Variational Methods for Potential Operator Equations: With Applications to Nonlinear Elliptic Equations 9783110809374, 9783110152692

Table of contents :
1 Constrained minimization
1.1 Preliminaries
1.2 Constrained minimization
1.3 Dual method
1.4 Minimizers with the least energy
1.5 Application of dual method
1.6 Multiple solutions of nonhomogeneous equation
1.7 Sets of constraints
1.8 Constrained minimization for Ff
1.9 Subcritical problem
1.10 Application to the p-Laplacian
1.11 Critical problem
1.12 Bibliographical notes
2 Applications of Lusternik-Schnirelman theory
2.1 Palais-Smale condition, case ρ ≠ q
2.2 Duality mapping
2.3 Palais-Smale condition, case ? = q
2.4 The Lusternik-Schnirelman theory
2.5 Case ρ > q
2.6 Case ρ < q
2.7 Case ρ = q
2.8 The p-Laplacian in bounded domain
2.9 Iterative construction of eigenvectors
2.10 Critical points of higher order
2.11 Bibliographical notes
3 Nonhomogeneous potentials
3.1 Preliminaries and assumptions
3.2 Constrained minimization
3.3 Application — compact case
3.4 Perturbation theorems — noncompact case
3.5 Perturbation of the functional a — noncompact case
3.6 Existence of infinitely many solutions
3.7 General minimization — case ρ > q
3.8 Set of constraints V
3.9 Application to a critical case ρ = n
3.10 Technical lemmas
3.11 Existence result for problem (3.34)
3.12 Bibliographical notes
4 Potentials with covariance condition
4.1 Preliminaries and constrained minimization
4.2 Dual method
4.3 Minimization subject to constraint V
4.4 Sobolev inequality
4.5 Mountain pass theorem and constrained minimization
4.6 Minimization problem for a system of equations
4.7 Bibliographical notes
5 Eigenvalues and level sets
5.1 Level sets
5.2 Continuity and monotonicity of σ
5.3 The differentiability properties of σ
5.4 Schechter's version of the mountain pass theorem
5.5 General condition for solvability of (5.11)
5.6 Properties of the function κ(t)
5.7 Hilbert space case
5.8 Application to elliptic equations
5.9 Bibliographical notes
6 Generalizations of the mountain pass theorem
6.1 Version of a deformation lemma
6.2 Mountain pass alternative
6.3 Consequences of mountain pass alternative
6.4 Hampwile alternative
6.5 Applicability of the mountain pass theorem
6.6 Mountain pass and Hampwile alternative
6.7 Bibliographical notes
7 Nondifferentiable functionals
7.1 Concept of a generalized gradient
7.2 Generalized gradients in function spaces
7.3 Mountain pass theorem for locally Lipschitz functionals
7.4 Consequences of Theorem 7.3.1
7.5 Application to boundary value problem with discontinuous nonlinearity
7.6 Lower semicontinuous perturbation
7.7 Deformation lemma for functionals satisfying condition (L)
7.8 Application to variational inequalities
7.9 Bibliographical notes
8 Concentration-compactness principle — subcritical case
8.1 Concentration-compactness principle at infinity — subcritical case
8.2 Constrained minimization — subcritical case
8.3 Constrained minimization with b ≢ φ const, subcritical case
8.4 Behaviour of the Palais-Smale sequences
8.5 The exterior Dirichlet problem
8.6 The Palais-Smale condition
8.7 Concentration-compactness principle I
8.8 Bibliographical notes
9 Concentration-compactness principle — critical case
9.1 Critical Sobolev exponent
9.2 Concentration-compactness principle II
9.3 Loss of mass at infinity
9.4 Constrained minimization — critical case
9.5 Palais-Smale sequences in critical case
9.6 Symmetric solutions
9.7 Remarks on compact embeddings into L2*(Q) and L2*K (R)n
9.8 Bibliographical notes
Appendix
A.1 Sobolev spaces
A.2 Embedding theorems
A.3 Compact embeddings of spaces W1,p(Rn) and D1,p(Rn)
A.4 Conditions of concentration and uniform decay at infinity
A.5 Compact embedding for H1R (Rn)
A.6 Schwarz symmetrization
A.7 Pointwise convergence
A.8 Gâteaux derivatives
Bibliography
Glossary
Index

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de Gruyter Studies in Mathematics 24 Editors: Heinz Bauer · Jerry L. Kazdan · Eduard Zehnder

de Gruyter Studies in Mathematics 1 Riemannian Geometry, 2nd rev. ed., Wilhelm P. Α. Klingenberg 2 Semimartingales, Michel Mitivier 3 Holomorphic Functions of Several Variables, Ludger Kaup and Burchard Kaup 4 Spaces of Measures, Corneliu Constantinescu 5 Knots, Gerhard Burde and Heiner Zieschang 6 Ergodic Theorems, Ulrich Krengel 7 Mathematical Theory of Statistics, Helmut Strasser 8 Transformation Groups, Tammo torn Dieck 9 Gibbs Measures and Phase Transitions, Hans-Otto Georgii 10 Analyticity in Infinite Dimensional Spaces, Michel Herve 11 Elementary Geometry in Hyperbolic Space, Werner Fenchel 12 Transcendental Numbers, Andrei B. Shidlovskii 13 Ordinary Differential Equations, Herbert Amann 14 Dirichlet Forms and Analysis on Wiener Space, Nicolas Bouleau and Francis Hirsch 15 Nevanlinna Theory and Complex Differential Equations, lipo Laine 16 Rational Iteration, Norbert Steinmetz 17 Korovkin-type Approximation Theory and its Applications, Francesco Altomare and Michele Campiti 18 Quantum Invariants of Knots and 3-Manifolds, Vladimir G. Turaev 19 Dirichlet Forms and Symmetric Markov Processes, Masatoshi Fukushima, Yoichi Oshima, Masayoshi Takeda 20 Harmonic Analysis of Probability Measures on Hypergroups, Walter R. Bloom and Herbert Hey er 21 Potential Theory on Infinite-Dimensional Abelian Groups, Alexander Bendikov 22 Methods ofNoncommutative Analysis, Vladimir E. Nazaikinskii, Victor E. Shatalov, Boris Yu. Sternin 23 Probability Theory, Heinz Bauer

Jan Chabrowski

Variational Methods for Potential Operator Equations With Applications to Nonlinear Elliptic Equations

W Walter de Gruyter G Berlin · New York 1997 DE

Author Jan Chabrowski Department of Mathematics The University of Queensland St Lucia, Qld 4072 Australia Series Editors Heinz Bauer Mathematisches Institut der Universität Erlangen-Nürnberg Bismarckstraße VA D-91054 Erlangen, FRG

Jerry L. Kazdan Department of Mathematics University of Pennsylvania 209 South 33rd Street Philadelphia, PA 19104-6395, USA

Eduard Zehnder ETH-Zentrum/Mathematik Rämi straße 101 CH-8092 Zürich Switzerland

1991 Mathematics Subject Classification: 35-02; 35D05, 35J20, 35J60, 35J65, 58E05, 58E30 Keywords: Nonlinear elliptic equations, critical and subcritical Sobolev exponents, min-max theory, variational principles ©

Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.

Library of Congress Cataloging-in-Publication Data Chabrowski, Jan, 1941 — Variational methods for potential operator equations : with applications to nonlinear elliptic equations / Jan Chabrowski. p. cm. — (De Gruyter studies in mathematics ; 24) Includes bibliographical references and index. ISBN 3-11-015269-X 1. Calculus of variations. 2. Differential equations, Elliptic-Numerical solutions. 3. Differential equations, Nonlinear-Numerical solutions. I. Title. II. Series. QA316.C485 1997 515'.64—dc21 97-8060 CIP

Die Deutsche Bibliothek — Cataloging-in-Publication Data Chabrowski, Jan: Variational methods for potential operator equations : with applications to nonlinear elliptic equations / Jan Chabrowski. - Berlin ; New York : de Gruyter, 1997 (De Gruyter studies in mathematics ; 24) ISBN 3-11-015269-X

© Copyright 1997 by Walter de Gruyter & Co., D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Typeset using the Author's T£X-Files: I. Zimmermann, Freiburg. Printing: Arthur Collignon GmbH, Berlin. Binding: Lüderitz & Bauer, Berlin. Cover design: Rudolf Hübler, Berlin.

Preface

In this book we are concerned with methods of the variational calculus which are directly related to the theory of partial differential equations of elliptic type. The methods which we discuss and describe here go far beyond elliptic equations. In particular, these methods can be applied to Hamiltonian systems, nonlinear wave equations and problems related to surfaces of prescribed mean curvature. In Chapter 1 we discuss minimization (maximization) of functionals subject to some constraints. The corresponding Euler-Lagrange equation has the form of the difference of two potential operators. This characteristic feature of the Euler-Lagrange equation is maintained throughout the whole book. In Chapter 1 we assume that both potential operators are homogeneous. Due to this assumption we can provide a rather complete description of relations between the various methods of minimization (maximization). We also emphasize here an efficient tool, an abstract version of the Sobolev inequality, which has been widely used in the theory of partial differential equations. In Chapter 2 we focus our attention on critical points which are not necessarily maximizers (minimizers) of the corresponding functionals. To study such points we use the Lusternik-Schnirelman theory of critical points. In this approach, critical values of a given functional F : X ->· Ε on a Banach space X are obtained as c = inf max7 F(u), ueA

AeJ

where JF is a suitable class of subsets of X. The choice of Τ depends on the nature of a variational problem and topological properties of level sets of T . We use this theory in the foifH presented in the work of Ambrosetti-Rabinowitz [10]. The reader may also consult the books by Berger [31], Chow-Hale [75] and Foucik-Necas-Soucek [121]. The essential condition that must be satisfied in order to use this theory is the PalaisSmale condition. The most difficult case is where both potential operators appearing in the Euler-Lagrange equation have the same degree of the homogeneity. Chapters 3 and 4 are devoted to some generalizations of the results presented in the first two chapters. Namely, in Chapter 3 we consider the situation where one of the potential operators is not homogeneous. We show that we can still apply the Lusternik-Schnirelman theory. Among the important results here are perturbation theorems. These theorems allow us to associate with a given variational problem another variational problem which is easier to solve and whose solvability implies the existence of a solution to an original problem. We return to this method in Chapter 8, where exactly the same role is played by the concentration-compactness principle at infinity.

vi

Preface

Chapter 4 is concerned with variational problems where both potential operators satisfy some covariance condition. Obviously, we can drop the homogeneity assumption on potential operators. In Chapter 5 we continue the investigation of constrained minimization. We consider constraints depending on a parameter and investigate dependence of eigenvalues and eigenvectors on this parameter. One of the most effective theorems applied in many variational problems is the Ambrosetti-Rabinowitz mountain pass theorem [10]. However, there are some limitations on the applicability of this theorem. In Chapter 6 we present some generalizations of this theorem. In Chapter 7 we concentrate on theorems of mountain pass type for locally Lipschitz functionals. The first proof of the mountain pass theorem, given by AmbrosettiRabinowitz [10], was based on a deformation lemma. In this chapter we present a different proof based on Ekeland's variational principle combined with techniques from the Clarke theory of generalized gradient. In the final Chapters 8 and 9 we concentrate on weak methods of convergence in Sobolev spaces. In particular, we discuss variational problems involving subcritical and critical Sobolev exponents. There are number of factors creating some difficulties in solving these problems: the lack of compactness, invariance of these problems with respect to a group of translations in subcritical case and additionally in the critical case with respect to a group of dilations. We discuss in these chapters the applicability of the first and second concentration-compactness principles both due to P. L. Lions [ 158], [ 159]. These two principles are complemented by quantitative information about possible loss of mass at infinity of weakly convergent minimizing sequences. This gives rise to the concentration-compactness principle at infinity in both the subcritical and critical case. The material discussed and covered by this book requires some knowledge of functional analysis, nonlinear partial differential equations and the theory of Sobolev spaces. For the reader's convenience, most of the required prerequisites are given in Appendix. The problems discussed in this book have an extensive literature and the bibliography for this book is not intended to be complete, but rather to supplement and illuminate the text. Finally, I would like to express my sincere gratitude to Dr K. R. Matthews for his help in improving the presentation of the material in this book. I am also indebted to Dr K. G. Smith for his help in surmounting various problems with the manuscript.

Contents

1

Constrained minimization 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12

2

Preliminaries Constrained minimization Dual method Minimizers with the least energy Application of dual method Multiple solutions of nonhomogeneous equation Sets of constraints Constrained minimization for Ff Subcritical problem Application to the p-Laplacian Critical problem Bibliographical notes

1 1 8 13 14 15 17 19 24 29 30 35 37

Applications of Lusternik-Schnirelman theory

39

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11

39 40 43 47 55 56 60 63 67 70 73

Palais-Smale condition, case ρ φ q Duality mapping Palais-Smale condition, case ρ = q The Lusternik-Schnirelman theory Case ρ > q Case ρ < q Case p = q The p-Laplacian in bounded domain Iterative construction of eigenvectors Critical points of higher order Bibliographical notes

3 Nonhomogeneous potentials 3.1 3.2 3.3 3.4 3.5 3.6

Preliminaries and assumptions Constrained minimization Application — compact case Perturbation theorems — noncompact case Perturbation of the functional a — noncompact case Existence of infinitely many solutions

74 74 76 79 81 85 88

viii

Contents 3.7 3.8 3.9 3.10 3.11 3.12

General minimization — case ρ > q Set of constraints V Application to a critical case ρ = η Technical lemmas Existence result for problem (3.34) Bibliographical notes

90 99 101 103 112 113

4

Potentials with covariance condition 4.1 Preliminaries and constrained minimization 4.2 Dual method 4.3 Minimization subject to constraint V 4.4 Sobolev inequality 4.5 Mountain pass theorem and constrained minimization 4.6 Minimization problem for a system of equations 4.7 Bibliographical notes

115 115 120 120 121 122 125 127

5

Eigenvalues and level sets 5.1 Levelsets 5.2 Continuity and monotonicity of σ 5.3 The differentiability properties of σ 5.4 Schechter's version of the mountain pass theorem 5.5 General condition for solvability of (5.11) 5.6 Properties of the function κ(ί) 5.7 Hilbert space case 5.8 Application to elliptic equations 5.9 Bibliographical notes

128 128 130 132 135 138 140 142 143 148

6

Generalizations of the mountain pass theorem 6.1 Version of a deformation lemma 6.2 Mountain pass alternative 6.3 Consequences of mountain pass alternative 6.4 Hampwile alternative 6.5 Applicability of the mountain pass theorem 6.6 Mountain pass and Hampwile alternative 6.7 Bibliographical notes

149 149 153 155 157 160 163 166

7

Nondifferentiable functionals 7.1 Concept of a generalized gradient 7.2 Generalized gradients in function spaces 7.3 Mountain pass theorem for locally Lipschitz functionals 7.4 Consequences of Theorem 7.3.1 7.5 Application to boundary value problem with discontinuous nonlinearity 7.6 Lower semicontinuous perturbation 7.7 Deformation lemma for functionals satisfying condition (L)

167 167 172 174 181 183 185 188

Contents 7.8 7.9

ix Application to variational inequalities Bibliographical notes

195 197

8

Concentration-compactness principle — subcritical case 198 8.1 Concentration-compactness principle at infinity — subcritical case . . 198 8.2 Constrained minimization — subcritical case 200 8.3 Constrained minimization with b φ const, subcritical case 205 8.4 Behaviour of the Palais-Smale sequences 211 8.5 The exterior Dirichlet problem 215 8.6 The Palais-Smale condition 218 8.7 Concentration-compactness principle I 221 8.8 Bibliographical notes 223

9

Concentration-compactness principle — critical case 9.1 Critical Sobolev exponent 9.2 Concentration-compactness principle II 9.3 Loss of mass at infinity 9.4 Constrained minimization — critical case 9.5 Palais-Smale sequences in critical case 9.6 Symmetric solutions 9.7 Remarks on compact embeddings into L2*(Q) and 9.8 Bibliographical notes

224 224 228 229 233 237 244 250 252

Appendix A.l Sobolev spaces A.2 Embedding theorems A.3 Compact embeddings of spaces W^OR") and A.4 Conditions of concentration and uniform decay at infinity A.5 Compact embedding for H} (R n ) A.6 Schwarz symmetrization A.7 Pointwise convergence A.8 Gateaux derivatives

253 253 254 255 259 261 264 264 266

Bibliography

270

Glossary

287

Index

289

Chapter 1

Constrained minimization

The purpose of this chapter is to describe various minimization techniques involving sets of constraints. We distinguish two cases: subcritical and critical minimization problems. In the next chapters, we illustrate both cases by considering variational problems from the theory of boundary value problems for nonlinear elliptic equations. We begin with two fundamental results of the modern variational calculus: Ekeland's variational principle and the Ambrosetti-Rabinowitz mountain pass theorem which are used throughout this book. The proof of the mountain pass theorem is postponed until Chapter 7, where we present several more general variants of this theorem for nondifferentiable functionals.

1.1 Preliminaries We introduce some terminology which is quite standard and for more details we refer to the monographs [231 ], [95], [203] and [9]. Let X be a Banach space equipped with norm || · ||. By {·, ·) we denote the duality pairing between X and its dual X*. We denote the weak convergence in X and X* by 1 a n d the strong convergence by "—> ". We always denote an open ball centered at x0 of radius R by B(x0, R) and the corresponding sphere by 5(λ:0, R), that is B(x0, R) = {χ· II* - JColl < R} and S(x 0 , R) = {*; ||x - x0|| = *}· Let X and Y be two real Banach spaces. A map F : X —>· Y is said to be Frechet differentiable at u0 e X if there is an F'(u0) € L(X, Y) such that F(u0 + h) = F(u0) + F'(u0)h + ω(κ0, h) anda>(w0, h) = ο(||Λ||) as/i 0. Herεω(« 0 , h) = ο(||Λ||) denotes Landau's symbol. If F is differentiable at every point u e X and F' : X L{X, Y) is continuous, then F is said to be continuously differentiable on X. We express this by writing F G Cl(X, Y). A functional a : X -»· Μ is said to be Gäteaux differentiable if there exists an Μ* € X* such that lim t~1 (a(u0 + th) - a(ua)) = {u*, h) = u*(h)

2

1 Constrained minimization

for all A e X. The functional u* is called the Gateaux derivative of a at u0 and we denote it by A(u0) = u*. We recall that if a : X -»· R has a Gateaux derivative A(m) at every point κ in a neighbourhood of a point u0 and A(u) is continuous at u0, then a is Frechet differentiable at u0 and the Frechet derivative of a at u0 is equal to A(u0). For simplicity, we have defined both derivatives for mappings defined on X. In the same way we can define Frechet and Gateaux differentiability for mappings defined on open set Ω C X. A mapping A : X —> X* is said to be a potential operator with a potential a : X — R , if a is Gateaux differentiable and lim t~l (a(u + tv) — a(u)) = (A(u), υ) for all u and ν in X. For a potential we always assume that α (0) = 0. A mapping A : X —• X* is called hemicontinuous if it is continuous on line segments in X and X* is equipped with the weak topology. A potential of a hemicontinuous potential operator can be represented in the form a(u) = f (A(tu),u)dt Jo

for all u € X.

Indeed, we set φ(ί) = a(v + t(u — ν)) for u and ν in X, 0 < t < 1, then φ'(ί) = (Λ(ι» + t(u — ν), u — ν). Integrating over [0, 1] and setting υ = 0, the integral formula follows. We say that the mapping Λ : X -> X* is homogeneous of degree β > 0 if for every u e X and t > 0 A(tu) = t^A(u). Consequently, for homogeneous hemicontinuous potential operator A of degree β with the potential a, we have ρ+ 1 A mapping A : X — X * is strongly monotone if there exists a continuous function κ \ [0, oo) [0, oo) which is positive on (0, oo) and lim/^oo/cC/) = oo and such that (A(u) — Α(υ), u — υ) > /c(||« — u||)||u — υ|| for all u and ν in X. A mapping A : X —• X* is said to satisfy condition (S)i if for every sequence {uj} in X with uj —^ u and A(uj) -»• υ in X* we have Uj u. Evidently, a strongly monotone operator satisfies the condition (5)i. We now recall weaker definitions of the monotonicity for potential operators. A mapping A : X X* is said to be monotone if (A(u) — Α(υ), u — ν) > 0 for all u and υ in X and strictly monotone if equality implies u = v. It is well known that a Gateaux differentiable functional a : X -»• R is convex if and only if its potential operator A is hemicontinuous and monotone. Moreover, potential α of a monotone potential operator A is convex.

1.1 Preliminaries

3

We say that a mapping A : X —>· X* is strongly continuous if for every Uj —^ u in X we have A(uj) A(u) in X*. We say that a mapping A : X X* is weakly continuous if for every uj —u in X we have A(uj) A(u) in X*. A functional a : X ->· R is said to be weakly continuous if uj —» u implies a(uj)

a(u).

A functional a : X —• R is said to be coercive if lim^H-»^ a(u) = oo. Let -Apu

= -D(\Vu\p~2Du),

1 < ρ < oo,

be the p-Laplacian defined for u in the Sobolev space bounded domain. Writing for u and ν in W0 (Ω) (A(u),v)=

where Ω c M" is a

\Vu\p~2DuDvdx

f Ja

we define a potential operator A : τν 0 1,ρ (Ω) - * potential a : W 0

l,p

± + j? = 1, with a

(ß) given by a{u) = - f Ρ Ja

\Vu\pdx.

The potential operator A is hemicontinuous, bounded, monotone and coercive. The monotonicity of A follows from the algebraic inequalities (\X\P~2X - \y\P-2y,

y\p

χ - y) > ax \x -

if 2 < ρ < oo and (\x\'-2x-\y\P-2y,x-y)>av

~ (\x\ +

^ \y\)2~p

if 1 < ρ < 2, for all χ and _y in M" and some constant a\ > 0. Applying these inequalities to a2\\Vu

-

Vv\\p

if ρ > 2, and IIVm - Vw||J (A(m) - A(v), u - v ) > a2-

(Ι|ν Μ || ρ + | | ν υ | | ρ ) Ρ - 2

if 1 < ρ < 2, for some constant a2 > 0. If 2 < p, the operator A is strongly monotone.

4

1 Constrained minimization

Lemma 1.1.1. Let X be a reflexive Banach space. If A : X —»• X* is a strongly continuous potential operator, with potential a, then A is uniformly continuous on bounded sets and a is weakly continuous. Proof. Arguing indirectly, suppose that there is a bounded set 5 C X such that A is not uniformly continuous on S. This means that there exists an € > 0 and sequences {uj},{vj} C S such that \\uj-vj\\€

for all j > 1. Since S is bounded we may assume that Uj u and vj v. Hence u = ν and since A is strongly continuous A(uj) — A(vj) -»· 0 in X* which is impossible. Hence A is uniformly continuous on bounded sets. To prove the second assertion, suppose that Uj u. Then b(u„)=

(A(aun), un) da

and ( A ( a u n , u n ) {A(au, «)) for every σ € [0, 1]. Since Λ is uniformly continuous on bounded sets, it must be bounded. Applying the dominated convergence theorem • we conclude that b(um) b(u). A mapping A : X —• X* is said to be bounded if A maps bounded sets into bounded sets. Obviously, if A : X — X * is strongly continuous, then A is bounded. A mapping A : X —• X* is said to be odd if A(u) = —A(—u) for all u e X. If A : X ->· X* is an odd potential operator, with a potential a, then a is an even potential. We shall frequently refer to Ekeland's variational principle ([109], [212]). Theorem 1.1.1 (Ekeland's variational principle). Let (M, d) be a complete metric space with metric d and let F : Μ —• R U {oo} be lower semicontinuous, bounded from below and φ oo. Then for every € > 0 and λ > 0 and every u e Μ such that F(u) < inf F + e Μ there exists an element ν € Μ such that F(v) < F(u), d(v, u) < — λ'

and for each w φ ν in Μ F(w) > F(v) — ekd(w,

ν).

5

1.1 Preliminaries

Proof. It is sufficient to prove our assertion for λ = 1. The general case is obtained by replacing d by an equivalent metric kd. We define the relation on Μ:

w


o Μ·

{um}

As an immediate consequence one obtains the existence of a minimizing sequence for C 1 -functional F on a Banach space for which F'(um) — 0 in X*.

1 Constrained minimization

6

Theorem 1.1.2. If X is a Banach space and F e Cl(X, R) is bounded from below, then there exists a minimizing sequence {um} for F such that F(um) —> inf χ F and F'(um) 0 in X* asm oo. To obtain the existence of a minimizer we need a compactness condition on F which guarantees the convergence of a subsequence of a minimizing sequence. Let F g Cl(X, R). We say that F satisfies the Palais-Smale condition (the (PS) condition for short) if for every sequence {um} e X such that {F(wm)} is bounded and F'(um) -»· 0 in X*, then {um} possesses a convergent subsequence. We say that F 6 C 1 (X, R) satisfies the (PS)C condition, c e R, if every sequence {um} C X such that F(um) -»• c and F'(um) 0 in X* contains a convergent subsequence. Corollary 1.1.1. If F e C ^ X . R ) is bounded from below and satisfies the (PS) condition, then there exists u e X such that F(u) = inf F(u). ueX It is worth mentioning that a C Afunctional bounded from below and satisfying the (PS) condition must be coercive. Proposition 1.1.1. Let F € Cl(X, R) be bounded from below. If F satisfies the (PS) condition then F is coercive. Proof. If the conclusion of the theorem were not true, there would exist c e R such that limn«ιι-^οο F(m) = c. Then for every integer m > 1 there exists um such that F(u m ) < c + ^ and ||um|| > 2m. By virtue of Theorem 1.1.1, applied with € = c + ^ — inf χ F and λ = ^ , we get the existence of vm e X such that F(vm)

< F(um)

< c + —, m

llWmll > \\um II - ll"m - vm || > ||um|| - m > m and F ( w ) > F ( v m ) - - ( c + - - inf F)||u; - um || m m x for each w e X. The last inequality implies that HF>m)|| R are potentials of potential operators A : X X* and Β : X X*, respectively. This minimization problem is very often used to solve the eigenvalue problem A(u) = XB(u), λ G R (1.1) If A and Β are continuously Fr£chet differentiable, then a solution of (/) satisfies (1.1), where λ is a corresponding Lagrange multiplier. In our approach we assume that A and Β have at u Gateaux derivatives not necessarily at all directions ν e X. In Theorem 1.2.1 below we assume that b is defined on a set D(b) C X which contains a linear subspace Ε c X. Obviously, it is understood that the minimum in (/) is taken over u e D(b) satisfying b(u) — 1. This specific assumption on the domain of definition of b is made in this chapter in Theorems 1.2.1 and 1.4.1. Otherwise we shall always assume that D(b) = X. Throughout this chapter we assume that X is a reflexive Banach space. We say that problem (/) has a solution if there exists u 6 D(b) such that a (w) = I and b{u) = 1. In the sequel we use notation b+(u) = max(fc(«),0) and b~(u) = min(t(M), 0). Theorem 1.2.1. (i)

Suppose that A is a hemicontinuous and homogeneous potential operator ofdegree ρ — 1 with potential a. Moreover, suppose that problem ( / ) has a solution u and thatb has a linear continuous Gateaux derivative {B(u), v) at all directions υ e Ε and that Β is homogeneous of degree q — 1. Then (A(u),v) =

I?-{B(u),v) 0, then the " scaled minimizer" u = au, where σ = y f j j P

?

if Ρ Φ Q satisfies the equation (A(ü), ν) = (B(ä), ν)

(1.3)

for all υ ξ. Ε. If ρ = q we have a typical eigenvalue problem. (ii) Additionally, suppose that A is a strongly monotone potential operator b~ is weakly lower semicontinuous and that b+ is weakly continuous. suppose that all weak limit points of every bounded subset of the level D(b)\ b{u) = 1} belong to D(b). Then the constrained minimization ( / ) has α nontrivial solution with I > 0.

and that Finally, set {u € problem

Proof, (i) The proof is similar to that of Proposition 2.1 in[48]. Wesetd = (B(u), υ), ν e Ε, and for σ > 0 we have by the definition of the Gateaux derivative that b(au + €σν) = aq (1 + ed + o(e)). There exists e 0 > 0 such that 1 + ed + o(e) > 0 for all |e| < eQ. Also, for such € > 0 there exists σ = σ(€) such that b(pu + eav) = 1. Consequently, if we set h = (A(m), ν), ν e E, we get I < a(au + σεν) = apa(u + ev) = a(u) + e(h - -Id) Since this holds for all |e| < €0, we must have h — ^Id Since for σ > 0 (.B(au),v)

=aq{B(u),a~xv),

{A{au),v)

+ o(c).

= 0 and this proves (1.2).

= ap(A{u),

σ~ι υ)

we see that (A(au), v) = -Iap~q{B{au), q

υ).

Therefore, if / > 0, then ü = σιι, with σ = ( ^ j j P~" satisfies equation (1.3). (ii) Let {uj} be a minimizing sequence. Since a(uj) = \\uj\\P{A(uj\\uj\\-l,Uj\\uj\\-1))

>K(l)\\uj\\P,

we see that the sequence is bounded. Therefore, we may assume that Uj — u in X. According to our assumption on D(b), u € D(b) and a(u) < I. We now have b~(uj) = b+(uj) - 1

b+(u) — 1 as j -»• oo,

and consequently b~(u) < liminf b~(uj) = b+(u) - 1. 7-»·οο

1 Constrained minimization

10

Therefore b(u) > 1 and u Φ 0. Obviously a(u) > 0 and hence I > 0. Assuming that b(u) > 1, we have a^biu) = 1 with 0 < aq < 1, since a(u) < I < a(pu) = apa{u) which implies that 1 < σ ρ and we have arrived at the contradiction and this completes the proof. • In this theorem we have assumed that the Gateaux derivative of b exists in all directions υ e E, where £ is a linear subspace of D(b). This requirement may look artificial, but as we shall see later, it is useful in applications. If we assume in Theorem 1.2.1 that b(u) > 0 on D(b) — {0}, then our assumption means that b is weakly continuous. If this assumption is replaced by a weaker condition: b is continuous on D(b), we may not be able to claim the existence of a solution of problem (/). However, we can still show that / > 0. In fact, we only need the continuity of b at 0. This leads to an abstract version of the Sobolev inequality. For the future use we formulate these two observations separately, in the case where D(b) = X. Proposition 1.2.1. Suppose that A is a hemicontinuous, strongly monotone and homogeneous potential operator of degree ρ — 1 and that Β is a positive homogeneous, strongly continuous potential operator of degree q — 1. Then the assertions (i) and (ii) of Theorem 1.2.1 hold with Ε = X. Proposition 1.2.2 (Sobolev inequality). Suppose that A is a strongly monotone, hemicontinuous and homogeneous potential operator of degree q — 1 and that Β is a positive hemicontinuous and homogeneous potential operator of degree q — 1 whose potential b is continuous at 0. Then there exists an absolute constant Spq > 0 such that Sp,q{B{u),u)*


-oo for each m > 1. By the strong monotonicity we have ll"mlk(ll"mll)
oo. This in turn implies that (5(0), 0) — 1, which is impossible. For a given « ^ O w e set V=

u (B(u),u)1

p.

1.2 Constrained minimization

11

It is easy to see that (B(v), v) = 1, so Sp,q < (Α(υ), υ) and this is equivalent to (1.4).

•

Remark 1.2.1. If we define the best Sobolev constant Sp 0 on (0, oo) and limf^.ooÄ-(i) = oo, such that {A(u), u) > *:(||m||)||u|| for each u G X. The inequalities established in Proposition 1.2.1 (or in Remark 1.2.1) correspond to the Sobolev inequalities from the theory of Sobolev spaces (see Theorem 1.2.1 in Appendix) with potentials a : Τνο1,ρ(Ω) ->· Κ and b : wl'p{Q.) R given by a(u) = - f \Vu\pdx Ρ Ja

and

b(u) = - [ \u\q dx, 1. On the other hand aqK = b(au) < K, that is σ < 1 and we arrive at the contradiction. • We now compare both methods and show that basically both methods give rise to the same result. For simplicity we assume that D(b) = Ε = X. Proposition 1.3.1. Let ρ φ q. Suppose that A is a hemicontinuous, strongly monotone and homogeneous potential operator of degree q — 1. Suppose further that Β is a

14

1 Constrained minimization

positive, strongly continuous and homogeneous potential operator of degree q — 1. Then both problems (K) and ( / ) have nontrivial solutions and

'' = Λ· κϊ

Moreover, ifu is a solution ofproblem (I), then ü = -j-w is a solution ofproblem (Κ). IP Similarly, ifu is a solution ofproblem (Κ), then ü = —η-u is a solution of problem (/). κΐ Proof Let u be solution of problem ( Κ ) , that is, Κ = b(u) and a(u) = 1. We choose σ > 0 so that b(au) = 1 = aqK. Consequently, I < a(au) = apa{u) = σρ and i , ι σ > I ρ , that is, - η - > I ρ . Let us now assume that u is a solution of problem ( / ) , κ* that is, a(u) = I and b(u) = 1. We choose ρ > 0 so that ppI = a{pu) = 1. Hence , ι b(pu) < K, and this implies that pq < Κ and - j < Ki and the first part of our IP

assertion follows. The second part is routine and is omitted.

•

1.4 Minimizers with the least energy If μ is a nontrivial solution of (1.1), with ρ φ q, then μ must belong to the set V defined by

V = {u 6 X - {0} : (A(u), u) - (B(u), u) = 0}. Therefore, it is natural to consider the following minimizing problem

m = inf F(u),

(m)

ueV

where a functional F : X - » R is defined by F(u) = a(u) - b(u). Assuming that A and Β are homogeneous potential operators of degree ρ — 1 and q — 1, respectively, we see that the set V is nonempty. Indeed, for every u φ 0 there exists t > 0 such that

{A(tu), tu) - (B(tu), tu) = 0. In the literature on variational methods the minimizing problem (m) is often referred

to as the minimizing problem with artificial constraint (see [171]). We show that a minimizer to problem ( / ) is also a minimizer of problem (m).

Proposition 1.4.1. Let ρ < q and suppose that A is a strongly monotone hemicontinuous and homogeneous potential operator of degree ρ — 1. Moreover, assume that Β is a positive, strongly continuous and homogeneous potential operator of degree q — 1. I f ü = öu, where σ =

9 p

and u is a solution to problem (I), then

F(w) = inf F(u) > 0. U€V

1.5 Application of dual method

15

Proof. Let υ e V,then aq = b(v) for some a > 0. Wesetu 0 = a~lv,thenb(v0) = 1. Multiplying the equation pa(v) — qb(v) = O b y a - ' ' we obtain pa(v0) — qa~p+e} = 0 P

and consequently σ = F(v) = (l -

• Hence

= (l - ?-)σρα(υ0)

= (l - - ) (-a(Uo)V~P

a(v0).

On the other hand, we have for ü

Since b(v0) = 1, we must have / < a(v0) and consequently F(Ä) < (l -

Qa(v0ή*'"

a(v0) = F(v).

We now observe that μ as a solution of (1.1) must belong to V and the result follows.

• Proposition 1.4.1 says that a solution to problem (/) is a solution with the least energy among all nontrivial solutions to (1.1). In applications we call it a ground-state solution (see [30] or [22]). Finally, combining Propositions 1.3.1 and 1.4.1 we get Corollary 1.4.1. Under the assumptions of Proposition 1.4.1, if u is a solution of problem (K), then —— ί ι ι q-p 1

W

=

I — / I

v

«

'

j-u

ΚΪ

is a solution of problem (m).

1.5 Application of dual method The dual method described in Section 1.3 is particularly applicable to a minimization subject to constraint V in case ρ > q. Proposition 1.5.1. Let ρ > q. Suppose that Ais a hemicontinuous, strongly monotone and homogeneous operator of degree ρ — 1 and that Β is a positive strongly continuous ι ? and homogeneous potential operator of degree q — 1. Ifü = du, where σ = and u is a solution to problem (K), then F{ü) = inf F(v) < 0. ueV

16

1 Constrained minimization

Proof. Let ν e V, then a{v) = σρ for some σ > 0 and set υ0 = σ Obviously, a{v0) = 1 and multiplying equation pa(v) — qb{v) = 0 by a~q we obtain ι P ρσ-ι+Ρα(ν0) - qb(v0) = 0. Hence σ = ~" and

On the other hand we have

and since Κ > b(v0) we get

Since ü e V, the result follows.

•

To proceed further we observe that due to the homogeneity of A and Β we have the following estimate for the functional F -Hull« sup b(u)+K\\u\\P < F(u) < \\u\\P sup a(u)-\\u\\q Nl=i ll«ll=i

inf b(u), II« ||=1

where κ = /c(l). One can give examples of strongly continuous potential operators Β with inf||tt||=i b(u) = 0. Therefore, if ρ < q F is in general neither bounded from below nor from above. However if ρ > q, F is bounded from below and we can expect the existence of an absolute minimum. Proposition 1.5.2. Let ρ > q. Suppose that A is a continuous, strongly monotone and homogeneous potential operator of degree ρ — 1 and that Β is a positive, strongly continuous and homogeneous potential operator of degree q — 1. Then there exists w φ 0 such that F(w) = inf F(u) < 0 ueX

and w satisfies (1.1). Proof

By virtue of the Sobolev inequality we have F(u) = a(u) - b(u) > —(A(u), u) --S~\ p q

{A(u), u)p.

Since ^ < 1, we see that F is bounded from below. Let u0 φ 0. Then for t > 0 sufficiently small, F(tu0) = a(tu0) - b(tu0) = tpa(u0) - tqb{u0) < 0

1.6 Multiple solutions of nonhomogeneous equation

17

and h e n c e - c o < infuex F(u) < 0. By Theorem 1.1.2 there exists a sequence [um] C X such that F(um) —»· inf„ e x F(u) and F'(um) 0 in X* as η oo. Under our assumptions, the Palais-Smale condition holds (see Theorem 2.1.1 in Chapter 2). This means that there exists a subsequence of {um} converging strongly to w and this completes the proof. • Let w be an absolute minimum obtained in Proposition 1.5.2. Since w g V, as a solution of (1.1), we get the following inequalties F(w) = inf F(u) < inf F(u) < F(w). ueX ueV Consequently, we deduce the following result Corollary 1.5.1. Let ρ > q and suppose that A is a continuous, strongly monotone and homogeneous potential operator of degree ρ — 1 and that Β is a positive strongly continuous potential operator of degree q — 1. Then inf F{u) = inf F(u) U€X U&V and both infima are attained.

1.6 Multiple solutions of nonhomogeneous equation The methods of the preceding sections provide a surprisingly simple proof of the existence of at least two solutions of the nonhomogeneous equation A(u)-B(u)

= f,

(1.5)

with / G X* — {0}. One solution is obtained by a local minimization and a second solution is a consequence of the mountain pass theorem without the Palais-Smale condition (Theorem 1.1.4). In Chapter 2 we shall use the Lusternik-Schnirelman theory of critical points to obtain the existence of multiple solutions of a homogeneous equation. We define a functional Ff : X R by Ff(u) = a(u) - b(u) - ( / , u). Lemma 1.6.1. Let q > p. Suppose that A is a continuous strongly monotone and homogeneous potential operator of degree ρ — 1 and that Β is a positive strongly continuous potential operator of degree q — 1. If

/ ** 0 and u0 e 5 ( 0 , p) such that Ff{u0)

=

min

Ff(u)

< 0

and u0 is a solution of equation (1.5). Proof. First, we show that there exists a constant ρ > 0 such that F/(u) u g 5(0, p). We have Ff(u)

> 0 for

> Μ Ι ^ ω ί Ν Ι * - 1 - N l * " 1 sup b(u) - II/H*,) II« II=1

and setting ||u|| = t, we write this estimate as Ff(u)>t(h(t)-\\f\\x*), where h(t) = atp~l

max A 0 0 as m —>• oo. This implies that lim Ff(um)= m-voo J

inf Ff(u) ueB(0,p)

and

lim F'(um) m->-oo

= 0 in X*.

The last two conditions yield that {wm} contains a strongly convergent subsequence which we again denote by {um}. Therefore um u0, with u0 e 5 ( 0 , p). Here, we have used the fact that the Palais-Smale condition holds (see Theorem 2.1.1 in Chapter 2). • We are now in a position to establish the existence of a second solution.

19

1.7 Sets of constraints

Theorem 1.6.1. Let q > p. Suppose that A is a continuous strongly monotone and homogeneous potential operator of degree ρ — 1 and that Β is a positive strongly continuous homogeneous potential operator of degree q — 1. If (1.6) holds then equation (1.5) has at least two nontrivial solutions. Proof Let u0 be a solution constructed in Lemma 1.6.1. According to (1.6) we may assume that Ff(u)>c for all ||u\\ = ρ for some constant c > 0 and obviously w e B(0, p). We take «ι # 0 and choose ta so that Ff{tu\) < 0 for all t > t0. We may choose t0 so that ||ί 0 "ι II > Ρ· We set for a fixed t > t0 Γ = {g e C([0,1], X); s(0) = 0, g( 1) = tux), then 0 < c < «ο = inf max £€Γί€[0,1]

Ff(g(t)).

By Theorem 1.1.3 there exists ü such that Ff(ü) = c and F'^(ü) = 0. Since Ff(ü) = c > 0, μ is distinct from the solution uQ constructed in Lemma 1.6.1. •

1.7 Sets of constraints In this section we investigate a set of constraints for the nonhomogeneous equation (1.15). By examining the second order derivative of F f , we obtain some information about the localization of critical points of (1.5). Throughout this section we assume that the potential operator A : X X* satisfies assumptions of Lemma 1.6.1 and that potential operator Β : X -»· X* is positive, continuous and homogeneous of degree q — 1. Its potential is denoted by b : X —^ K. Moreover, we assume that q > p, where ρ is the degree of the potential a : X E. If u satisfies (1.5), then (A(m), u) - (B(u), u) - (/, u) = 0. Therefore, we introduce a natural set of constraints Λ given by A = {u g X; {A(u), u) - (B(u), u) - (/, u> = 0}. We aim to show that there exists u0 g X such that Ff(u0) = inf Ff(u), Κ€Λ and that uQ is a local minimum of F. From Section 1.6 it is clear that this is true under some restrictions on ||/||χ». Suppose that μ is a local minimum of F. Then h"{t) | i = i > 0,where/i(0 = Ffitu). This yields that p(p - I M « ) - q(q - \)b(u) = (Ρ - 1)(Λ(μ), U) - (q - 1 )(B(u), u) > 0.

20

1 Constrained minimization

Consequently, we can split Λ into three parts: Λ+ = {u e A; (p - l){A(u), u) - (q - 1 )(B(u), u) > 0}, Λ„ = ( « 6 Λ; ( ρ - 1 )(A(u), u) — (q — 1 )(B(u), u) = 0},

Λ_ = {u G Λ; (p - 1)(A(m), U) - (q - 1 )(B(u), u) < 0}. We now impose an assumption on / : ( f , u ) γ μο

> ο

for all m. On the other hand by virtue of (1.14) we have ^E-{B{um),um) ρ- 1

+ o(\) =

{f,um)

and consequently, taking a subsequence if necessary, we see that

^ F(t |7/#— u)\ F(su)

r 0η < s < ,i m a x tor

, vr - Ό 1)

(B(u),u)

where t u e Λ+. It follows from the final part of Proposition 1.7.2 that a solution ua obtained in Theorem 1.8.1 must belong to Λ+, so we have

t

\ 1 , ( ρ - 1 ) (A(M 0 ),M 0 )\?-P = t (m0) = 1 < τ τ τ ^ τ — Λ — • (q- 1) {B(u0),u0)J

Taking e > 0 sufficiently small we may also assume that 1 < ι (P ~ 1) (a(uq ~ w), u0-w)\*h (q - 1) B(u0 - w), m0 - w)

( U 5 )

1 Constrained minimization

28

for w e B(0, e). By Lemma 1.7.2 there exists a function t : B{0, e) -> (0, oo) such that t(w)(u0 Since t(w)

1 as ||u»||

- w) € A

for w € B(0, e).

0 we may also assume that

/ (ρ - 1) {A(u0 - w), U0-W)\I=P t(w) < ( } V {q - 1) (B(uο - w), u0 w)) Consequently, we have Ff(s(u0

- w)) > Ff(t(w)(u0

- w)) >

Ff(u0)

ι " " · A c c o r d i n g t o (1-15) we can take s = 1 in the

for 0 < ί < last inequality to obtain

Ff(u0 - w) >

Ff(u0).

• Under assumption (1.8) the space X can be disconnected by A_ into the sum of two connected components. Indeed, by Proposition 1.7.2 for every ||u|| = 1, there exists t+(u) > 0 such that t+(u)u e Λ_ and Ff{t+(u)u) = max,> imax Ff{tu) and t+ is a continuous function of u. We now set U\ = {m = 0 o r ||u|| ί + ( « | | Μ | Γ 1 ) } and X — Λ_ = U\ U U2. We see that Λ+ c U\. It is clear that Λ_ is a closed set. Proposition 1.8.1. Suppose that f satisfies either (1.9) or (1.8) with Β being strongly continuous. Then inf Ff(u) = inf Ff(u) < inf Ff{u). utA ueΛ+ ueΛ_ Proof. In the contrary case infA Ff(u) — infA_ Ff(u). As in the proof of Theorem 1.8.1 we can construct a minimizing sequence {um} C A_ w i t h F / ( « m ) —> infA Ff < 0 and Ff(um) —> 0 in X*. Then {um} contains a subsequence , denoted again by {um}, such that um —> w in X. We necessarily have { f , w) > 0, so w G A-)-. On the other hand, since A_ is a closed set we must have w e A_ and this gives a contradiction. • The existence results in Sections 1.6-1.8 for equation (1.5) have been established assuming that the potential operator Β is strongly continuous. This is the case for the nonhomogeneous Dirichlet problem in Ψ 0 1,2 (Ω) -Au

= \u\p~2u +f u = 0

on 3Ω,

in Ω,

1.8 Minimization for Ff

29

where Ω c Μ", η > 2, is a bounded domain and f e ΐ ^ _ 1 · 2 ( Ω ) with / φ 0. If 2 < ρ < then defining potential operators A : Ψ 0 ! · 2 (Ω) -»• W~l·2(Ω) and Β : Wl' 2

ΐ ν " 1 · 2 ( Ω ) by, (A(u),v)

= / JQ

DuDvdx

(B(u),v)=

I Jn

\u\p~2vdx

and

for u and υ in W 0 1,2 (fi), we see that A is continuous, strongly monotone and that Β is strongly continuous. The strong continuity of Β is the consequence of the Sobolev compact embedding theorem. Consequently we can apply results of Sections 1.6-1.8 to obtain the existence of at least two distinct solutions provided the norm of / in W~1'2 is small enough. If ρ = then Β is not strongly continuous and the above problem is characterized by the lack of compactness. In this case Tarantello [222] established the existence of at least two distinct solutions by showing that μ0 > 0 using the intrinsic properties of the extremal function ([43], [44]) characterizing the Sobolev inequality in W 1 , 2 (R' 1 ) (see Section 9.1 in Chapter 9).

1.9 Subcritical problem We now consider a slightly more general equation than (1.1), namely, A(u)-kB(u)

= C(u)

(1.16)

with 0 < λ < λ ι , where λ\ is the smallest eigenvalue for the homogeneous problem (1.1). We assume that C : X X* is a strongly continuous potential operator. A problem of this nature will be called the problem with a subcritical potential operator C (or a problem with compactness). We set h = inf{a(u)-kb(u);

c(u) = 1}

(1.17)

Theorem 1.9.1. Suppose that A is a strongly monotone, hemicontinuous and homogeneous potential operator of degree ρ — 1. Further, we suppose that Β and C are positive strongly continuous and homogeneous potential operators of degree ρ — 1 and q — \, respectively. If0 a(uj) - kb(uj) > a(Uj)( 1 ~ γ ) > IM"(l λ]/ ρ

-

λ

γ)· Μ

Consequently, the sequence {uj} is bounded in X and we may assume that Uj u in X. Since C is strongly continuous we have c(w) = 1. By the lower semicontinuity of a and strong continuity of Β we get a(u) - Xb(u) < h and this implies that a(u) — Xb{u) = /χ. The fact that u satisfies (1.16) follows from Theorem 1.2.1(i). • Remark 1.9.1. For a functional Ff : X

Μ given by

Ff{u) = a(u) - Xb(u) - c(u) - ( / , u) associated with the equation A(u) - XB(u) - C(u) = / , where / € X* — {0}, we have the following estimate from below P-1 _ llwlli-1 sup c(u) - II/II a:* ll«ll=l

Ff(u)>\\u\

Also, if μ φ 0, then Ff (tu) < 0 for t sufficiently large. Therefore, assuming that q > ρ and additionally that A is continuous and using arguments from the proof of Lemma 1.6.1 and Theorem 1.6.1 we can prove the existence of at least two distinct solutions provided Ei X. < „ ( ! ) ( ! - A ) and / e r

(q -

1 ) SUp||

tt||=l

c(u)

(q-p) (q- 1)

- {0}.

1.10 Application to the /7-Laplacian Our aim is to apply the methods described in this chapter to the problem -ApU u e

+ C\U\P~2U WL-P(RN),

= r(x)\u\«-2u

in E" (1.18)

31

1.10 Application to the p-Laplacian where — Δ^ is the p-Laplacian, that is,

-Apu =

Di(\Vu\p-2Diu).

We assume that c > 0 is a constant and 2 < ρ < η. We write r ( x ) = r + (;c) — r_(x) and assume that r g L ^ C K " ) and that ϊ±ί (αϊ) r+ e Llo* (R"), where pl)

r

lim^i^oo JQ^x

£+£

r+(;y) « dy = 0 for some l > 0, where

I Q(x,l) = {y\ Iyt - χι \ < -, ι = 1, ..., n} A: W-]'p'(Rn), p' = ^ y , by {A(u),v)= f (\Vu(x)\p-2Diu(x)Div(x) + c\u(x)\p-2u(x)v{x))dx (1.19) J R" for all u, ν € W ^ i R " ) . Let us set D(b) = {«; fRn lu(x)^r(x)dx < oo}, then we We define a potential operator

define a potential b : £>(b)

R, by

b(u) = b+(u) - b-(u) = - f r+CxOlMCx)!* dx - - f r-(x)\u(x)\q dx. JRn JR"

- q,

hence u G D(b). Since b is Gateaux differentiable at u in every direction, the equation (1.18) is satisfied in the distributional sense, that is,

f (\Vu(x)\p~2DuDv Jw

+ c\u(x)\P-2u(x)v)dx

= f Jw

for every υ in C ^ R " ) .

\u(x)\q~2uvdx •

If c = 0, then the appropriate Sobolev space for problem (1.18) is ι Ώ ·Ρ(β. η ) which is defined as the closure of C£°(R") with respect to the norm

\u\\PDUp= f Jw

\Wu(x)\pdx.

We can now state

Theorem 1.10.2. Let ρ < η and c = 0. We assume that r+ satisfies (a\) and that r+ G L1 (R n ) and r_ G Lj1 (R n ), then problem (1.18) admits at least one nontrivial solution in Dhp(W). To prove this result we apply Theorem A.3.2 with

a(u) = — f \Vu(x)\pdx. Ρ Jw

1.10 Application to the p-Laplacian

33

The weak continuity of b+ is a consequence of Theorem A.3.2. If r{x) = 1 on R", then Theorem 1.2.1 is not applicable since is not q n compactly embedded into L (R ) (see Appendix A.5). This difficulty can be overcome if we use instead of W 1,p (IR") the subspace of radially symmetric functions. To explain this in more details we consider the following problem + cu = \u\p~2u inR" ,, u G Wl-2QBLH),u > 0 onR", -Au

(1-20)

where c > 0 is a constant and 2 < ρ < Let H}{Rm) be a subspace of radially symmetric functions equipped with norm from the space W1 ' 2 (R n ). The space H} (R n ) is compactly embedded in L p ( R n ) for 2 < ρ < ^ ^ (see Appendix, Theorem A.5.2). Therefore, there exists a positive function ν e Η) (M") such that f {\Vv(x)\2+ Jw

cv(xf)dx

= inf{ f (|VM(JC)| 2 + cu(x)2) dx\ I Jw

(1.21)

u e H* (R") and / \u(x)\p dx = 1} JR" * If {um} is any minimizing sequence in W1,2(R")

/

n

'R

(|V«m(x)|2

+ cum(x)2) dx

such that I

as m o o ,

where / = inf I I n (\Vu(x)\2+ IJu

cu(x)2)dx\

u eWh2(Rn),

I \u(x)\p dx = JR"

l|, '

then using the Schwarz symmetrization (see Appendix, Section A.6) we have f \u*m(x)\P dx = f \um(x)\p J R" J R"

dx

and / < / {\Vu*m(x)\2 + cu*m(x)2)dx< J R"

f (\Vum(x)\2 J R"

+

cum(x)2)dx.

Therefore, the infimum in (1.21) can be taken over W 1 , 2 (R n ) and is equal to I. This discussion leads to the following existence result for problem (1.20). Proposition 1.10.1. There exists a positive solution of problem (1.20) in Hrl (R n ). This result will be used in Chapter 3. In Chapter 5 we consider a generalization of problem (1.20).

34

1 Constrained minimization Finally, let us consider the nonhomogeneous problem — ApU + c\u\p~2u u G

= r(x)\u\y~2u

+ f { x ) in

(1.22)

Wl'P(]R"),

where f £ Wl 0 is the first eigenvalue of the problem —Apu = k\u\p~2u

in Ω,

1.11 Critical problem

35 u(x) = 0

on 3Ω.

The multiplicity of λι is 1 and the corresponding eigenfunction can be taken positive (see [14] and [155]). Applying Theorem 1.9.1 to the minimization problem (h)

jnf {Un{\Vu(x)\P νν0''ρ(Ω)

+ X\u(x)\P)dx·,

I /Ω \u{x)\* dx = 1}

for 0 < λ < λι, we obtain the existence of a nontrivial solution u to problem (1.23). Since |m| together with u is a minimizer of (/χ), we can assume that u is positive (see [16]).

1.11 Critical problem Our aim here is to consider problem (1.16) under a weaker assumption on C. We say that problem (1.16) is critical (or a problem without compactness) if the potential operator C : X X* is weakly continuous. This problem is more complex than the problem discussed in Section (1.9) and has attracted a lot of attention in recent years. We give here a simple existence result for problem (1.16). We shall investigate this problem in more detail in Chapter 9. We denote by Sp,q the best Sobolev constant for operators A and C, that is, SPtg = inf{fl(w); c(u) = 1} (see Proposition 1.2.2 and Remark 1.2.1). Finally, let λι > 0 be the smallest eigenvalue for operator equation (1.1) (see Proposition 1.2.3). Theorem 1.11.1. Suppose that potential operators A and Β satisfy assumptions of Theorem 1.9.1 and that A is a bounded potential operator. Moreover, we suppose that C is a monotone, weakly continuous and homogeneous potential operator of degree q — \, with q > p, where ρ — 1 is a degree of Β and A. Let {uj} be minimizing sequence for problem (1.17), that is, a(uj) — kb(uj) —• Ιχ as j —>- oo, with c(uj) = 1 for every j. If Ιχ > 0, then either c(u) = 1 or u = 0. If Ιχ < Sp>q andO < λ < λ ι, then c(u) = 1 and Ιχ = a(u) — kb(u). Proof. It is easy to see that {uj} is bounded and we may assume that uj u. Suppose that u φ 0. Let υ e X — {0} and t > 0. Since potential c is convex we have c(uj

+ tv) - c{Uj)

> t(C(Uj),

v)

and hence c(uj+tv)

> 1 +t(C(uj),

v).

36

1 Constrained minimization

For potentials a and b we have fl d a(uj + tv) — a(uj) = I —a(uj+stv)ds Jo ds

fl = I (A(uj + stv),tv) Jo

= / (A(uj + stv) - A(uj),tv)ds Jo

+ t(A(uj),

ds v)

and b(u •j+tv) We also have

- b(uj) = [ (B(uj+stv) Jo

- B(uj), tv)ds + t(B(uj),

υ).

£ a(uj + tv) - Xbiuj + tv) > hc(uj + tv)» .

Therefore, a(uj) - kb(uj)

+ t ({A(uj), v) - k(B(uj), > /λ(ΐ

+ (C(uj),b))i

v}) + Aj(t) - kBj(t)

(1.24)

,

where Aj(t)=l

(A(uj + stv) - A(uj), tv) ds Jo

and Bj(t) = I (B(uj + stv) - B(uj), tv) ds. Jo The strong continuity of Β implies that lim Bj(t) = t f (B(u + stv) - B(u), v) ds. j^-oo Jo Since A is bounded on bounded sets, using the dominated convergence theorem we obtain that lim Aj(t) = t I (A(u +stv) j-> oo Jo

- A(u),

v)ds.

Letting j -»· oo in (1.24) we get h + t ((A(M), v) -

k(B(u),v))

-tk - tk fI i{B(u + stv) - B(u)) ds !o Jo

+ t I (A(u + stv) - A(u), v) ds > Ιλ ( 1 + t{C(u), v))* Jo

1.12 Bibliographical notes

37

We write this in the form

{(1 +i

I\c(u)i.

£ Hence c(u) > c(u)i. Since 0 < c(u) < 1 and ρ < q, we see that c{u) = 1. To prove the second assertion we observe that if {uj} is a minimizing sequence, then a(uj) - kb(uj) > Sptqc(uj) - kb(uj) = Sp Sp 0 and by the previous part of the proof we must have c(u) = 1 and this completes the proof. •

1.12 Bibliographical notes Results presented in Section 1.1 are standard and can be found in the monographs by Vainberg [231], Chow-Hale [75], Deimling [95], Aubin-Ekeland [19], AmbrosettiProdi [9] and Schwarz [203], The example of the p-Laplacian as a potential operator is taken from Lions' monograph [155]. The algebraic inequalities used to show the monotonicity of A can be found in Glowinski-Marrocco [128] (see also [210]). The proof of Theorem 1.1.1 is taken from Ekeland's paper [110] and it also appeared in a similar form in Penot's paper (see also Ekeland [109] and monographs [164] and [212]). Theorem 1.1.1 can also be derived from a more general result due to Brezis and Browder [39]. Extensions of Theorem 1.1.1, and variations in the method,

38

1 Constrained minimization

were given by Danes [54], Penot [176], Georgiev [123] and Deville-Godefroy-Zizler [98], [99]. In papers [54], [176] and [123], it is proved, among other things, that the Ekeland's variational principle is equivalent to the Drop Theorem. The paper [123] contains a strengthened version of Theorem 1.1.1 also based on the Drop Theorem. The papers [176], [98] (see also a monograph [99]) offer a quite different approach based on properties of Banach spaces admitting a smooth Lipschitzian bump function. Theorem 1.1.3 is taken from Ambrosetti-Rabinowitz's paper [10]. Theorem 1.1.4 can be traced in Brezis-Nirenberg's paper [42]. Proofs of Theorems 1.1.1-1.1.4 can be found in the monographs by Struwe [212] and Mawhin-Willem [164], Proposition 1.1.1 appears in Calkovic, Shujie and Willem [51] and Goeleven [129]. Constrained minimization methods, as well as relations among them described in Sections 1.2-1.5, are scattered throughout the literature and have been used in many different problems in variational calculus and in boundary value problems for nonlinear ordinary and partial differential equations. In particular, the constrained minimization subject to constraint V (Proposition 1.4.1) originates in Nehari's paper [171]. We have attempted to provide a unified approach based on the use of potential operator equations in Banach spaces. For further aspects of the dual method we refer to the papers [8], [9], [11], [12] and [87]. To prove Theorems 1.2.1 and 1.3.1 and Proposition 1.4.1 we have used some ideas from Briining's paper [48]. Theorem 1.6.1 is an abstract version of results obtained by Xi Ping Zhu [244] and Xi Ping Zhu and Huan Song Zhou [245]. Results of Sections 1.7 and 1.8 are patterned after Tarantello's paper [222]. A different approach to the nonhomogeneous elliptic equation involving a critical Sobolev exponent can be found in [161], [246] and [60]. Theorems 1.10.1 and 1.10.2 are due to Chabrowski [65], [66].

Chapter 2

Applications of Lusternik-Schnirelman theory

The main purpose of this chapter is to study the existence of critical points via the Lusternik-Schnirelman theory of critical points. The essential advantage of this approach is that it requires a C 1 -smoothness of a given functional. We distinguish constrained and unconstrained cases. The latter case is described on some selected situations. The two essential ingredients in the Lusternik-Schnirelman theory are the (PS) condition and the existence of a deformation mapping. This mapping allows to deform the set {F(u) < c + e} into (F(m) < c — e} for small e > 0 provided the level set {F(u) = c} does not contain any critical point. This chapter contains a deformation lemma for C 1 functionals. Variants of this lemma are presented in Chapters 5, 6 and 7. In particular, in Chapter 7 we present a deformation lemma for C 1 -functionals perturbed by a lower semicontinuous convex functional.

2.1 Palais-Smale condition, case ρ Φ q Throughout this chapter we assume that X is a reflexive Banach space. Let F : X -> Μ be a functional defined in Section 1.5. We know that under some regularity assumptions on potential operators A and B, any critical point of F is a solution of equation (1.1). In this chapter we examine the existence of critical points via the Lusternik-Schnirelman theory of critical points. The fundamental step in applying this theory is to show that the Palais-Smale condition holds, that is, if for a sequence {um} in X, F{um) c and F'(um) —• 0 in X* as m oo, then {um} has a strongly convergent subsequence in X (see [174]). If the degrees of A and Β are distinct, it is not difficult to show that F satisfies the (PS) condition. However, if the degrees of A and Β are the same, the situation is more complex. In this case we use the variant of the (PS) condition established by Amann [13] (see also [243]) which holds for sequences {um} along which b{um) > const > 0. It turns out that this is sufficient to establish the existence of infinitely many critical points. We point out here that in Amann's original version of the Palais-Smale condition, the homogeneity of operators A and Β is not needed. Theorem 2.1.1. Suppose that A is a hemicontinuous, strongly monotone and homogeneous potential operator of degree ρ — 1 and that Β is a strongly continuous potential operator of degree q — 1, with ρ φ q. Let {um} be a sequence in X such

2 Applications of Lusternik-Schnirelman theory

40

that F(um) c for some constant c and F'(um) —>• 0 in X* as m —> oo, then the sequence {um} contains a subsequence strongly convergent in X. Proof.

According to the homogeneity of A and Β, F takes the form

Ρ

0, we can find m0 such that -€
m 0 . Moreover, we have -M

1 < —(A(um), Ρ

1 um) - —(B(um), um) < Μ q

for all m and some constant Μ > 0. These two inequalities yield - M - - < ( - - - ) (B(um), um) < Μ + — ρ \p qJ ρ for m > m0. This means that the sequence {(B(um), um)} is bounded. m)i u m)} is also bounded. By the strong monotonicity of A we have l|KmllPK(l|K«ll) < (Mum), Um) < € + -^—M q - ρ

+

Hence,

q - ρ

for all m > m 0 . Since l i m ^ o o κ(ί) = oo, we see that the sequence {um} is bounded. Therefore, we may assume that um —u in X. Since Β is strongly continuous and F'(um) —> 0 in X* we see that {A(um)} is convergent in X*. We now observe that A satisfies condition (S)i, hence {um} converges strongly in X. • Remark 2.1.1. Theorem 2.1.1 remains true, if F is replaced by Ff(u) = a(u)—b(u) — ( / , u), with / 6 X*. This fact has already been used in Lemma 1.6.1.

2.2 Duality mapping In this section we recall some properties of a duality mapping that will be needed in this chapter. Let X be a Banach space. A set-valued map J : X J(u) = {«* € X*; ||K*|| = ||u|| is called the duality mapping.

and

2X

defined by

(u*, u) = ||«||2}

2.2 Duality mapping

41

Proposition 2.2.1. Let J be a duality mapping, then (a)

J is a map if and only ifX* is strictly convex; in particular, J = I (identity map), ifX is α Hilbert space.

(b) X* is uniformly convex if and only if J : Χ —»· X* is uniformly continuous on bounded sets. (c)

X* is uniformly convex if and only if || · || is Frechet differentiate (uniformly on a unit sphere). Consequently, J can be viewed as Frechet derivative of a functional j(u) = J || u ||2. The norm is Gateaux differentiable on X — {0} if and only if X* is strictly convex. For the proof we refer to [95] (Chapter 3).

Let Φ : [0, oo) —• [0, oo) be a continuous strictly increasing function with Φ (0) = 0 and linv-yoo ΦΟ) = oo. Following J. L. Lions (see [156], Chapter 2) we say that a mapping Jφ : X —> X* is a duality mapping relative to Φ if (7Φ(Μ), U) = ||/Φ (Μ) || || Μ ||

f o r all Μ e X

and ||/Φ(Μ)||

= Φ(|Μ).

Proposition 2.2.2. Every duality mapping 7φ is monotone. Indeed, by a straightforward computation, using the above definition, we easily check that ( Φ ( | Μ ) - Φ(ΙΜ|))(||η|| - Ν | ) > 0. Proposition 2.2.3. I f X is strictly convex, J is strictly monotone, that is, {J(u) — J 0 the level set Ma = {u; a(u)

= a}

is bounded and each ray passing through the origin intersects Ma. there exists ρ = p(a) > 0 such that {A(u), u) > ρ for all u € Ma.

Moreover,

A mapping A satisfying (a) must be bounded. Moreover, we can define a mapping ρ : X — {0} (0, oo) by a(p{u)u) = a. This mapping is well defined. Indeed, suppose that for some 0 < t\ < t j we have a(t\u) = a f a u ) = a. Then by assumption (a) we have 0 = a(t2u)

— a{t\u)

f" dt = I (A(tu),tu}—> t Jt\

0,

which is impossible.

Lemma 2.3.1. The

fiinctional

ρ has the following

(i)

ρ is bounded

(ii)

ρ has Frechet derivative ρ' : X —> X* which on bounded sets which are bounded away from

(iii) If for u G Ma,

on sets bounded

(A(u),

away from

properties: zero.

v) = 0 for all ν € X, then

is odd and uniformly zero.

continuous

( p ' ( u ) , v) = 0 for all ν € X.

Proof. It follows from (a), that Ma is symmetric, bounded and bounded away from zero. It is easy to see that ρ inherits these properties; that is, ρ is even and bounded on sets which are bounded away from zero and (i) holds. Since Λ is a continuous potential operator, its potential a is Frechet differentiable. For every u e X — {0} and every t > 0 we have d —a(tu) dt

= {A(tu),

u) = t

_, 1

(A{tu),

tu) > 0.

The implicit function theorem yields that ρ is continuously differentiable and 0 = ( a ( p ( u ) u ) ) ' = (A(p(u)u),

u)p'(u)

+

p(u)A(p(u)u),

that is, Ρ (") =

P(uΫ Λ (A(p(u)u,

, , A(p(u)u). p(u)u)

(2.1)

44

2 Applications of Lusternik-Schnirelman theory

This representation for p' implies that p' is bounded on sets which are bounded away from zero because ρ and A have this property. Finally, using the relation p(u)-p(v)=

f (p'(v + Jο

t(u-v)),u-v)dt

which holds for u, ν e X and with 0 £ {v + t(u — ν), 0 < t < 1}, we see that ρ is uniformly continuous on bounded sets which are bounded away from zero. Consequently, by virtue of (2.1) p' satisfies (ii). Property (iii) follows from (2.1) and from the fact that p(u) = 1 for all u e Ma. • To proceed further we assume that X is uniformly convex Banach space. Hence X is reflexive Banach space. Let 7 : X* —> X be a duality mapping, that is, for each u € X* we have (u,J(u)) = \\u\\2 and ||7(||)|| = ||«||. By Proposition 2.2.1(b) 7 is uniformly continuous on bounded sets. Suppose that Β : X —> X* is an odd potential operator which is uniformly continuous on bounded sets. We define mappings D : Ma —• X* and Τ : Ma —• X by Diu) = B(u)

-

{A(u), u)

and TV λ = Junt T{u) (D(u))*

{Mu)

/(D(m))) >——— u.

(A(u), «)

By a straightforward computation we check that (B(u),T(u)) and

= \\D(u)\\2

(A(u), T(u)) =0

for u € Ma

for u G Ma.

(2.2) (2.3)

Moreover, Τ is odd and uniformly continuous and hence is bounded. Consequently, there exist constants τ 0 > 0 and y0 > 0 such that \\u + TT(U)\\ >

Vo

for all u g Ma and r € [—r 0 , r 0 ]. We now define a mapping Η : Ma χ [—τ0, r 0 ] —> M a by H(u, R) = ρ (μ + τΤ(μ))(μ

+ ΤΓ(Μ)).

We now formulate and prove Lemma 2.3.2 and 2.3.3 which will be used in Chapter 3. Lemma 2.3.2. Let X be a uniformly convex Banach space. Suppose that A : X —»· X* satisfies (a) and that Β : X X* is an odd potential operator which is uniformly

2.3 Palais-Smale condition, case ρ = q

45

continuous on bounded sets. Then there exists a mapping r : Ma χ [—τσ, τ0] —• Μ, with limr_>o r(u, r) = 0 uniformly in u £ Ma, such that b(H(u, t)) ~ b(u) 4- f {\\D(u)\\2 + r(u,x))dx Jo

(2.4)

for all u e Ma and t e [—r0, r0]. (Here, b denotes the potential of B). Proof

Let («, t) e Ma χ [-r0, τ0], then b(H(u, 0) = b(u) + f (B(H(u, τ)), Jo

9τ

H(u, r)> dx.

It follows from Lemma 2.3.1 and (2.3) that 3r

τ) = {p\u + xT(u)), T(u))(u + τΤ(τ)) = T(U) +

+ p(u +

τT(u))T(u)

R(U,T),

where R(U,T)

= (P(U + TT(U))~

We see that limr_>.o the following form

P(U))T(U)

+ (P'(U + TT(U))~

P'(U),T(U))(U

+

TT(U)).

r ) = 0 uniformly in Ma. Therefore, we write b(H(u, t)) in

b(H(u, t)) = b(u) + f ((B(u), T(u)) + r(u, τ)) dx, Jo where r(u, r) = (B(u), R(u, x)) + {B(H(u, τ)) - £(u), ^ - / / ( « , τ » , dx and this implies that limr_>.o r(u, r ) = 0 uniformly in Ma. Now, by (2.2) we have (B(u), T(u)) = (Diu), J(D(u))} = ||D(m)||2 and this completes the proof.

•

For every β > 0 w set Nß = {ueMa:

\b(u)\ > β).

Using Lemma 2.3.2 we can prove the deformation lemma: Lemma 2.3.3. Let the assumptions of Lemma 2.3.2 hold. Suppose that for β > 0 there exist an open set U C Ma and constants δ > 0 and 0 < ρ < β such that ||D(«)|| > δ for Vp = {ue M a ; u#U,\\b(u)\

- β\ < ρ}.

Then, there exists an e > 0 and an odd continuous mapping H€ : Nß-€ —

U^-Nß+€.

46

2 Applications of Lusternik-Schnirelman theory

Proof. Since limr_,.o Κ", τ) = 0 uniformly in u e Ma, we can find τ\ e (0, rQ] such [2 that Ir(u, r)| < y for (w, τ) e Ma χ [—ri, ri]. We now set t(u, r) = r · sgnfo(u) and by Lemma 2.3.2 we have \b(H(u,t(u,T))\>\b(u)\

«52τ + —

for all (Η, τ) e VP χ [0, rj]. Setting e = min(p, we see that this inequality becomes Ib(H(u, t(u, r))| > \b(u)\ +2t>ß +€ for u g Vp Π Nß- e . It follows from (2.4) that for every fixed u e Vp \b(H(u, t(u, ·))| is strictly increasing on some interval [0, σ] D [0, τι]. Therefore, for every u e VP, the functional t€(u) = min{r > 0; \b(H(u, t(u, τ))| = β + e} sgnb(u) is well-defined, continuous on Vp and 0 < |i e («)| < τ\ for u e Vp. To complete the proof we define a mapping H€ : Nß-€ — U —• Nß+e by Hf(u) =

H(u, t€(u)) for u e Vp, u

for u e Nß-c - (U U Vp).

• We are now in a position to formulate Amann's version of the (PS) condition. Theorem 2.3.1. Suppose that X is uniformly convex real Banach space, A: X —• X* satisfies (a) and that Β : X —• X* is an odd potential operator which is uniformly continuous on bounded sets. Further, we suppose that b(u) φ 0 implies B(u) φ 0. Let β > 0 be fixed and let {uj} be a sequence in Ma such that b(uj) > β for all j and D(uj) —• 0 in X* as j —• oo. Then {Uj} contains a strongly convergent subsequence whose limit is an eigenfunction u e Ma of( 1.1). Proof Let ß j = . Since Ma is bounded and X is reflexive we may assume that uj —^ u in X and ßj —> μ as j —• oo. It follows from the weak continuity of b that \b(uj)\ \b(u)\ > β. Hence u Φ 0 and B(u) φ 0. Since D(uj) 0, we see that ßjA(uj) = B(uj) - D(uj) B(u) φ 0. Hence μ φ 0 and A(uj) A(u) = XB{u), where

μ~{B(u). λ =μ

Now, by condition (S)ι Uj —1

=

u. Therefore,

( A ( M ) , u)

(B(u),u)

.

•

2.4 The Lusternik-Schnirelman theory

47

2.4 The Lusternik-Schnirelman theory We establish the existence of infinitely many solutions of equation (1.1) using the

Lusternik-Schnirelman theory of critical points. Let Σ ( Χ ) be the family of sets A C X — {0} such that A is closed in X and symmetric with respect to 0, i.e., χ e A implies — χ € A. For A e Σ (X), we define the genus of A to be n, denoted by γ (A) = n, if there is a continuous and odd map φ : A - » R " — {0} and η is the smallest integer with this property. If such η does not exist we set γ {A) = oo and moreover y ( 0 ) = 0. The following properties of genus are well known (see [10], [185], [184] or [86]).

Lemma 2.4.1. Let A and Β be in Σ(Χ). Then: (a) I f f e C(A, B) is odd, then y(A) < y{B). (b) If Α dB,

then γ (A)

IR" — {0}. Since h φ 0 in A, the compactness of A implies that h φ 0 on Ng(A) for some δ > 0. Consequently, γ (Ng(A)) < k = γ {A). The reverse inequality follows from (b). (i) Let Ρ be a projection of X onto Ε along E 1 . If Λ Π = 0 , then Ρ is an odd continuous mapping on A with Ρ (Α) φ 0 on A. Therefore, by (c) γ (A) < k, which is impossible. • We note that if A e Σ ( Χ ) and γ (A) > 1, then A contains infinitely many distinct points. If A e Σ ( Χ ) and Ω is a bounded neighbourhood of 0 in R" and there exists a mapping h e C(A, 3Ω), where h is an odd homemorphism, then γ (A) = n. The proof of the existence of infinitely many critical points of a functional F ( u ) = a(u) — b(u), introduced in Section 1, will be based on theorems of min-max type. The simplest result from the Lusternik-Schnirelman theory of critical points is the following result (due to Lusternik): Let / e C ^ R " , R) be an even function. Then / = /|s(o,i) possesses at least η distinct pair of critical points. To explain the main idea of the proof, which we will follow in the proof of the unconstrained case, we set η = { A C 5(0, 1); y ( A ) > * } . The critical points are obtained by the min-max characterization

bk = inf max f(x). A€yk X€A By Lemma 2.4.1 γk Φ 0 for k = 1 , . . . , η and it is easy to check that the sequence {bk} is nondecreasing. Since A = —x) e γ\, b\ = infs(o.i) f ( x ) . On the other hand y„ = {5(0, 1)}, so bn = max^o.i) f ( x ) . The proof consists of two steps. First, using a deformation mapping we show that each bk is a critical value. In the next step we prove that if b = bk = • • • — bk+p~\ for some ρ > 1, then γ (Kb) > p, where Kb = {*; f'(x) = 0, f ( x ) = b}. This result, as well as the method, has an extension to a real infinite dimensional Hilbert space H. If / € CX(H, R) is even, satisfies the (PS) condition and is bounded from below, then for each r > 0 /|s(o,r) has infinitely many distinct pairs of critical points. We now turn our attention to a deformation lemma, the proof of which is straightforward for C 2 -functionals. The deformation mapping is then obtained by considering a gradient flow of F which in the case of C 1 -functionals is only continuous. To overcome this difficulty we replace a gradient by a pseudogradient. Let U be a subset of a Banach space X and let F e C 1 (X, M). We say that a vector

υ is a pseudogradient of F at u £ U, if

49

2.4 The Lustemik-Schnirelman theory (i)

||f|| < 2||F / (m)|| x ., (F'(U),V)>\\F'(U)\\2X..

(ii)

If F g Cl(X, R) and Y = {u e X\ F'(u) φ 0}, then υ : Y X is apseudogradient vector field on Y if υ is locally Lipschitz continuous and υ(.χ) is a pseudogradient vector of F at χ for all χ e Y. It is easy to check that if F € C2(X, M), where X is a Hilbert space, then F' is a pseudogradient vector field of F on Y. On the other hand, if F G Cl(X, R), then F' in general is not a pseudogradient vector field of F. Lemma 2.4.2. Let X be a Banach space and F G Cl (X, R). Then there exists a locally Lipschitz pseudogradient g : Y —> X. Proof. In what follows we use || · || to denote norms in X and X*. Let ü e Y, then there exists w G X such that ||w|| = 1 and (F'(w), W) > | | | F ' ( u ) | | . Letting ν — 11|F'(ü) II w we get INI < 2 | | f " ( S ) | |

||^(«)||2

(2.5)

for u g NÜ. The collection of of neighbourhoods Μ — {NQ\ Ü g Y} forms an open covering of Y. Since Y is a metric space it is also a paracompact space. Consequently, the open covering Ν has a locally finite open refinement Μ = {A/, ; i G /}. This means that for each i g / , there exists ü e Y such that M, c NQ. This also implies that there exists a vector υ = υ, such that inequalities (2.5) are satisfied on M,·. We now define ρ,(μ) = dist(«, X - Mi) and

•

It is easy to check that g is a pseudogradient of F on Y. To formulate a deformation lemma we need the following notation FA

= {«; F(m) < a] and Ka

= {m; F'{U)

=

0 and F(u) = a}.

Lemma 2.4.3 (Deformation lemma). Let e e l and F G C 1 (X, M). Then for every 0 < 8 < I there exists a continuous mapping (deformation) η : [0, 1] χ Χ —» X such that

50

2 Applications of Lusternik-Schnirelman theory

(i) η(0, u) = ufor all u £ X, (ii) η(ί, ·) is a homeomorphism ofX onto X for each t e [0, 1], (iii) τ](t,u) = uforallt € [0, 1] ifu & F _ 1 ( ( c — 2S,c + 28)) Π {u; | F » | > VS}, (iv) 0 < F(u) - F(n(t,u)) < 48 for all u e X and all t e [0,1], (ν) \\η(ί, u) — u\\ < leVSforallu € X andt G [0, 1]. (vi) Ifu G Fc+S, then either (a.) η(1, u) e Fc~s, or (b) for some t\ € [0, 1], we have \\F'(ji{t\, u)|| < 2\f&. More generally, let τ e [0, 1] and Ν = {utX\\F{u)-c\ V & }.

The sets Ν and X — Ν are disjoint and closed. Let g : X —> [ 0 , 1 ] be a locally Lipschitz continuous function defined by

g(u)

=

dist(«, X — N ) dist(w, X - N ) + dist(w, N )

Let φ be a pseudogradient of F defined on a set {m; F'(m) φ 0} and set V(«) =

0

ifueX-N.

It follows from the definition of the pseudogradient that ||0(«)|| > ||F'(m)|| > *Jh for u e Ν and hence || VC")II
denotes the duality pairing between spaces ^ ' ^ ( Ω ) and Ψ~ι,ρ'(Ώ), with ρ' = The potentials operators A and Β are homogeneous of degree ρ — 1 and q — 1, respectively. If ρ < q
Ρ ( Ω ) - {0}; A closed, symmetric} and for every integer k > 1 we set Σ* = { Α ε Σ ,

(A)>k)

Y

where y is a genus. In the sequel we shall always assume that 1 < p,q < Applying Theorem 2.5.1 and Corollary 2.5.1 (case ρ > q), Theorems 2.6.1 and 2.6.2 (case ρ < q) and Theorem 2.7.1 (case ρ = q) we obtain the following result.

2.8 The p-Laplacian in bounded domain

65

Theorem 2.8.1. Let ρ > q and for every integer k > 1 we set Ck = inf sup U€A

F(m),

then for each c* is a critical value of F. If c = c* = · · · = c*+r and Kc = {u G W0 (Ω); F(u) = c], then y(Kc) > r + 1. Moreover, the set of critical values is compact. To formulate the next result we need some notation: F0 = {u g

0 < F(u) < oo}

and the set of all odd homeomorphisms h in C f W o ' ^ ) ; W0 and h(B(0, 1)) C Fa is denoted by Γ.

(Ω)) such that Λ (0) = 0

Theorem 2.8.2. Let ρ < q and for each integer m > 1 let Tm be a family of all compact and symmetric subsets of ^ ' ^ ( Ω ) such that γ(Κ Π h(dB(0, 1))) > m for each h € Γ. Then bm = inf sup F(u) ueK is a critical value of F such that 0 < c < bm < bm+1. I f b = bm= bm+1 = · • · = bm+r, then y(Kb) > r + 1. The fact that in this case there exist infinitely many critical values is consequence of Theorem 2.6.3 which takes the following form Theorem 2.8.3. Let {Em} be an increasing sequence of subspaces of W0''Ρ(Ω) such that dim Em = m and U Em = Ψ*·Ρ(Ω). m> 1 Let Εbe

the topological and algebraic complement of Em and let cm = sup inf F{h{u)). ΛεΓ* «69ΰ(0,ΐ)η£^_[

Then cm is a critical value of F, the sequence {cm} is nondecreasing, 0 < cm < bm for each m > 1 and limm_».oo cm — oo. Finally, the case ρ = q is a typical eigenvalue problem. Following the notation of Section 2.1 we set Ma = {n G ^ ( Ω ) ; -

f \Vu(x)\Pdx

Ρ Ja

= a}

for a > 0 and by Ck we denote a collection of all compact and symmetric subsets C of Ma with y(C) > k.

2 Applications of Lustemik-Schnirelman theory

66

Theorem 2.8.4. Let ρ = q and set for every integer k > 1 ßk = sup min£(w). CeC* Then ßk > 0 for each k and there exists Uk € Ma such that /Ω \uk(x)\p dx = ßkp and Uk is a solution of the problem (2.14), (2.15) with λ = λ* = j^. Moreover, lim^oo ßk = 0 (lim^oo λ* = oo). Theorems 2.8.1-2.8.4 were obtained by Garcia Azorero and Peral Alonso [16]. Combining Theorems A.3.1 and 2.7.1 we can now establish the existence of a sequence of eigenvalues for a problem corresponding to (1.18). For each a > 0 we set Ma = {ue

W1·^);

- f (|Vu(x)\p + c\u(x)\p dx = a} Ρ JRn

and by Ck we denote the collection of all compact and symmetric subsets Κ of Ma with y(C) > k. Theorem 2.8.5. Let ρ = q and let r > 0 on R" with r φ 0. If η > ρ, we assume that £±1 r 6 (IR"), with 1 < ρ < ρ + € < j ^ f o r some € > 0 and -p I y lim I r(y) r(y * dy=0 \x\->°°JQ(x,l) Jo ι c

for some I > 0.

If η — ρ, we assume that L^ oc (M")/or some s > 1 and

/

lim I r(y)s dy = 0 for some I > 0. x|->oo .ln( r /) If η < ρ, we assume that r e L|'oc(R") and lim I r(y)dy= \x\-+°°JQ(x,[) Jo

0 for some I > 0.

For each integer k > 1 we set ßk = sup min I r(x)\u(x)\p dx. CeCti CeCt " e C JlV Then there exists a sequence of eigenfunctions [uk] and corresponding eigenvalues {λ^} of the problem -Apu

+ c\u\p~2u = Xr{x)\u\p~2u

in R",

such that r{x)\u{x)\p dx = ßk

Moreover, lim^oo

q IR" Jw λ* = oo.

and

λ* =

a —. ßk

sequence

of

2.9 Iterative construction of eigenvectors

67

In an obvious way one can formulate the existence results for problem (1.18) using Theorem 2.5.1 if ρ > q and Theorems 2.6.1, 2.6.2 and 2.6.3 if ρ < q. According to Theorem A.3.1 and Proposition A.3.1 one has to distinguish cases: ρ < η, ρ = η and ρ > η.

2.9 Iterative construction of eigenvectors We shift our attention from a general existence theory to numerical methods of approximating eigenvectors obtained via min-max theory of critical points (see [150], [141], [142], [171] and [170]). We describe here a method due to Lehtonen [150], which can be applied to an eigenvalue problem in a uniformly convex Banach space. For eigenvalue problems in Hilbert spaces we refer to Kratochvfl-Necas' papers [141] and [142]. Let ßj(u)

= B(u)

in X,

where μ e R and X is a uniformly convex Banach space equipped with norm || · ||. A mapping J : X —> X* is a duality mapping of X, which has the properties ||./(u)|| = ||w|| and {J(u), u) = ||m||2 for all u e X. J can be interpreted as the Gateaux derivative of the functional j { u ) = ^ |Μ || 2 . Throughout this section we assume that Β : X X* is a strongly continuous potential operator with a potential b : X -*• R. We assume that 6(0) = 0, B(0) = 0, if b(u) φ 0 and ||m|| < 1, then B(u) φ 0. Moreover, (B(u) - Β (υ), u-v)> —h(\\u - υ||) (2.16) for all ||u|| < 1 and ||u|| < 1, where h : [0, oo) —• [0, oo) is a continuous function. Following Lehtonen [150], we define a measure of uniform convexity in X by Sr(€) = inf{/? - λ-\\u + υ||; ||u - υ|| >e,u,ve

B(0,R)},

where e > 0 and R > 0 are constants. Similarly, 2 ^

=

if 1

< ρ 0

L e t u \

and

let

4 n Λ ί 1 0 < 0 < mini , U ( l ) c 0 + 2h{\)

W?

d e f i n e a sequence

{ u

m

}

1 1. J

by J ( u

) +

m

9 B ( u

)

m

J(um+1) = — — ——. ||7(« m ) + 0fi(w m )|| 77ien there such

e x i s t s a subsequence

t h a t u

m

—•

u

, with u

0

o f { u 0

e

m

} , r e l a b e l l e d a g a i n by

5 ( 0 , 1 ) , and

ß J ( u

) =

0

B ( u

Since J is the duality mapping, we have 1 = C 5. Setting r = || J ( u ) + 9 B ( u ) II, we get

u

m )

m

m

0

{ u

m

} , and

a number

μ

R

€

) .

||/(w m +i)ll =

P r o o f . {

(2.20)

ll"m+ill. so

m

m > { / ( « m ) + ö ß ( H m ) , « m ) = ||«ml|2+ö(B(um),«m) > 1 - 0 A ( 1 ) .

r

Since

c

5 ( 0 , 1 ) , we may assume that um —" u0. b ( u ) —• b ( u ) . T o s h o w t h a t u 0 φ Owe prove that { b ( u Since Β is a potential operator it follows from (2.16) that m

0

b ( u

m

m

+ \ ) -

b { u

m

)

=

I

{ B ( u

m

+

t ( u

\ — u

m +

)

Obviously, we have that } is an increasing sequence.

) ) , um-i-i

m

— u

m

)

dt

Jo >

where e =

\\um+i

e ( B ( u

— u

m

) ,u

m +

m

1

( B ( u

m

) ,M

i

m +

-

u

m

)

-

f Jo

d t , t

||. According to Proposition 2.9.1 and (2.20) w have -

u

m

)

=

r

m

+

( J ( u ( J ( u

m

m

1 ) , u

+

) , u

m

m +

i

-

u

-

um+1>

m

)

> 2 ( r m + 1)Ä(€).

Combining the last two inequalities we get 2

Γ

σ

Jo

A(f) t

It follows from assumption (2.19) and the restriction on θ that { b ( u ) } is an increasing sequence. Moreover, due to the reflexivity of X* we can assume that J(um) —- u* in X * . Since {|| J ( u m ) + 9 B ( u m)II} is bounded we may also assume that r —> r . Consequently, the iterative formula (2.20) yields that m

m

( r

0

- \ ) u * =

9 B ( u

0

0

) .

On the other hand b{uQ) > b{um) > 0 and consequently B(u0) φ 0. This implies that u0 φ 0, u* φ 0 and r0 φ 1. It remains to show that um —> u0. To achieve this we write (1 - r ) J ( u ) = r ( J ( u ) - J ( u ) ) - 6 B { u ) (2.21) m

m

m

m

+

l

m

m

70

2 Applications of Lusternik-Schnirelman theory

and show that J(um+1) — J(um) —• 0. Indeed, applying Proposition 2.9.1 to u = J(um+1) and ν = J(um) on X* we have for η > 0 sufficiently small that G(b{um+1) - b(um))

> r m (7(M m + i), u m +i - n m ) + (J(u m)ι um ~ um+1 > 4T7 0 and by the previous part of the proof we know that b(um+1) — b{um) 0. Consequently, J(um+\) — J{um) 0. Now the strong convergence of {um} to uQ follows from (2.21) and from the fact that rm ->• rQ Φ 1, B{um) —• B(u0) and the continuity of the mapping : X* —• X. • In case when X = Η is a Hilbert space and assuming that condition (2.16) has the form (B(u) - B(v), u-v)>

-c\\u - v\\2

for some constant c > 0, the iterative sequence can be determined by = m+1

"

um+0B(um) \\um + 0B{um)\\

for some constant 0 < θ < £ (see Necas-Lehtonen-Neittaanmaki [170] and NecasKratochvil [141]).

2.10 Critical points of higher order To iterate higher order critical points we use Amann's approach described in Section 2.7. We additionally assume that b{u) > 0 for u φ 0 and that Β is odd on S. It follows from Theorem 2.7.1 that critical values {ßk} are given by formula (2.10). According to that theorem, for an eigenvalue ßk and the corresponding eigenvector w*, we have b{uk) = ßk with ßk —• 0 and Uk 0 in X (see Proposition 2.7.2). Let ß\ and ß2 be the first and the second critical values of b and suppose that there exists a constant € > 0 such that there are no critical values in the interval (#2 — e, βι)· Let us choose a compact set K\ G CI such that ßl — € < min b(u) < βι and define a mapping ψ : S

S by

2.10 Critical points of higher order

71

We see that φ is an odd continuous function. We now select a point u g Κ ι and form the η-fold composition φ^ = φο · • · ο φ. For each integer m > 1 let u°m be a vector in AT] such that min b( 2. Consequently, it follows from (2.22) and (2.23) that

lim min b(p(m)(u)) m^ooueK\

< ß2.

Let us set

κ = lim min b(oo u°)) is a critical value of b. Since, by our assumption, there are no critical values in (ß2 — €, ß2), we obtain lim m-»oo

> ß2.

Given η > 0, there exist m0 and mι such that

b{ß2-n for all m >m\.

This and (2.23) implies that

b{ b(β2~η

2 Applications of Lusternik-Schnirelman theory

72

for all m > max (mi, m 0 ). This means that κ = lim b^m\u°m))>ß m-yoo

2-η

and assertion (i) follows. The proofs of remaining assertions (ii), (iii) and (iv) are patterned after the proof of Theorem 2.9.1. • We now formulate a parallel result for higher order critical values. Theorem 2.10.2. Suppose that ßl > · • • > ßk > ßk+1 = · · · = ßk+l > ßk+l+l are eigenvalues ofb restricted to 5(0, 1) and moreover assume that there are no critical values in the interval (ßk+l — ßk+l)· Let K\ be a compact set in Ck+\ such that ßk+ι - e < min b(u) < ßk+t u&Ki and let {u^} be a sequence defined by (2.22). Then lim b(• R belongs to C 1 (X, R) and its potential operator Β : X X* satisfies

(B(u), u) > qb(u) > 0

(A3)

on X — {0} with ρ < q.

Let b(u) = (B(u), u) — pb(u) and we assume that *(")< * ( N I ? + ll«f) for all u G X and some constants ρ < q < r and Κ > 0.

(A4)

The functional qb(u) for all u e X - {0}, where Β : X derivative of b.

We now list the consequences of the hypotheses (Αι) later. It follows from (A2) that b(u) = (B(u), u) - pb(u) >{q-

X* is the Frechet

(A5) that will be needed

p)b(u) > 0

(3.2)

for all u e X — {0}. Hence by (A3) we have b(u) < —^—(\\u\\q q - ρ

(3.3)

+ ||h|D

for all μ € X. Now combining (3.2), (3.3) and (A3) we get 0 < (B(u), u) < — ( Ι Μ Ι * + Μ Γ ) q - ρ

(3.4)

for all u G X — {0}. Moreover, it follows from (A5) and (A2) that (Bx(u),u)

= (B(u),u)

+p{B(u),u)

(3.5)

> qb(u) + qpb(u) = q{B(u), u) = Let us now set for a fixed u G X — {0} k(t) = {B(tu), tu)t~q Then

for all t > 0.

{B\{tu), tu) - q(B(tu),

tu)

qb\(u).

76

3 Nonhomogeneous potentials

for all t > 0 and (B(tu), tu)t~p is strictly increasing function on (0, oo). Hence \im(B(tu), t-> 0

tu)t~p

= 0

lim {B(tu), tu)t~p f->oo

and

= oo.

A similar statement can be derived for b. Let us now set for a fixed u G X — {0} h{t) = b{tu)t~q Since h (t) =

for all t > 0.

(B(tu), tu) — qbjtu) >- 0 tq+1

for all t > 0,

we see that b(tu) < tqb(u)

for 0 < ί < 1,

and b(tu) > tqb{u)

f o r i > 1.

(3.6)

Also, we have 5(0) = 0 and fc(0) = 0.

3.2 Constrained minimization We minimize the functional F(u) = a(u) — b(u) subject to the constraint V = {« 6 X - {0}; (A(u), u) - (B(u), u) = 0}. It will be shown in Lemma 3.1 that V φ 0. Setting g(u)=(A(u),u)-(B(u),u), we have pF(u) = (A(u), u) - {B(u), u) + b(u) = g(u) + b(u) and p(Ff(u),

u) = p(A(u), u) - p(B(u), u) = p((A(u), u) - (B(u), «>) = pg(u) = (gf(u),u)

+

(B(u),u)

for all μ e X. Therefore, for ν e V we have PF(u)

= b(u)

and

(g'(u), u) = ~(B(u), u)

and moreover (Μ)

Μ = inf{F( M ); « e V ) = inf{^fc(u); u e V}.

(3.7)

3.2 Constrained minimization

77

L e m m a 3.2.1.

(i)

There exists 8 > 0 such that {B(u),u)

(ii)

> pk\ ||μ ||p > 5 for all u e V,

infueVF(u)>S^R,

(iii) -^F{u)>kMpforue

V,

(iv) If u e V and F(u) = M, then u satisfies (1.1) and

pq Proof

q

(i) If ν e V, then

pkiM?

< (A(u),u) = (B(u),u)

< -^-{\\u\\q + ΙΙ«ΙΓ)· q-p

Since u φ 0 and ρ < q < r, ||«|| > const > 0 for u e V and (i) follows, (ii) If ueV, then by (3.7), pF(u) = b{u) and

(B(u), u) = b(u) + pb(u) < b(u) + —£—b(u) = q-p Consequently,

-2—b(u). q-p

according to (i) we have

1F(u) = -b(u) Ρ

q — ρ > -—~(B(u), pq

q — ρ u) > -——8 q

and this implies (ii). The estimate (iii) follows from the last inequality and step (i). (iv) If u e V and F(u) = M, then there exists λ e R such that

(F'(u), v) = k(g'(u), v) for all υ 6 X. It follows from (3.7) and ( A 5 ) that (g'(u), u) < 0. Since (F'(u), u) = 0, we must have λ = 0, that is ( F ' ( u ) , υ) = 0 for all υ € X and u satisfies (1.1). •

Lemma 3.2.2. There exists a unique s : X — {0} —• (0, oo) in Cl(X — {0}, R) with the following property: ifu e X — {0} and t > 0 then tu e V if and only if t = s(u). The gradient of s satisfies the estimate , „ ^ (p||A(^(M)«)|| + ||gi(.y(M)»)||>(«)2 I Vj(M) I
R defined by f ( t , u) = pa(u) - (B(tu), tu)t~p

= (A(u), u) - (B(tu),

tu)t~p.

According to (3.5) we have (Bi(tu),tu)

- p(B(tu),tu)

=

(q -

^n

p)(B(tu),tu)

^

also { f u { t , u), v) = p{A(u),

ν)

^Ti

(B\(tu), —

υ)

·

.

Since a(u) is homogeneous of degree p, it is clear that for u e X — {0}, tu 6 V if and only if ψ{ί, u) = 0. Since { B ( t u ) , tu)t~p is strictly increasing function, there exists a unique value t = j(m) such that \(r(s(u), u) = 0. We also have , f , \ \ -ft(s(u), u)
F(um)

- m _ 1 1| w - um ||

for all if g V. Since by Lemma 3.2.1, {um} is bounded, it follows from assumption (A4) that {B1 (um)} is also bounded. We now write for each ν € X — {0} F(v) - F{um)

= F ( v ) - F(s(v)v)) >

Since s(um)

+ F(s(v)v)

- F{um)

(3.8)

1

F(u)-F(i(u)u)-w- ||i(M)i;-£iin||.

= 1, applying Lemma 3.2.2 we get

| | i ( i 0 u - u m | | < Ci\\v-um\\

and

\s(v) - 1| < Ci||v - um\\

f o r s o m e C i > 0, provided || v — um\\ is sufficiently small. To estimate F(v) we observe that |F(v) -

< \s(v) - l||(F'(0(w)v), υ)|,

(3.9) —F(s(v)v)

3.3 Application — compact case

79

where θ (ν) lies between 1 and s(v). um e V and consequently

On the other hand (F'(u m ), um) = 0 , since

| F ( v ) - F ( i ( i ; ) i ; ) | 0, provided ||υ — um\\ is sufficiently small. Combining (3.8), (3.9) and (3.10) we obtain rp / \ - F(u ) V> ^- C ^ F(v) m 2

Ik-"mil m

for some constant C2 > 0. Taking ||z|| = 1 we get for t > 0 small F(um+tz)~

F(um) t

>

~

C2 m'

which implies that {F'(u m ), z) > — ^ and replacing ζ by — ζ we derive that \(F'(um),z)\
0 on

Γ,·(Χ)|Μ|«_2Μ

in

(3.11)

80

3 Nonhomogeneous potentials

where 1 < ρ < oo, c > 0 is a constant, ρ < qi < if η > ρ and ρ < qi if ρ > η. We assume that q\ < qi < · ·· < qN· T h e functions r, (i = I,... ,N) satisfy the following hypotheses: Case ρ < n: ( R n ) , where ρ
0. R and b : Wl-P(Rn)

We define functionals a :

a(u) = — f (\Vu(x)\p Ρ J R" and

,

Ν , b(u) = T qi Jw>

R by

+c\u(x)\p)dx

n(x)\u(x)\*'dx.

By virtue of Theorem A.3.1 the functional b is well defined. T h e potential operator Β : Ψ ι · Ρ φ » ) - * W - 1 ^ ^ " ) is given by Ν f (B(u), v) = T i=l J®·" H e n c e the potential b : W ^ ^ R " )

2

ri(x)\u(x)\n-

Μ is given by

f b{u) = Σ ( 1 - - ) / n(x)\u(x)\* fbt V J Jr" ν

/

D

\

u(xMx)dx.

N

dx = Y

a

-

l, where 9m lies between 1 and sm. We also have (A(um),um)

=

(B(um),um)

and (A(u m)i um) — sm

mum)i

If sm > 1 we have ( B * ( s m u m ) , Jm«m) > s^{B*(um), t^- v-p ^ (Mum),um) 1 < Sm
s*m">

{B(um),um) / Λ, (B*(um),um)

(3.16)

m m m ), U ml (B(um), um) + [(ß*(Mm), um) - (B(um), Mm>] It follows from Lemma 3.2.1(i) that (B(u m ), um) > 8 > 0 for some constant δ and all m. Since u m 0 and bi —fr*is weakly continuous at 0, we deduce from (3.15) and (3.16) that sm 1.

3.4 Perturbation theorems — noncompact case

83

Since 5 * ( u ) -

b ( u )

=

b \ { u )

-

b

( u )

x

-

p ( b * ( u ) -

b ( u ) ) ,

the relation (3.14) easily follows.

•

In the case where Β and B* are homogeneous of degree q — 1, it is convenient in applications to write Theorem 3.4.1 in the following form:

Theorem 3.4.2. and

s u p p o s e that

Let

Β

:

X

—•

X*

assumptions

and

Β

:

o f Theorem

.

.

inf

X

equation

( 1 . 1 ) has

a { u ) i ~ p

.

be

U

homogeneous

Then,

o f degree

q

—

1

if

a ( u ) i ~ p

- j - < inf

α nontrivial

X*

3.4.1 hold.

b { u ) " - p

the

—>

-ρ-,

(3.17)

* ° b * ( u ) ^

solution.

First of all, if u φ 0 then there exists a unique t > 0 such that p a ( t u )

-

This unique t is given by * = ( j j |(Π)) ?

q b ( t u )

P

=

0.

· Then q - p

F ( t u ) = a ( t u ) - b ( t u )

=

(

^

(

l

-

\ p j

?

a

)

(

-

\

q J

u

)

9

bh( u ) i ~ p

and the inequality Μ < Μ* relating the constrained minima from Theorem 3.4.1 is equivalent to (3.17). To apply Theorem 3.4.2 we consider two problems - A u u

+

u

=

Q(X)\U\P~2U

€

u

>

in

R \

(3.18)

0 on M",

and - A u

+ u

=

Q * ( x ) \ u \

p

~

2

u

in

M \

(3.19)

u G W 1 · 2 ^ " ) , μ > 0 on R", where Q and Q* are positive functions in L°°(M n ) and 2 < ρ < -^zj. We put ι P M

Q

.

P

=

( J

Q

t

o

W

d

x

)

'

a n d

M


d x )

84

3 Nonhomogeneous potentials

Proposition 3.4.1. Suppose l i m ^ ^ o o Q(x) sesses a solution if

= lim^i-»^ Q*(x).

Then (3Λ&) pos-

Μ inf < inf 2 2 «eW'. (K")-{0} II«||Q,p ΗεΗ^· (Κ")-{0} II"II Q*,p

(3.20)

where ||u || is a norm of u in Indeed, in our situation, condition (3.17) can be written as

& inf tteW'^iR")—{0}

< Iß.P

inf κ€νν'·2(Κη)-{0}

Μ Λ* Q*.P

and the result follows. Proposition 3.4.2. If solution.

lim| J C |_>. 0 0

Q(x) = infjteR" Q(x), then problem (3.18) admits a

Proof Let infm* Q(x) = m. If Q(x) = m, then by Proposition 1.10.1 problem (3.18) has a radial solution. So we can assume that Q φ m and we then have ä = =

sup I Q(x)\u(x)\p ||u||=l JW f m\v{x)\p Jr"

dx > am = sup I m\u(x)\p ||u|| = l JR"

dx

dx,

since am is attained by a positive radial function v, which is a solution of (3.19) "modulo rescalling" with Q*(x) ξ m. It is now clear that (3.20) holds with Q*(x) = m and problem (3.18) has a solution. • If lim iri —voo Q(x) > Q(x) f ° r solution. Indeed, we have

€ R", the problem (3.18) does not have a

Proposition 3.4.3. Let lim|x|_>.oo Q{x) > Q{x)forallx all u satisfying f (\Vu\2 + u2)dx J R« Proof

G R". Then F(u) > Μ for

= f Q(x)\u\p J R"

dx.

Suppose that there exists u G V such that Μ = inf F{u) = inf U(u) U€V uev 2

= ( ] - - - ) [ \2 ρ/ Jw

Q{x)\u\pdx.

3.5 Perturbation of the functional a — noncompact case

85

Since F{u) = F(|w|), we may assume that u(x) > 0 on R w . Let t e R and e\ = ( 1 , 0 , . . . , 0 ) and set ut(x) = u(x - te i). Then ||u,|| = \\u\\ for all t € R and b(jut) = ( l - ~ ) [ \

Ρ J J R"

Q(x)\u(x

= (1--)/" Q(x + Ρ J R"

- tei)\P dx tei)\u(x)\Pdx.

By the Lebesgue Dominated Convergence Theorem we get lim b(u t ) = ( 1 - - ) f lim Q(y)\u(x)\p t-too \ ρ) JRn |>|->00

dx > b(u).

This means that there exists t > 0 such that b(ut) > b(u) and setting s(t) = s(ut), we get that s(t)ut e V with < 1. Hence Μ < F(s(t)u{)

X

= =

-b(s{t)ut)

2

Κ·

Ρ J Jw

QM\mut(x)\pdx

- G - ä||s(,)"'»2 < G - i)Ι|ϋ»2' which is impossible.

•

The above observation is due to Stuart (see [215]).

3.5 Perturbation of the functional a — noncompact case In Theorem 3.4.1 we have obtained the existence result by perturbating the functional b. We now derive the existence theorem by perturbating the functional a. Given a potential operator A* : X —• X* and its potential a* : X - > I w e set V* = {u € X - {0}; (A*(u), u) - (B{u), u) = 0},

F*(u) = (a*(u) -

b(u))

and M* = inf{F*(u); " e V*}. Theorem 3.5.1. Suppose that A and Β are weakly continuous and b is weakly lower semicontinuous. Let A* : X —> X* be strongly monotone operator satisfying (Ai) and such that a — a* is weakly continuous. If Μ < Μ*, then there exists u Ε V such that F(u) = Μ and u is a solution of (1.1).

86

3 Nonhomogeneous potentials

The proof is similar to that of Theorem 3.4.1. Let {um} be a minimizing sequence for F constructed in Lemma 3.2.3. We now apply Lemma 3.2.2 to F* and obtain a function s € Cl(X — {0}, (0, oo)) such that s(um)um e V*. Setting sm — s (um) we show that Proof.

lim

0.

[b(smum)-b(um)]=

(3.21)

To prove this we show that sm -*• 1 as m —• oo. If sm > 1, then the relations {A(um),um)

=

(B(um),um}

and (A*(um),

um)

= SmP(B(smum),

smum))

> s%~p{B(um),

um)

imply that 1
u m)

.

Similarly, if sm < 1 we get 1>

q

st,

-

p

> -

(A*(um),

um)

{B(u m)ι

um

)

= 1Η

(A*(um),um)

(B{u

(A(um),um)

),

um)

.

By Lemma 3.2.1 {B(um), um) > 8 for some constant 8 > 0 and all m and consequently in both cases we get that sm ->• 1 as m -*• oo. Having established that sm 1, we get (3.21) from the mean value theorem Ib(SmUm)

-

H"m)I

< \sm ~ 1| (B(ßmUm),

Um),

where 1 < θη < sm. Finally, we have M*

< F*(smum)

=

=

-b(smum) Ρ

- \b(smum) Ρ

-

b(um)]

+

-b(um) Ρ

and letting m - > oo we see that Μ* < M, which is impossible and this completes the proof. • We reformulate Theorem 3.5.1 in a way which is more suitable for applications.

Theorem 3.5.2.

Suppose

that assumptions

inf

'

of Theorem

< inf

U^°b(u)i~p

then

equation

(1.1)

has α nontrivial

3.5Λ

' b(u)i~p

solution.

,

hold.

If

(3.22)

87

3.5 Perturbation of the functional a — noncompact case

The proof is identical to that of Theorem 3.4.2. As an application we consider the following problems = \u\P~2u

-Au+q(x)u

in R "

(3.23)

u e Wu(Rn),

u > 0,

and + qoou = \u\P~2u

-Au

1 2

u e W ' ^"),

in

R\

(3.24)

u > 0,

where q is a nonnegative function in L°°(R") and