the p-Laplacian problems in the resonant case with respect to Fuˇcik spectrum. SUT J Math 2009;45(2):149–66. [223] Tanaka M. The antimaximum principle and the existence of a solution for the generalized p-Laplace equations with indefinite weight. Differ Equ Appl 2012;4:581–613. [224] Tanaka M. Generalized eigenvalue problems for (p, q)-Laplacian with indefinite weight. J Math Anal Appl 2014;419(2):1181–92. [225] Tanaka M. Existence of the Fucík type spectrums for the generalized p-Laplace operators. Nonlinear Anal 2012;7:3407–35. [226] Tanaka M. Existence of a positive solution for quasilinear elliptic equations with nonlinearity including the gradient. Bound Value Probl 2013;2013(173). 11 pp. [227] Tolksdorf P. Regularity for a more general class of quasilinear elliptic equations. J Differential Equations 1984;51(1):126–50. [228] Troianiello GM. Elliptic differential equations and obstacle problems. New York: Plenum Press; 1987. [229] Vázquez JL. A strong maximum principle for some quasilinear elliptic equations. Appl Math Optim 1984;12(3):191–202. [230] Wei J, Wang J. Infinitely many homoclinic orbits for the second order Hamiltonian systems with general potentials. J Math Anal Appl 2010;366(2):694–9. [231] Winkert P. Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values. Adv Differential Equations 2010;18(5–6):561–99. [232] Winkert P. Multiple solution results for elliptic Neumann problems involving set-valued nonlinearities. J Math Anal Appl 2011;377(1):121–34. [233] Wu M, Yang Z. A class of p–q-Laplacian type equation with potentials eigenvalue problem in RN . Bound Value Probl 2009:185319. [234] Yin H, Yang Z. A class of p–q-Laplacian type equation with concave–convex nonlinearities in bounded domain. J Math Anal Appl 2011;382(2):843–55. [235] Yin H, Yang Z. Multiplicity of positive solutions to a p–q-Laplacian equation involving critical nonlinearity. Nonlinear Anal 2012;75(6):3021–35. [236] Yosida K. Functional analysis. Classics Math. Berlin: Springer-Verlag; 1995. [237] Zeidler E. Nonlinear functional analysis and its applications. I. Fixed-point theorems. New York: Springer-Verlag; 1986. [238] Zeidler E. Nonlinear functional analysis and its applications, vol. II B. Berlin: Springer; 1990. [239] Zeidler E. Nonlinear functional analysis and its applications. IV. Applications to mathematical physics. New York: Springer-Verlag; 1988. [240] Zhang J, Li S, Whang Y, Xue X. Multiple solutions for semilinear elliptic equations with Neumann boundary condition and jumping nonlinearities. J Math Anal Appl 2010;371(2):682–90. [241] Zhang Q, Motreanu D. Existence and blow-up rate of large solutions of p(x)-Laplacian equations with large perturbation and gradient terms. Adv Differential Equations 2016;21(7–8):699–734. [242] Zou HH. A priori estimates and existence for quasi-linear elliptic equations. Calc Var Partial Differential Equations 2008;33(4):417–37.

Index

Symbols

E

β-condensing multivalued map, 327 (k, β1 , β2 )-boundedness, 326 m-accretive operator, 21 (p, q)-Laplacian, 80 p-Laplacian operator, 25 (S)+ -property, 24

Eigenfunction of p-Laplacian, 26 Eigenvalue of p-Laplacian, 25 Ekeland’s variational principle, 74 Evolution system, 308 Evolutionary quasivariational inequality, 292 Evolutionary variational inequality, 286 Extremal solutions, 166

A Accretive operator, 21 Ambrosetti–Rabinowitz condition, 40 Approximate solutions, 174 Approximately controllable system, 309 Asymptotic property, 235 Asymptotically (p − 1)-homogeneous near +∞, 109 Asymptotically (p − 1)-homogeneous near zero, 104

B Baire’s category theorem, 30 Biggest negative solution, 153 Bochner’s theorem, 1 Boundary point result, 50

C Carathéodory function, 43 Concave–convex nonlinearities, 226 Constant-sign solution, 204 Control function, 307 Controllability map, 313 Convection term, 191 Critical point, 19, 30

F Filippov implicit function lemma, 330 First deformation theorem, 19 First nontrivial curve, 61 Fréchet differentiability, 11 Fredholm operator, 31 Fredholm operator of index zero, 29 Fuˇcík spectrum, 59 Fundamental theorem for pseudomonotone operators, 25

G Gâteaux differentiability, 11 Generalized directional derivative, 18 Generalized gradient, 18 Generalized pseudomonotone multivalued map, 283 Generic property, 31 Gradient, 12 Gradient dependence, 154 Gronwall’s inequality, 313

D

H

Directional derivative, 11 Dirichlet boundary condition, 59

Hahn–Banach separation theorem, 2 Hardy–Sobolev inequality, 266 343

344 Index

Hausdorff MNC, 326 Hemivariational inequality, 301 Higher order Sobolev spaces, 5 Homoclinic solution, 34 Hybrid solutions, 263

I Implicit function theorem, 12 Infinitesimal generator of a C0 -semigroup, 321 Integrable function, 1

K KKM mapping, 321 Kluge’s fixed point theorem, 291

Mountain pass theorem, 19 Mountain pass theorem on manifolds, 70 Multivalued reaction term, 214

N Negative solution, 153 Nemytskii operator, 157 Neumann boundary condition, 61 Nodal solution, 159 Nondegenerate critical point, 30 Nonhomogeneous operator, 43 Nonlinear comparison principle, 280 Nonlinear elliptic problem, 42 Nonlinear elliptic system, 229

O

L

Optimal control, 313

Lagrange multiplier rule with equality constraints, 14 Lagrange multiplier rule with inequality constraints, 15 Laplacian operator, 23 Lebourg’s mean value theorem, 19 Linear boundary value problem, 10 Local minimizers, 200 Locally Lipschitz function, 18 Location property, 222 Lusternik–Schnirelman eigenvalues, 25

P

M Marcus–Mizel theorem, 6 Maximal monotone multivalued map, 283 Maximal negative solution, 251 Measurable function, 1 Measure of noncompactness, 326 Mild solution, 309 Mild solution for evolution equation driven by variational inequality, 321 Mild trajectory, 321 Minimal positive solution, 251 Minimization problem, 14 Minty’s formulation for variational inequalities, 290 Monotone multivalued map, 283 Monotone operator, 23 Morse function, 57 Moser’s iteration technique, 46

Palais–Smale condition, 19 Palais–Smale condition on manifolds, 69 Palais–Smale sequence, 74 Poincaré’s inequality, 7 Positive solution, 116 Pseudomonotone multivalued map, 283 Pseudomonotone operator, 24

R Rayleigh quotient, 81 Reachable set, 309 Regular domain, 7 Regular value, 30 Regularity up to the boundary, 46 Rellich–Kondrachov embedding theorem, 8 Residual set, 30 Resolvent, 21 Resonant cases, 129 Riesz representation theorem, 8 Robin boundary condition, 66 Robin eigenvalue problem, 66

S Sard–Smale theorem, 31 Schaefer’s fixed point theorem, 151 Schauder’s fixed point theorem, 271 Second deformation theorem, 20 Second eigenvalue, 26 Second-order Hamiltonian system, 33

Index 345

Semilinear elliptic boundary value problem, 39 Semilinear evolutionary inclusion, 307 Singular quasilinear elliptic system, 265 Singularities in the gradient, 272 Singularity at the origin, 265 Smallest positive solution, 144 Sobolev critical exponent, 155 Sobolev space of vector-valued functions, 4 Sobolev spaces on multi-dimensional domains, 6 Sobolev’s embedding theorem, 8 Steklov boundary condition, 64 Strong maximum principle, 56 Subhomogeneous condition, 260 Subsolution, 157 Subsolution–supersolution for systems, 237 Superpositionally measurable, 324 Supersolution, 157

Time derivative, 304 Trapping region, 251 Truncation functions, 199 Truncation-perturbation, 221

U Unbounded linear operator, 21 Uniform bounds, 180 Unilateral growth condition, 230 Uniqueness of solution, 233 Upward directed ordered set, 169

V Variational control, 321 Variational form for systems, 248 Variational selection, 292

W Weighted eigenvalue problem for p-Laplacian, 25

T

Y

Tangent space, 99

Yosida approximation, 21