This book is devoted to the study of elliptic second-order degenerate quasilinear equations, the model of which is the p

*235*
*93*
*3MB*

*English*
*Pages 476
[473]*
*Year 2017*

- Author / Uploaded
- Laurent Veron

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Names: Véron, Laurent. Title: Local and global aspects of quasilinear degenerate elliptic equations : quasilinear elliptic singular problems / by Laurent Véron (Université François Rabelais, France). Description: New Jersey : World Scientific, 2017. | Includes bibliographical references and index. Identifiers: LCCN 2017007723 | ISBN 9789814730327 (hardcover : alk. paper) Subjects: LCSH: Differential equations, Elliptic. | Differential equations, Parabolic. | Quasilinearization. | Eigenvalues. Classification: LCC QA377 .V44 2017 | DDC 515/.3533--dc23 LC record available at https://lccn.loc.gov/2017007723

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

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To the memory of Stanislav Ivanovich Pohozaev (1935–2014) and James Burton Serrin (1926–2012), two outstanding mathematicians, great minds and sincere friends.

v

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Preface

The study of local properties of solutions of quasilinear elliptic equations −divA(x, u, ∇u) + g(x, u, ∇u) = 0

(0.0.1)

has been initiated by James Serrin ﬁfty years ago when he introduced genuinely nonlinear methods which did not belong to the world of classical harmonic analysis. However, many of his results were of a form that had made them considered as extensions of those already obtained in the case of linear equations. The two seminal articles he published in 1964 and 1965 contain in particular the proof of Harnack inequality and the H¨older continuity of the solutions of (0.0.1) via the Moser’s method, the description of positive solutions with an isolated singularity and the removability of compact sets with zero Sobolev W 1,p capacity. They had a tremendous inﬂuence on the development of the local theory of solutions of a large range of quasilinear equations. A few years after, a new proof of Harnack inequality for a wider class of equations was published by Trudinger. Later on the study was focused on more speciﬁc types of operators A, the model case of which being the p-Laplacian deﬁned by Δp u = div(|∇u|

p−2

∇u)

(0.0.2)

where p > 1. The C 1,α regularity was proved by several authors with more and more general assumptions and the a priori estimates obtained in these regularity results have been the key-stone in the development, in the mid eighties, of the ﬁne description of isolated singularities of larger and larger classes of solutions of various types of perturbations of the p-Laplace equation in a punctured domain, the main examples being the p-Laplace vii

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Local and global aspects of quasilinear degenerate elliptic equations

equation which deals with solutions of −Δp u = 0,

(0.0.3)

the p-Laplace equation with a zero-order reaction with strong growth, q−1

−Δp u + |u|

u = 0,

(0.0.4)

or of Hamilton-Jacobi (or Riccati) type q

−Δp u + |∇u| = 0,

(0.0.5)

where q > p − 1 and = 1 (the absorption case) or = −1 (the source case). The study of the behavior of singular solutions depends on the value of q with respect to some critical values of q and these threshold are typical of problems with strong reaction. The precise description of this behavior is linked to the construction of separable solutions of (0.0.3), (0.0.4) and (0.0.5), that are solutions under the form u(x) = |x|β ω(|x|−1 x). The spherical p-harmonic functions correspond to the separable p-harmonic functions. They satisfy a quasilinear equation on the unit sphere S N −1 in RN p−2 p−2 2 2 2 2 −div β ω + |∇ ω| ∇ ω − βΛp (β) β 2 ω 2 + |∇ ω|2 2 ω = 0, (0.0.6) where Λp (β) = β(β(p − 1) + N − p)). When p = 2 the solutions of (0.0.6) are the spherical harmonics. When N = 2 this equation reduces to a separable ordinary diﬀerential equation on S 1 and all the solutions are known. When N > 2 the problem became much harder but inﬁnitely many solution couples (β, ω) have already been obtained associated to regular tesselations of S N −1 . Two diﬀerent types of positive separable solutions in any cone vanishing on its boundary exist and they play an important role in the characterization of positive p-harmonic functions therein. In some range of values of p = q + 1, there exist separable solutions of (0.0.4). Then p and ω satisﬁes β := −βq = − q+1−p −div

βq2 ω 2 + |∇ ω|2

p−2 2

− βq Λp (βq )

∇ ω

βq2 ω 2

+ |∇ ω|

2

p−2 2

(0.0.7) ω + |ω|

q−1

ω = 0.

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Preface

ix

p−q There exist also separable solutions of (0.0.5) with β = −γq = − q+1−p and ω satisﬁes p−2 2 2 2 2 ∇ω −div γq ω + |∇ ω| (0.0.8) p−2 q 2 2 2 2 2 2 2 2 ω + γq ω + |∇ ω| = 0. − γq Λp (γq ) γq ω + |∇ ω|

These solutions turned out to have a key role in the classiﬁcation of solutions with isolated boundary singularity. Besides the problem of singularities, a second direction of research was open in the same period dealing with quasilinear equations with measure data, using, as a starting point, Stampacchia’s truncation method. The ﬁrst studies concerned the problem −Δp u = μ

in Ω (0.0.9)

u=0

on ∂Ω,

where Ω ⊂ RN is a bounded smooth domain and μ a bounded Radon measure in Ω. Initially the solutions where understood in the sense of distributions, but neither uniqueness nor any kind of stability could be considered. An important progress is made with the introduction of renormalized solutions, in the framework of which the strong convergence of the gradient is obtained and then the stability. Simultaneously, another approach inherited from the classical potential theory was developed in particular by the Finish school of mathematics. The key role was assigned to the psuperharmonic functions and the associated p-harmonic measures. It was proved later on that these two approaches are in fact equivalent. These results allows to treat problems such as q−1

−Δp u + |u|

u=μ

in Ω

u=0

on ∂Ω

(0.0.10)

in the subcritical range of q > p−1, which means that all bounded measures can be considered. These diﬀerent approaches apply also to more general equations such as −Δp u + g(x, u, ∇u) = μ u=0

in Ω (0.0.11) on ∂Ω.

The supercritical range was considered much later, using the Wolﬀ potential W1,p as a key tool. The associated operator appeared to play a

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Local and global aspects of quasilinear degenerate elliptic equations

role similar to the one of the Newtonian potential in order to provide an explicit representation of p-superharmonic functions instead of mere harmonic functions. Then, necessary and suﬃcient conditions for the existence of renormalized solutions to the problem −Δp u = uq + μ u=0

in Ω (0.0.12) on ∂Ω,

were obtained under a Lipschitz continuity condition of μ with respect q . In the last ﬁve years progresses were to the Bessel capacities cp, q+1−p obtained in the study of the regularity properties of the Wolﬀ potential of a Radon measure belonging to some Lorentz space. These results are expressed in terms of Lorentz-Bessel spaces. Combined with estimates of Riesz or Bessel potential, they allowed to provide suﬃcient conditions for existence of renormalized solutions to −Δp u + g(u, ∇u) = μ u=0

in Ω (0.0.13) on ∂Ω,

with g nondecreasing satisfying some integrability condition, or q−1

−Δp u + |u|

u=μ

in Ω

u=0

on ∂Ω,

(0.0.14)

with q in the super supercritical range. Problems with an absorption term λ such as g(u) = ea|u| sgn(u) can also be treated, always in the framework of nonlinear potential estimates. As a by product of these techniques, necessary or suﬃcient conditions of existence of large solutions of quasilinear equations with strong absorption in a domain in connexion with the regularity of its boundary expressed via capacitary Wiener tests, have been proved. The study of quasilinear equations with measure data, in the new framework of nonlinear potential theory has become a mature subject of research. Many fascinating problems remain unsolved, and many new ones even more fascinating will appear, the solution of which will require new innovating techniques. We hope this book will incite new generations to participate to the progress of this branch of mathematics. Laurent Veron

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Contents

Preface 1.

vii

Regularity

1

1.1 Notations . . . . . . . . . . . . . . . . . . . . . 1.1.1 Domains and ﬂow coordinates . . . . . . 1.1.2 Riemannian manifolds . . . . . . . . . . 1.2 Distributions, Sobolev and Besov spaces . . . . 1.2.1 Lp and Lorentz spaces . . . . . . . . . . 1.2.2 Hausdorﬀ measure . . . . . . . . . . . . 1.2.3 Weak derivatives and distributions . . . 1.2.3.1 Distributions . . . . . . . . . . . 1.2.3.2 Weak derivatives . . . . . . . . 1.2.3.3 Sobolev spaces . . . . . . . . . . 1.2.3.4 Fractional Sobolev spaces . . . 1.2.4 Bessel potentials . . . . . . . . . . . . . 1.2.4.1 Bessel potential spaces . . . . . 1.2.4.2 Bessel-Lorentz potential spaces 1.2.4.3 Bessel-Lorentz capacities . . . . 1.2.4.4 Riesz and Sobolev capacities . . 1.3 General quasilinear elliptic equations . . . . . . 1.3.1 The Serrin’s results . . . . . . . . . . . . 1.3.2 Regularity . . . . . . . . . . . . . . . . . 1.3.2.1 Local regularity . . . . . . . . . 1.3.2.2 Boundary regularity . . . . . . 1.3.3 Maximum principle . . . . . . . . . . . . 1.4 Existence result . . . . . . . . . . . . . . . . . . xi

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

1 1 2 6 6 9 10 10 13 13 16 17 18 18 19 21 22 23 29 29 31 31 33

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1.4.1 Method of monotonicity . . . . . . . . . . . . . . . 1.4.2 Method of super and sub solutions . . . . . . . . . 1.5 The p-Laplace operator . . . . . . . . . . . . . . . . . . . 1.5.1 Comparison principles . . . . . . . . . . . . . . . . 1.5.2 Isotropic singularities of p-harmonic functions . . . 1.5.3 Rigidity theorems for p-harmonic functions . . . . 1.5.4 Isolated singularities of p-superharmonic functions 1.6 Notes and open problems . . . . . . . . . . . . . . . . . .

. . . . . . . .

2. Separable solutions

65

2.1 Eigenvalue and eigenfunctions . . . . . . . . . . . . . . . . . 2.1.1 Dirichlet problem . . . . . . . . . . . . . . . . . . . . 2.1.2 The case p → ∞ . . . . . . . . . . . . . . . . . . . . 2.1.3 Higher order eigenvalues . . . . . . . . . . . . . . . . 2.1.4 Eigenvalues on a compact manifold . . . . . . . . . . 2.2 Spherical p-harmonic functions . . . . . . . . . . . . . . . . 2.2.1 The spherical p-harmonic eigenvalue problem . . . . 2.2.2 The 2-dimensional case . . . . . . . . . . . . . . . . . 2.2.3 The case p → ∞ . . . . . . . . . . . . . . . . . . . . 2.2.3.1 The spherical inﬁnite harmonic eigenvalue problem . . . . . . . . . . . . . . . . . . . . 2.2.3.2 The 1-dim equation . . . . . . . . . . . . . . 2.3 Boundary singularities of p-harmonic functions . . . . . . . 2.3.1 Boundary singular solutions . . . . . . . . . . . . . . 2.3.2 Rigidity results for singular p-harmonic functions in the half-space . . . . . . . . . . . . . . . . . . . . . . 2.4 Notes and open problems . . . . . . . . . . . . . . . . . . . 3.

33 37 52 53 56 60 63 64

Quasilinear equations with absorption

65 65 70 74 77 87 87 99 106 106 111 113 113 121 124 129

3.1 Singular solutions with power absorption in R . . . . 3.1.1 Removable singularities in RN . . . . . . . . . . 3.1.2 Classiﬁcation of isolated singularities . . . . . . 3.1.3 Global solutions in RN . . . . . . . . . . . . . . 3.1.4 Large solutions . . . . . . . . . . . . . . . . . . 3.2 Singular solutions in RN + . . . . . . . . . . . . . . . . . 3.2.1 Separable solutions in RN + . . . . . . . . . . . . 3.3 Classiﬁcation of boundary singularities . . . . . . . . . 3.3.1 The general gradient estimate near a singularity N

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

129 130 134 151 154 167 167 173 174

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3.3.2 Removable singularities . . . . . . . . . . . . . . . . . 178 3.3.3 Classiﬁcation of boundary singularities . . . . . . . . 184 3.4 Singularities of quasilinear Hamilton-Jacobi type equations 196 3.4.1 Comparison principles . . . . . . . . . . . . . . . . . 196 3.4.2 Radial solutions . . . . . . . . . . . . . . . . . . . . . 201 3.4.3 Gradient estimates and applications . . . . . . . . . 203 3.4.4 Isolated singularities . . . . . . . . . . . . . . . . . . 207 3.4.4.1 Removable isolated singularities . . . . . . . 207 3.4.4.2 Isolated singularities of positive solutions . . 211 3.4.4.3 Global singular solutions . . . . . . . . . . . 216 3.4.4.4 Isolated singularities of negative solutions . 217 3.4.5 Geometric estimates . . . . . . . . . . . . . . . . . . 222 3.4.5.1 Gradient estimates on a Riemannian manifold 222 3.4.5.2 Growth of solutions and Liouville type results 226 3.5 Boundary singularities of quasilinear Hamilton-Jacobi type equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 3.5.1 Separable solutions . . . . . . . . . . . . . . . . . . . 231 3.5.2 Boundary isolated singularities . . . . . . . . . . . . 233 3.5.2.1 Removable singularities . . . . . . . . . . . . 233 3.5.2.2 Construction of singular solutions . . . . . . 235 3.5.2.3 Global solutions . . . . . . . . . . . . . . . . 241 3.5.2.4 Entire singular solutions . . . . . . . . . . . 245 3.6 Notes and open problems . . . . . . . . . . . . . . . . . . . 247 4. Quasilinear equations with measure data 4.1 Equations with measure data: the framework 4.2 p-superharmonic functions . . . . . . . . . . . 4.2.1 The notion of p-superharmonicity . . . 4.2.2 The p-harmonic measure . . . . . . . . 4.2.3 The Peron’s method . . . . . . . . . . 4.3 Renormalized solutions . . . . . . . . . . . . . 4.3.1 Locally renormalized solutions . . . . . 4.3.2 The stability theorem . . . . . . . . . . 4.3.3 Further stability results . . . . . . . . 4.4 Notes and open problems . . . . . . . . . . . 5.

249 . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

Quasilinear equations with absorption and measure data

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

249 250 250 254 256 260 260 267 286 288 291

5.1 The subcritical case . . . . . . . . . . . . . . . . . . . . . . 291

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Contents

xiv

5.2

5.3

5.4

5.5

5.6

5.1.1 General nonlinearity with data in W −1,p (Ω) . . . 5.1.2 Subcritical nonlinearity with measure data . . . . 5.1.3 Applications . . . . . . . . . . . . . . . . . . . . . Removable singularities . . . . . . . . . . . . . . . . . . 5.2.1 Local estimates for locally renormalized solutions 5.2.2 The power case . . . . . . . . . . . . . . . . . . . 5.2.3 The Hamilton-Jacobi equation . . . . . . . . . . . Supercritical equations . . . . . . . . . . . . . . . . . . . 5.3.1 Estimates on potentials . . . . . . . . . . . . . . . 5.3.2 Approximation of measures . . . . . . . . . . . . Solvability with measure data revisited . . . . . . . . . . 5.4.1 The general case . . . . . . . . . . . . . . . . . . 5.4.2 Power growth nonlinearities . . . . . . . . . . . . 5.4.3 Exponential type nonlinearities . . . . . . . . . . Large solutions revisited . . . . . . . . . . . . . . . . . . 5.5.1 The maximal solution . . . . . . . . . . . . . . . 5.5.2 Potential estimates . . . . . . . . . . . . . . . . . 5.5.3 Applications to large solutions . . . . . . . . . . . 5.5.4 The case p = 2 . . . . . . . . . . . . . . . . . . . Notes and open problems . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

6. Quasilinear equations with source 6.1 Singularities of quasilinear Lane-Emden equations 6.1.1 Separable solutions in RN . . . . . . . . . . 6.1.2 Isolated singularities . . . . . . . . . . . . . 6.1.3 Rigidity theorems . . . . . . . . . . . . . . . 6.1.4 Separable solutions in RN + . . . . . . . . . . 6.1.5 Boundary singularities . . . . . . . . . . . . 6.2 Equations with measure data . . . . . . . . . . . . 6.2.1 The case p = 2 . . . . . . . . . . . . . . . . 6.2.2 The subcritical case . . . . . . . . . . . . . . 6.2.2.1 Suﬃcient condition . . . . . . . . . 6.2.2.2 The case 1 < p < N . . . . . . . . . 6.2.2.3 The case p = N . . . . . . . . . . . 6.2.3 The supercritical case . . . . . . . . . . . . 6.2.3.1 Necessary and suﬃcient conditions 6.2.3.2 Removable singularities . . . . . . . 6.3 Quasilinear Hamilton-Jacobi type equations . . . .

291 297 304 305 305 311 317 320 320 334 335 335 340 342 343 344 346 354 356 365 367

. . . . . . . . . . . . . . . .

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. . . . . . . . . . . . . . . .

367 367 369 374 388 398 403 403 404 405 409 415 425 425 428 428

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Contents

xv

6.3.1 Separable solutions of quasilinear Hamilton-Jacobi equations . . . . . . . . . . . . . . . . . . . . . . . . 428 6.3.2 Quasilinear Hamilton-Jacobi equations with measure data . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 6.4 Notes and open problems . . . . . . . . . . . . . . . . . . . 440 Bibliography

443

List of symbols

453

Glossary

455

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Chapter 1

Regularity

1.1 1.1.1

Notations Domains and ﬂow coordinates

We denote by RN the N -dimensional real vector space with Euclidean structure , (or more simply . ) and associated norm |.|. An open ball of center x0 and radius r > 0 is denoted by Br (x0 ) (or Br if there is no ambiguity on its center). Its canonical basis is Bcan := {e1 , ..., eN } and the coordinates of x, (x1 , ..., xN ). The positive half space RN + is the set of x := (x , xN ) N −1 N and xN > 0 and its boundary ∂R+ can with x = (x1 , ..., xN −1 ) ∈ R be identiﬁed with RN −1 . A domain, i.e. a connected open set, Ω ⊂ RN , is of class C k (k ∈ N∗ ) if its boundary ∂Ω is an imbedded submanifold of dimension N − 1. If a ∈ ∂Ω we denote by Ta (∂Ω) the tangent hyperplane to ∂Ω at a and by na the outward normal unit vector to Ω at a. If x ∈ Ω, ρ(x) is the distance function from x to ∂Ω. The signed distance function ρ(x) ˙ is equal to ρ(x) if x ∈ Ω and −ρ(x) if x ∈ Ωc . By the triangle inequality, ρ˙ is a contraction mapping from RN to R. For β > 0 we put Ωβ = {x ∈ Ω : ρ(x) < β} Σβ = ∂Ωβ = {x ∈ Ω : ρ(x) = β} Ωβ

(1.1.1)

= {x ∈ Ω : ρ(x) > β}.

Then Σ0 = ∂Ω and Ω0 = Ω. If Ω is a bounded C k domain with k ≥ 2, there exists β0 > 0, depending on the curvature of ∂Ω (see next section), such that for any x ∈ Ωβ0 there exists a unique σ(x) ∈ ∂Ω verifying ρ(x) = |x − σ(x)|. Furthermore the function ρ is of class C k−1 . All the points x in Ωβ can be represented in a unique way by the couple (ρ(x), σ(x)) ∈ (0, β0 ) × ∂Ω, and the mapping x → Π(x) := (ρ(x), σ(x)) is a C k−1 diﬀeomorphism from 1

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Local and global aspects of quasilinear degenerate elliptic equations

Ωβ0 into (0, β0 ) × ∂Ω with inverse Π−1 ((ρ, σ)) = σ − ρnσ . This change of variables, called ﬂow coordinates, plays an important role in the description of solutions of equations near the boundary. 1.1.2

Riemannian manifolds

A Riemannian manifold (M d , g) is a d-dimensional diﬀerentiable manifold (always connected) endowed with a symmetric deﬁnite positive 2-tensor g = (gij ) with inverse matrix g −1 = (g ij ). The tangent space to M at x is denoted by Tx M , and the tangent bundle T M , i.e. the union of the tangent spaces, can be viewed as the set of pairs (x, v) : x ∈ M, v ∈ Tx M . It is endowed with a natural structure of a 2 d-diﬀerentiable manifold of class C k−1 . If X and Y belong to Tx M , we denote X, Y g = g(X, Y ) = gij X i Y j and |X|g = X, Xg . (1.1.2) i,j

A point x on M can be represented in any chart by a system of local coordinates on M , say (xj ), j=1, ...d. The length L of a C 1 curve γ with parametric representation {γ(t) = (xj (t)) : t ∈ [a, b], j = 1, ..., d} is expressed by b b |γ(t)|g dt := gij x˙ i x˙ j dt. (1.1.3) L= a

a

ij

A curve γ with minimal length between two points x and y is a geodesic, and this length is the geodesic distance between x and y denoted by dg (x, y); it is a metric on M . If M endowed with this metric is a complete metric space it is called a complete manifold. The Euler equation corresponding to the minimization of L in (1.1.3) is a second order diﬀerential equation. To any x in M and X ∈ Tx M there exists a curve γ deﬁned on some interval (−a, b) (a, b > 0) such that γ(0) = x, γ (0) = X which has the property that for any t, t ∈ (−a, b), it minimizes the distance between γ(t) and γ(t ) provided |t − t | is small enough. It is called the exponential map and denoted by γ(t) = expx (tX). The set of points y = expx (tX) ∈ M such that the geodesic starting from x in some direction X and reaching y ceases to be minimizing on [0, t + ) for any > 0 is called the cut locus of x. The distance function y → dg (x, y) is smooth except on the cut locus. The least distance from x to its cut locus is the injectivity radius at x. A subset O of M is convex if for any x, y ∈ O there is a unique geodesic curve

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Regularity

3

joining x to y. The convexity radius rM (a) of some a ∈ M is the supremum of all the r > 0 such that the ball Br (a) is convex. The exponential map X → expx (X)) may or may not be deﬁned for all x ∈ M and X ∈ Tx M , however since the diﬀerential D expx (0) is the identity in Tx M , the exponential map is a local diﬀeomorphism from a neighborhood of x into Tx M . An important result in Riemannian geometry is the Hopf-Rinow theorem. Theorem 1.1.1 Let (M d , g) be a Riemannian manifold. Then following statements are equivalent: (i) M is a complete metric space. (ii) Every non-empty bounded closed subset of M is compact. (iii) M is geodesically complete, i.e. for every x ∈ M , X → expx (X) is deﬁned for all X ∈ Tx M . An important class of manifolds are the ones which possess a pole. Deﬁnition 1.1 A Riemannian manifold (M d , g) possesses a pole o if expo is a diﬀeomorphism from To M to M . We will see later that the distance function x → r(x) from the pole o to the point x possesses important properties associated to the curvature of the manifold.

Set |g| = det(gij ), then dvg = |g|dx1 dx2 ...dxd is the volume element on M . The curvature Riemann tensor is a (1-3)-tensor deﬁned in the local coordinates (xj ) by i Riemg = (Rjkl )=

∂Γilj ∂Γikj i n Γkn Γlj − Γiln Γnkj , − + ∂xk ∂xl n

(1.1.4)

where the Christoﬀel symbols Γilj are linear combinations of the derivative ∂ of the gij . If , j=1,...,d, denotes the vector ﬁeld in Tx M associated with ∂xj the xj coordinates, then ∂ ∂ ∂ ∂ i Riemg , = Rjkl . (1.1.5) i ∂xk ∂xl ∂xj ∂x i If X, Y and Z are vector ﬁelds on M , their components in Tx M ∂ are expressed by linear combination of the ; this allows to deﬁne ∂xj Riemg (X, Y ) Z as an element of Tx M . The Riemann curvature tensor describes completely the curvature of g, however many computations linked

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to analysis of partial equations equations on manifolds, are based upon i , such as the sectional curvature, the speciﬁc linear combination of the Rjkl scalar curvature, the Gaussian curvature and the Ricci curvature. The sectional curvature is the natural extension to manifold of the Gaussian curvature of surface which is the product of the principal curvature of this surface. If X and Y are independent vectors in Tx M , they span a 2-plane π tangent to M at x. Then the sectional curvature secg (π) is expressed by secg (π) =

Riemg (X, Y ) Y, Xg . |X|g |Y |g − X, Y 2g

(1.1.6)

Similarly to the Gaussian curvature, the sectional curvature secg (π) is welldeﬁned, independent of the choice of basis {X, Y } for π. The Ricci curvature Riccg is a (0, 2)-tensor computed on any couple X, Y in Tx M . It is the trace of the endomorphism Z → Riemg (X, Z) Y.

(1.1.7)

If X and Y are expressed in an orthonormal basis {ej } of Tx M , then Riccg (X, Y ) =

Riemg (X, ej ) Y, ej g .

(1.1.8)

j

Finally, the scalar curvature Scalg is the trace of the Ricci curvature. It is expressed by Riccg (ej , ei ) = Riemg (ei , ej ) ej , ei g . (1.1.9) Scalg = i=j

i=j

If u is a C 1 function on M , its gradient, quoted by ∇u, is the vector ∂u ﬁeld with covariant components ∇j u = and contravariant components ∂xj ∂u ∇j u = g j,k k . Therefore ∂x k

∇u 2 = ∇u, ∇ug =

∂u ∂u . ∂xj ∂xk j,k

The divergence of a vector ﬁeld X = (X j ) is deﬁned by 1 ∂

divg (X) =

|g|X k . k |g| k ∂x

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The Laplacian Δg of a C 2 function is the divergence of the covariant gradient of u. It is expressed locally by

1 ∂ j,k ∂u . (1.1.10) |g|g Δg u =

∂xj ∂xk |g| j,k

The following formulas hold for any C 2 real valued function f deﬁned on R and u and v on M , (i)

Δg (uv) = uΔg v + vΔg u + 2∇u, ∇vg

(ii)

Δg f ◦ u = f ◦ uΔg u + f ◦ u ∇u . 2

As a special case of (ii), if w is a C 2 function, radial with respect to some point a ∈ M , which means w(x) = w(r(x)) ˜ where r(x) := dg (x, a), then ˜ + w ˜ Δg r, Δg w = w

(1.1.11)

on the set of x where r is regular. It holds in particular if r(x) is smaller than the injectivity radius of a. If (M d , g) is a compact Riemannian manifold without boundary (resp. with a nonempty boundary), then −Δg has a discrete spectrum in the Sobolev space W 1,2 (M ) (resp. W01,2 (M )) denoted by σ(Δg ; W 1,2 (M )) (resp. σ(Δg ; W01,2 (M ))). Then (i)

σ(Δg ; W 1,2 (M )) = {λ0 < λ1 ≤ λ2 ≤ ... ≤ λk ≤ ...}

(ii)

σ(Δg ; W01,2 (M ))

= {λ1 ≤ λ2 ≤ ... ≤ λk ≤ ...}

with λ0 = 0

with λ1 > 0.

Hk = N (−Δg − λk I) is a ﬁnite dimensional subspace of W 1,2 (M ) (resp. W01,2 (M )) and W 1,2 (M ) = ⊕k Hk (resp. W01,2 (M ) = ⊕k Hk .) The second covariant derivatives of u ∈ C 2 (M ) are the following ∇jk u =

∂u ∂2u − Γjk . ∂xj ∂xk ∂x

(1.1.12)

The Hessian of u, denoted D2 u is the covariant

symmetric

2 2-tensor with components (D2 u)jk = ∇jk u and its norm is D2 u = jk (∇jk u)2 . The following formula is valid for a C 2 function w radial with respect to a, with w(x) = w(r(x)) ˜ and with the same restriction on r(x) as above, ˜ dr ⊗ dr + w ˜ D2 r. D2 w = w

(1.1.13)

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The Ricci curvature is involved in many estimates of functions on manifolds via the Weinstenb¨ock formula, Theorem 1.1.2 Let (M d , g) be a Riemannian manifold, then for any u ∈ C 2 (M ) there holds

2 1 2 Δ |∇u| = D2 u + ∇u, ∇Δug + Riccg (∇u, ∇u). 2

(1.1.14)

An important tool for performing analysis on Riemannian manifold are Nash imbedding theorems [Nash (1954)], [Nash (1956)]. Theorem 1.1.3 Any closed d-dimensional Riemannian manifold (M d , g) has a C 1 isometric imbedding i into R2d+1 . It has a C k isometric imbedding i into RD with D ≤ 12 d(d+1)(3d+11) provided 3 ≤ k ≤ ∞. If M is compact D turns out to be less or equal to 12 d(3d + 11). The interest of this result lies in the fact that a Riemannian manifold can be seen as a submanifold of the Euclidean space RD , the tangent space Tx M is a d-dimensional linear subspace of RD and for any X, Y ∈ Tx M , X, Y g = i(X).i(Y ) where . denotes the Euclidean scalar product in RD . If f : M → R is a C 1 function with diﬀerential dfx identiﬁed with the vector δx f thanks to Riesz theorem via the scalar product ., .g , it is convenient to consider δx f as a tangent vector to i(M ) in RD . 1.2 1.2.1

Distributions, Sobolev and Besov spaces Lp and Lorentz spaces

Let dx be the Lebesgue measure in the Euclidean N-space RN . If E ⊂ RN is a measurable set, we set |E| = χE dx, (1.2.1) where χE is the characteristic function of E. The space L1 (E) is the space of measurable functions f : E → R such that f L1 (E) := |f |χE dx < ∞. (1.2.2) It is actually a quotient space of classes with respect to the equivalence property between two function of equality except for a set of zero Lebesgue measure. Then the mapping f → f L1 (E) is a norm on L1 (E). For 1

0, we set Sf (t) = {x ∈ E : |f (x)| > t} and λf (t) = |Sf (t)|m where |E|m = mχE dx. (1.2.4) The decreasing rearrangement f ∗ of f is deﬁned by f ∗ (t) : inf{s > 0 : λf (s) ≤ t}.

(1.2.5)

If Φ : R+ → R+ is a continuous nondecreasing function it is classical that Φ(f ∗ ) = (Φ(f ))∗ . We deﬁne f ∗∗ by 1 t ∗ f (τ )dτ, (1.2.6) f ∗∗ (t) = t 0 and, for 1 ≤ s < ∞ and 1 < p ≤ ∞, we set ⎧ ∞ s dt 1s ⎪ 1 ⎪ ∗∗ ⎨ p t f (t) t f Lp,s (E) = 0 ⎪ ⎪ 1 ⎩ ess sup t p f ∗∗ (t) t>0

if s < ∞

(1.2.7)

if s = ∞.

Then Lp,s (E) is the space of measurable functions such that f Lp,s (E) is ﬁnite. It is complete for the norm f → f Lp,s (E) . The following relation links the norm in Lp,s (E) and the decreasing rearrangement (see [Grafakos (2010)])

1 p

p1 ∗

p ∗ ≤ f Lp,s (E) ≤ .

t f s

t f s (1.2.8) dt p−1 L (R+ , t ) L (R+ , dt t )

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Furthermore, if Ω ⊂ RN is a domain, 1 ≤ s < ∞ and 1 ≤ p < ∞, any function in Lp,s (Ω) is the limit of a sequence of C ∞ functions with compact support in Ω. The space Lp,∞ (E) is often called Marcinkiewicz space or weak Lp space. The space Lp,s can be deﬁned by the real interpolation method [Lions, Peetre (1964)] between L1 and L∞ . The next result is due to Hardy and Littlewood, Theorem 1.2.1 For any measurable functions f, g in E ⊂ RN vanishing at inﬁnity, there holds ∞ |f (x)g(x)| dx ≤ f ∗ (t)g ∗ (t)dt. (1.2.9) E

0

This inequality allows to characterize the dual of a given Lorentz space [Grafakos (2010)]. For any 1 < p, s < ∞

Theorem 1.2.2

(Lp,s (E)) = Lp ,s (E),

where as usually p =

p p−1 ,

s =

(1.2.10)

s s−1 .

If f ∈ Lp,s (E) we can write s f Lp,s

:= 0

∞

1 s dt js 2j+1 s ∗ p ∼ t f (t) 2 p−1 (f ∗ (t)) dt, t 2j

(1.2.11)

j∈Z

where ∼ stands for equality up to two universal positive multiplicative constants, and

2j+1

(f ∗ (t)) dt =

s

2j

s

|f (x)| dx, Ωj

where Ωj := {x ∈ E : f ∗ (2j+1 < |f (x)| < f ∗ (2j }. Thus f = Since |{x : |f (x)| > t}| = |{s : f ∗ (s) > t}|, then

j∈Z

f χΩ j .

{x : f ∗ (2j+1 ) < |f (x)| < f ∗ (2j )} = {s : f ∗ (2j+1 ) < f ∗ (s) < f ∗ (2j )} = 2j . This implies |Ωj | = 2j (when non-empty otherwhile f ∗ (2j ) = 0). Therefore j∈Z

sj

2 p (f ∗ (2j+1 ))s ≤ f sLp,s ≤

j∈Z

sj

2 p (f ∗ (2j ))s .

(1.2.12)

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9

There leads to an alternative deﬁnition of Lorentz spaces based upon diadic decomposition. For s > 0, we denote s |c|j < ∞}, sZ = {{cj }j∈Z : j

with norm {cj } s Z

⎛ ⎞ 1s s =⎝ |c|j ⎠ , j∈Z

and with the usual modiﬁcation if s = ∞. The next result can be found in [Grafakos (2010)]. Theorem 1.2.3 Let 1 ≤ p, s < ∞. A function f : E → R belongs to Lp,s (E) if and only if it can be decomposed as j cj 2 p f j , f= (1.2.13) j∈N

where {cj } ∈ sN and the functions fj : E → R are measurable, have their support mutually disjoint, sup |fj | ≤ 1, |supp( fj )| ≤ 2j . 1.2.2

Hausdorﬀ measure

Let 0 ≤ s < ∞ and A ⊂ RN . For 0 < δ ≤ ∞ we denote by Hsδ (A) the (s, δ)-dimensional Hausdorﬀ capacity of A deﬁned by ⎧ ⎫ ⎨ ⎬ rjs : A ⊂ Brj (aj ), rj ≤ δ . (1.2.14) Hsδ (A) = inf ⎩ ⎭ j

j

The function δ → Hsδ (A) is decreasing and Hs (A) = lim Hsδ (A) = sup Hsδ (A) δ→0

(1.2.15)

δ>0

is the s-dimensional Hausdorﬀ measure of A. Then (i) Hsδ is an outer measure (or a capacity), Hs is a Borel measure on RN . (ii) For every set A ⊂ RN , Hs (A) = inf{Hs (G) : G ⊃ A, G open}.

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(iii) For every open set G ⊂ RN , Hs (G) = sup{Hs (K) : K ⊂ A, K compact}. (iv) For every measurable set A ⊂ RN such that Hs (A) < ∞ and every > 0, there exists a compact set K ⊂ A such that Hs (A \ K) < . (v) If s > N , then Hs (A) = 0 for every set A ⊂ RN . Furthermore, if Hs (A) < ∞ for some s ∈ [0, ∞), then Ht (A) = 0 for every t > s. On the opposite, if Hs (A) > 0 for some s ∈ (0, ∞], then Ht (A) = ∞ for every t > s. This allows to deﬁne the Hausdorﬀ dimension of a set A ⊂ RN by dimH (A) = inf{s > 0 : Hs (A) = 0}. Proposition 1.2.4

(1.2.16)

(i) H0 is the counting measure in RN

(ii) If N = 1, H1 is the Lebesgue measure on R. (iii) If N ≥ 2, the normalized N -dimensional Hausdorﬀ measure deﬁned n by HN = ωN HN coincides with the N -dimensional Lebesgue measure, where we have denoted by ωN the volume of the unit ball in RN for the N -dimensional Lebesgue measure. If X is a locally compact topological space, we denote by M(X) (resp. Mb (X)) the set of Radon measures on X (resp. the set of bounded Radon measures on X). Its positive cone is M+ (X) (resp. M+ b (X)). If μ ≥ 0, the Borel measure extension of μ is denoted in the same way. If K ⊂ X is some closed set, M(K) denotes the measures μ with support in K and M+ (K) = M(K) ∩ M+ (X). 1.2.3 1.2.3.1

Weak derivatives and distributions Distributions

If Ω ⊂ RN is a domain and k ∈ N, we denote by C k (Ω) the set of functions φ : Ω → R which are k-times continuously diﬀerentiable in Ω. If k = 0, C 0 (Ω) = C(Ω) is the space of continuous functions. By extension, C ∞ (Ω) is the set of functions which are indeﬁnitely continuously diﬀerentiable. We also denote by C0k (Ω), C0 (Ω) and C0∞ (Ω) the subspaces of C k (Ω), C(Ω) and C ∞ (Ω) of functions with compact support in Ω. If K is a compact subset of Ω, C0k (K), C0 (K) and C0∞ (K) denotes the respective subspaces of C0k (Ω), C0 (Ω) and C0∞ (Ω) of functions with support in K. An element α = (α1 , ..., αN ) ∈ NN is a multi-indice, |α| = j αj , α! = Πj αj ! and we

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denote Dα φ =

∂ |α| φ αN 1 ∂xα 1 ...∂xN

,

and ∇φ =

φxj ej =

j

∂φ ej . ∂xj j

|α|

If α ∈ NN and φ ∈ C0 (Ω) we set φ α = sup |Dα φ(x)| . x∈Ω

|α|

It is a norm on C0 (Ω). Deﬁnition 1.2 A distribution T in Ω is a linear form φ → T (φ) on the space C0∞ (Ω) which possesses the following property: for any compact set K ⊂ Ω, there exist a constant CK and a ﬁnite subset IK of NN such that for any φ ∈ C0∞ (K) there holds |T (φ)| ≤ CK φ αj . (1.2.17) j∈IK

The distributions form a linear space denoted by D (Ω). For any α ∈ N, the Dα derivative of a distribution T is deﬁned by Dα T (φ) = (−1)|α| T (Dα φ)

∀φ ∈ C0∞ (Ω).

(1.2.18)

It is a distribution. The multiplication of a distribution T by a C ∞ (Ω) function ζ is the distribution ζT deﬁned by ζT (φ) = T (ζφ)

∀φ ∈ C0∞ (Ω).

(1.2.19)

The classical Leibnitz rule of diﬀerentiation applies to ζT at any order. A sequence {Tn } ⊂ D (Ω) converges weakly to a distribution T if lim Tn (φ) = T (φ)

n→∞

∀φ ∈ C0∞ (Ω).

(1.2.20)

Although there exists a strong topology on D (Ω), this weak convergence is suﬃcient for treating most of the problems encountered in the study of partial diﬀerential equations, inasmuch the mappings T → Dα T and T → ζT are stable for this sequential topology.

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If Ω = RN the Fourier transform of φ ∈ C0∞ (RN ) is deﬁned by ˆ F(φ)(x) = φ(x) = e−2iπx,y φ(y)dy ∀x ∈ RN . (1.2.21) RN

Then φˆ ∈ C ∞ (RN ) satisﬁes ˆ =0 lim P (x)Dα φ(x)

|x|→∞

∀P ∈ R[x1 , ..., xN ] , ∀α ∈ NN ,

(1.2.22)

where R[x1 , ..., xN ] denotes the set of real polynomial in N real variables. The space S(RN ) is the space of C ∞ functions φ : RN → R which satisfy lim|x|→∞ P (x)Dα φ(x) = 0, for all P ∈ R[x1 , ..., xN ] and α ∈ NN . It is a real algebra stable under diﬀerentiation, and the expression φ P,α := sup |P (x)Dα φ(x)| ,

(1.2.23)

x∈RN

is a norm on S(RN ). Noticing that the Fourier transform has actually complex value and can be extended on the complexiﬁed space S C (RN ) S C (RN ) = {φ + iψ : φ, ψ ∈ S(RN )}, by setting F (φ + iψ) = F (φ) + iF (ψ), it deﬁnes an isomorphism of S C (RN ) 2 having for ﬁxed point the function x → e−π|x| . The inverse of F is the ˜ where ˜. is the symmetry with respect to 0, mapping φ → F −1 (φ) = F (φ) ˜ φ(x) = φ(−x). Furthermore this isomorphism is continuous for the topology deﬁned by the norms . P,α . The space of tempered distributions S (RN ) is the subspace of distributions T in RN such that there exist C > 0, a ﬁnite subset I ⊂ NN and a ﬁnite set of polynomials {Pj }j∈I satisfying |T (φ)| ≤ C φ Pj ,αj ∀φ ∈ C0∞ (RN ). (1.2.24) j∈I

If T is tempered it can be naturally extended as a linear form on S(RN ), still denoted by T , which satisﬁes the same inequality (1.2.24), but for all φ ∈ S(RN ). The Fourier transform F is deﬁned on S (RN ) by F (T ) = T (F (φ))

∀φ ∈ C0∞ (RN ).

(1.2.25)

The mapping F is an isomorphism from S (RN ) into itself, which inverse is the operator F −1 deﬁned by F −1 (T )(φ) = T (F −1 (φ)). This isomorphism is continuous with respect to the topology deﬁned by duality with S(RN )

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endowed with the norms . P,α . We recall below some standard results: (i) If φ ∈ L1 (RN ), F (φ) is continuous and bounded, F (φ) L∞ (RN ) ≤ φ L1 (RN ) and lim|x|→∞ F (φ(x) = 0. (ii) F can be naturally extended as an isometric isomorphism from L2 (RN ) into itself. αN N 1 (iii) If for α ∈ N, we set M α (x) = xα 1 ...xN , then for all φ ∈ S(R ) there holds

F(Dα φ) = (2iπ)|α| M α F (φ) and Dα F (φ) = (2iπ −|α| )F (M α φ). (1.2.26) (iv) Identities (1.2.26) are valid if φ is replaced by a tempered distribution. 1.2.3.2

Weak derivatives

Let g ∈ L1loc (Ω), we say that gj ∈ L1loc (Ω) is the weak derivative of g following xj if ∂φ g = − gj φdx ∀φ ∈ C0∞ (Ω). (1.2.27) Ω ∂xj Ω The function gj is uniquely determined. More generally, if α ∈ NN , a function gα ∈ L1loc (Ω) is the weak derivative of order α of g if gDα φ = (−1)|α| gα φdx ∀φ ∈ C0∞ (Ω). (1.2.28) Ω

Ω

The weak derivative gα is, as usually, denoted Dα g. 1.2.3.3

Sobolev spaces

Deﬁnition 1.3 Let Ω ⊂ RN be a domain, 1 ≤ ∞ and k ∈ N∗ . The Sobolev space W k,p (Ω) is the space of functions g ∈ Lp (Ω) such that Dα g ∈ Lp (Ω) for any α ∈ NN , with |α| ≤ k. It is endowed with a structure of complete normed space with the norm ⎧ p1 p ⎪ α ⎨ if p < ∞ |α|≤k D g Lp (Ω) (1.2.29) g W k,p (Ω) = ⎪ ⎩ α sup|α|≤k D g L∞ (Ω) if p = ∞.

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The closure of C0∞ (Ω) in W k,p (Ω) is denoted by W0k,p (Ω). If Ω is bounded, Poincar´e inequality asserts 1

g Lp (Ω) ≤ (diamΩ) p ∇g Lp (Ω)

∀g ∈ C0∞ (Ω).

As a consequence, an equivalent norm on W0k,p (Ω) is ⎧ p1 p ⎪ α ⎨ D g if p < ∞ |α|=k Lp (Ω) g W k,p (Ω) = 0 ⎪ ⎩ sup|α|=k Dα g L∞ (Ω) if p = ∞.

(1.2.30)

(1.2.31)

Below are listed some of the main properties of Sobolev spaces (i) Gagliardo-Nirenberg inequality. Assume 1 ≤ p < N . For any φ ∈ C0∞ (RN ) there holds φ Lp∗ (RN ) ≤ where p∗ =

Np N −p .

p(N − 1) ∇φ Lp (RN ) , N −p

(1.2.32)

The Sobolev constant SN,p is deﬁned by

SN,p = inf ∇φ Lp (RN ) : φ ∈ C ∞ (RN ), φ Lp∗ (RN ) = 1 .

(1.2.33)

(ii) Gagliardo-Nirenberg interpolation inequality. Let 1 ≤ p, q, r ≤ ∞ and m ∈ N. If there exist k ∈ N and θ ∈ [0, 1] such that 1 k 1−θ 1 m k ≤ θ ≤ 1 and = + − + , m r N p N q then one can ﬁnd a constant c = c(N, p, q, m, j, θ) > 0 with the property that for any φ ∈ C0∞ (RN ) there holds.

j

D φ r N ≤ c Dm φ θ p N φ 1−θ (1.2.34) L (R ) Lq (RN ) , L (R ) with the exception that if m − j −

N r

∈ N, then θ < 1.

(iii) Morrey inequality. Assume N < p ≤ ∞, then there exists c = c(N, p) > 0 such that sup |φ(x)| + x∈RN

sup x=y∈RN

|φ(x) − φ(y)| ≤ c ∇φ Lp RN , |x − y|α

(1.2.35)

for any φ ∈ C ∞ (RN ), where α = 1 − Np . (iv) Meyers-Serrin approximation. Assume Ω ⊂ RN is a domain, k ∈ N∗ and 1 ≤ p < ∞. If g ∈ W k,p (Ω) there exists a sequence of C ∞ (Ω) functions

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{gn } which converges to g in W k,p (Ω). Moreover, if Ω is bounded and ∂Ω Lipschitz, there exists a sequence {gn } ⊂ C ∞ (Ω) which converges to g in W k,p (Ω) (v) Extension. Assume Ω ⊂ RN is a C k domain with a compact boundary and 1 ≤ p ≤ ∞. Then there exists a continuous linear operator P from W k,p (Ω) to W k,p (RN ) such that P (u)Ω = u. Furthermore for any domain Ω such that Ω ⊂ Ω , P can be constructed in order P (u) ≡ 0 outside Ω . (vi) Rellich-Kondrachov theorem. Let Ω ⊂ RN be a bounded domain with a C 1 boundary and 1 ≤ p < N . Then for any p ≤ q < p∗ the imbedding of W 1,p (Ω) into Lq (Ω) is a compact mapping. A consequence of this compactness theorem is another Poincar´e inequality. Assume Ω ⊂ RN is a bounded domain with a C 1 boundary and 1 ≤ p ≤ ∞. Then there exists c = c(p, Ω) > 0 such that φ − φΩ Lp (RN ) ≤ c ∇φ Lp (RN ) , where φΩ =

1 |Ω|

(1.2.36)

φdx. Ω

(vii) Gagliardo-Nirenberg-Poincar´e-Wirtinger inequality. It is a variant of (1.2.36) and (1.2.32). Assume Ω ⊂ RN is a bounded domain with a C 1 boundary, 1 ≤ p < N , q ≥ 1 and 0 ≤ θ ≤ 1. If φ ∈ W 1,p (Ω) ∩ Lq (Ω), then θ

1−θ

φ − φΩ Lr (Ω) ≤ c ∇φ Lp (Ω) φ Lq (Ω) ,

(1.2.37)

where c = c(Ω, p, q, θ) > 0 and 1 1−θ = +θ r q

1 1 − p N

=

1−θ θ + ∗, q p

(1.2.38)

with the convention that p∗ = ∞ if p = N . (viii) Boundary trace. Let Ω ⊂ RN be a bounded domain with a C 1 boundary and 1 ≤ p < ∞. Then there exists a continuous linear map γ0 from W 1,p (Ω) into Lp (∂Ω) such that γ0 (g) = g∂Ω for any g ∈ W 1,p (Ω) ∩ C(Ω). Furthermore ker(γ0 ) = W01,p (Ω).

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(ix) Co-area formula Let u ∈ Cc1 (RN ), w ∈ L1 (RN ) and E ⊂ RN be a measurable set. Then ∞

wdHN −1

w|∇u|dx = E

dt,

(1.2.39)

E∩s|u| (t)

0

where su (t) = {x ∈ RN : |u(x)| = t} (see e.g. [Federer (1969), p. 258]). Note that (1.2.39) deﬁnes a local change of coordinates in any region where ∇u does not vanish. In particular ∞ |∇u|dx = P er(Su (t))dt (1.2.40) RN

0

where Su (t) = {x ∈ RN : |u(x)| > t} (note that su (t) = ∂Su (t) with the notations of Section 1.2.1) and P er(Su ) denotes the perimeter of Su (t); it is the (N-1)-dimensional surface area if ∇u never vanish on su (t). Formula (1.2.39) extends to u ∈ W 1,p (RN ). 1.2.3.4

Fractional Sobolev spaces

Let Ω ⊂ RN be a domain and 1 ≤ p < ∞. If 0 < s < 1, the AronszajnGagliardo-Slobodeckij norm on C0∞ (Ω) is deﬁned by φ W˙ s,p (Ω) =

Ω×Ω

|φ(x) − φ(y)|p dxdy |x − y|N +sp

p1 .

(1.2.41)

The space W s,p (Ω) is the subset of functions φ ∈ Lp (Ω) such that φ W˙ s,p (Ω) < ∞. It is a complete normed space for the norm p1 p p φ W s,p (Ω) = φ Lp (Ω) + φ W˙ s,p (Ω) , and is called a fractional Sobolev space. A limit case when p = ∞ yields the H¨ older space C 0,s (Ω) with norm |φ(x) − φ(y)| . |x − y|s x=y∈Ω

φ C 0,s (Ω) = sup |φ(x)| + sup x∈Ω

We list below some useful results concerning these spaces: ˙ s,p -norm satisﬁes the Poincar´e inequality (i) If Ω is bounded, the W φ Lp (Ω) ≤ c(diam(Ω))s φ W ˙ s,p (Ω)

∀φ ∈ C∞ 0 (Ω).

(1.2.42)

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17

(ii) When sp < N , the Gagliardo-Nirenberg inequality holds with p∗ (s) = Np N −sp and c = c(N, p, s) > 0 under the form p

∀φ ∈ C0∞ (Ω).

p

φ Lp∗ (s) (RN ) ≤ c φ W˙ s,p (RN )

(1.2.43)

(iii) If sp > N and Ω has a compact Lipschitz boundary, the Morrey inequality is valid with α = s − Np , φ C 0,s (Ω) ≤ c φ pW s,p (RN )

∀φ ∈ W s,p (Ω).

(1.2.44)

(iv) If Ω is bounded and q < p∗ (s) the imbedding of W s,p (Ω) into Lq (Ω) is compact. If 1 ≤ p < ∞, k ∈ N∗ and 0 < s < 1 the fractional Sobolev spaces (Ω) is the subset of φ ∈ W k,p (Ω) such that Dα φ ∈ W s,p (Ω) for any W N α ∈ N with |α| = k. It is a complete space if equipped with the norm ⎛ ⎞ p1 p p φ W k+ss,p (Ω) = ⎝ φ W k,p (Ω) + Dα φ W˙ s,p (Ω) ⎠ . k+s,p

|α|=k

1.2.4

Bessel potentials

If α is a real number, the Bessel kernel of order α is deﬁned by − α Gα = F −1 1 + |ξ|2 2 ,

(1.2.45)

where F −1 is the inverse Fourier transform deﬁned in S (RN ) (see Section 1.2.3). Put Gα = (I − cα Δ)− 2 , α

(1.2.46)

for some normalizing cα > 0, then there holds f = Gα [g] = Gα ∗ g ⇐⇒ g = G−α [f ] = G−α ∗ f

∀f, g ∈ S (RN ). (1.2.47)

Moreover the group property holds Gα Gβ = Gα+β ,

(Gα )−1 = G−α

∀α, β ∈ R.

(1.2.48)

Finally, for any R > 0 the Bessel kernel satisﬁes a two-sided pointwise estimate (see e.g. [Stein (1971), V-3-1]) c−1

χB2R (x) χB (x) |x| ≤ Gα (x) ≤ c RN −α + ce− 2 N −α |x| |x|

∀x ∈ RN \ {0},

(1.2.49)

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where c = c(N, α, R) > 0. 1.2.4.1

Bessel potential spaces

The classical Bessel potential spaces are deﬁned for p ≥ 1 and α ∈ R by ! Lα,p := Lα,p (RN ) = f = Gα ∗ g, g ∈ Lp (RN ) , (1.2.50) with norm f Lα,p = G−α ∗ f Lp .

(1.2.51)

A classical result due to Calderon asserts that W α,p (RN ) = Lα,p (RN ) when α is an integer and 1 < p < ∞ (see e.g. [Adams, Hedberg (1999)]). Actually, the Lα,p (RN ) spaces share the same imbedding properties as the ones of Sobolev spaces when α > 0 and 1 ≤ p < ∞: (i) if αp < N and p ≤ q ≤ p∗α where Lq (RN );

1 p∗α

=

1 p

−

α N,

then Lα,p (RN ) ⊂

(ii) if αp = N and p ≤ q < ∞, then Lα,p (RN ) ⊂ Lqloc (RN ); (iii) if αp > N , then Lα,p (RN ) ⊂ C(RN ). 1.2.4.2

Bessel-Lorentz potential spaces

The scale of Bessel spaces can be reﬁned if we replace the Lp spaces by Lorentz spaces. If 1 ≤ s ≤ ∞, 1 < p < ∞ and α ∈ R ! Lα,p,s := Lα,p,s (RN ) = f = Gα ∗ g, g ∈ Lp,s (RN ) , (1.2.52) with norm f Lα,p,s = G−α ∗ f Lp,s .

(1.2.53)

The space Lα,p,p (RN ) is identiﬁed with Lα,p (RN ). When 1 ≤ s, p < ∞ the space C0∞ (RN ) is dense in Lα,p,p (RN ). The following duality holds α,p,s N L (R ) = L−α,p ,s (RN ). (1.2.54) The Bessel potential Gα is an isomorphism between Bessel-Lorentz potential spaces. Theorem 1.2.5 Assume α > 0, 1 < p < ∞, 1 < s < ∞. Then Gα is an isomorphism between L−α,p,s (RN ) and Lp,s (RN ).

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1.2.4.3

19

Bessel-Lorentz capacities

The Bessel capacity of a compact set K relative to RN is given by p cR α,p,s (K) = inf { φ Lα,p,s : φ ∈ ωK } ,

(1.2.55)

ωK = {φ ∈ S(RN ) : φ ≥ 1 on K}.

(1.2.56)

N

where

In this deﬁnition, it is equivalent to take test functions in C0∞ (RN ). If α > 0 the closure of ωK in Lα,p,s (RN ). Then and 1 < s, p < ∞ we denote by ω α,p,s K α,p,s there exists a unique ΦK ∈ ω K , called the capacitary potential of K, such that cR α,p,s (K) = ΦK Lα,p,s . N

p

This set function is extended to open sets by N N R (O) = sup c (K), K ⊂ O, K compact , cR α,p,s α,p,s

(1.2.57)

(1.2.58)

and to general sets by N N R cR (E) = inf c (O), E ⊂ O, O open . α,p,s α,p,s

(1.2.59)

When s = p and α is an integer, cR α,p,s is a Sobolev capacity while if s = p, RN cα,p,s it is a Bessel capacity. Thanks to Calderon’s theorem, the Sobolev capacity of a compact set can be directly computed by minimization of the energy ⎧ ⎫ ⎛ ⎞ ⎨ ⎬ N γ p⎠ ∞ N ⎝ cR (E) = inf |D φ| (R ) ∩ ω dx : φ ∈ C K . (1.2.60) 0 α,p,p ⎩ RN ⎭ N

|γ|≤α

The dual deﬁnition of a capacity is ! N p + cR α,p,s (K) = sup (μ(K)) : μ ∈ M (K), Gα ∗ μ Lp ,s ≤ 1 .

(1.2.61)

The following result is proved in [Adams, Hedberg (1999), Th 2.2.7] in the Lp framework, but the proofs are similar with Lp,s . Proposition 1.2.6 Let α > 0, 1 < s, p < ∞, K ⊂ RN be compact and the capacitary potential of K deﬁned by (1.2.57). Then the ΦK ∈ ω α,p,s K

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capacitary function φK of K is deﬁned by ΦK = Gα ∗ φK . It belongs to Lp,s (RN ) and by (1.2.53), p p cR α,p,s (K) = ΦK Lα,p,s = φK Lp,s . N

(1.2.62)

There exists a nonnegative measure μK , called the capacitary measure of K, such that μK (K c ) = 0 and

φK = (Gα ∗ μk )p −1 =⇒ ΦK = Gα ∗ (Gα ∗ μk )p −1 .

(1.2.63)

Furthermore μK ∈ L−α,p ,s (RN ) and RN p (Gα ∗ μk ) dx = ΦK dμK = μK (K). cα,p,s (K) = RN

(1.2.64)

K

The following very practical results are proved by parts in [GrunRehomme (1977)], [Adams, Hedberg (1999)], [Feyel, de la Pradelle (1977)] in the case of Sobolev spaces W α,p (RN ) (α ∈ N). The proof of (iii) in Bessel spaces is given in [V´eron (2004)] and in the case of Bessel-Lorentz spaces in [Bidaut-V´eron, Nguyen, V´eron (2014), Th 2.5]; therein the convergence in the narrow topology, which requires some new arguments, is also given. Proposition 1.2.7

Let α > 0 and 1 < s, p < ∞.

(i) If K is a compact set such that cR α,p,s (K) = 0 and O an open set containing K, there exists a sequence {ξn } ⊂ C0∞ (O) of functions such that 0 ≤ ξn ≤ 1, ξn = 1 in a neighborhood ok K such that ξn → 0 in Lα,p,s (RN ). N

(ii) If λ ∈ L−α,p ,s (RN ) ∩ M(RN ), then |λ| (E) = 0 for any Borel set such N that cR α,p,s (E) = 0. We say that λ is absolutely continuous with respect N RN to the cR α,p,s -capacity, or that λ does not charge subsets with zero cα,p,s capacity. (iii) If λ ∈ M+ (RN ) is absolutely continuous with respect to the cR α,p,s capacity, there exists a nondecreasing sequence of nonnegative measures with compact supports {λn } ⊂ L−α,p ,s (RN ) which converges to λ in the N weak sense of measures. Furthermore, if λ ∈ M+ b (R ), then the convergence of {λn } towards λ holds in the narrow topology. N

The following terminology will be useful in the sequel. Deﬁnition 1.4

Let α > 0 and 1 < s, p < ∞.

(i) A property is said to hold cα,p,s -quasieverywhere if it holds everywhere but on a set with zero cα,p,s -capacity.

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21

(ii) A function f deﬁned in RN is cα,p,s -quasicontinuous if for any > 0 N there exists an open set O ⊂ RN such that cR α,p,s (E) < and f is continuous on Oc . (iii) A set O ⊂ RN is cR α,p,s -quasiopen (quasiopen if there is no ambiguity) N if there exists a set E ⊂ O such that cR α,p,s (E) = 0 and O \ E is open. We N deﬁne similarly cR α,p,s -quasiclosed (quasiclosed) sets. N

1.2.4.4

Riesz and Sobolev capacities

In some cases it is convenient to use other capacities than the LorentzBessel ones. For 0 < α < N we deﬁne the Riesz kernel of order α, Iα by Iα (x) = c(N, α) |x|

α−N

∀x ∈ RN \ {0},

(1.2.65)

for some normalizing positive constant c(N, α) and the associated Riesz potential Iα [μ] of a positive measure μ by Iα (x − y)dμ(y). (1.2.66) Iα [μ](x) = Iα ∗ μ(y) = RN

It can be ﬁnite or inﬁnite. If μ is no longer positive Iα [μ](x) is well deﬁned if Iα [|μ|](x) < ∞. The Riesz kernel corresponds to the inverse operator of α (−Δ)− 2 . N For 1 ≤ p < ∞, the Riesz capacity c˙R is α,p of a compact set K ⊂ R deﬁned by N p . (1.2.67) (K) = inf I [φ] : φ ∈ ω c˙R p N α K α,p L (R ) N

This set function is extended to general sets by the same rules as for the N Bessel capacity cR α,p and there holds R cR α,p (K) ≤ c˙ α,p (K). N

N

(1.2.68)

The main advantage of the Riesz capacity is its homogeneity invariance under dilations: for any > 0 there holds N −αp R c˙α,p (E). c˙R α,p (E) = N

N

(1.2.69)

Similarly to Deﬁnition 1.4 it is possible to express that a property holds c˙α,p -quasieverywhere, that a function is c˙α,p -quasicontinuous and that a set is c˙α,p -quasiopen or quasiclosed.

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22

Ω It is also useful to deﬁne the fractional Sobolev capacity Cα,s of a comN pact set K ⊂ Ω relative to a domain Ω ⊂ R by Ω Cα,p (K) = inf { φ pW α,p : φ ∈ ωK,Ω }

(1.2.70)

where ωK,Ω = {φ ∈ C0∞ (Ω), φ ≥ 0 , φ ≥ 1 on K}. R If α is an integer Cα,p = cR α,p by Calderon’s theorem. N

1.3

N

General quasilinear elliptic equations

The purpose of this section is to present the historical framework of general quasilinear elliptic equations in a domain Ω ⊂ RN . Consider A : (x, r, ξ) ∈ Ω× R× RN → A(x, r, ξ) ∈ RN and B : (x, r, ξ) ∈ Ω× R× RN → B(x, r, ξ) ∈ R two functions which are Lebesgue measurable with respect to x and continuous in r and ξ (A and B are called Caratheodory functions). It is assumed that there exist constants p > 1 and c, c > 0 and measurable functions cj , (j=1,..., 7) such that the following inequalities hold for all (x, r, ξ) ∈ Ω × R × RN , (i)

A(x, r, ξ), ξ ≥ c|ξ|p − c1 |r|p − c2

(ii)

|A(x, r, ξ)| ≤ c |ξ|p−1 + c3 |r|p−1 + c4

(iii)

|B(x, r, ξ)| ≤ c5 |ξ|p−1 + c6 |r|p−1 + c7 .

(1.3.1)

A weak solution in Ω of −div A(x, u, ∇u) + B(x, u, ∇u) = 0, is a function u ∈ W 1,p (Ω) which satisﬁes (A(x, u, ∇u), ∇φ + B(x, u, ∇u)φ) dx = 0 Ω

∀φ ∈ C0∞ (Ω).

(1.3.2)

(1.3.3)

By extension, a weak subsolution (resp. supersolution) of (1.3.2) is a function u ∈ W 1,p (Ω) which satisﬁes (A(x, u, ∇u), ∇φ + B(x, u, ∇u)φ) dx ≤ 0 Ω

∀φ ∈ C0∞ (Ω), φ ≥ 0, (1.3.4)

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23

respectively (A(x, u, ∇u), ∇φ + B(x, u, ∇u)φ) dx ≥ 0 Ω

∀φ ∈ C0∞ (Ω), φ ≥ 0.

(1.3.5) The key result dealing with positive weak solutions is the fact that they satisfy Harnack inequalities as harmonic functions do. The constants in these inequalities depend in a crucial way of the assumptions on the Lebesgue spaces to which the cj belong. A classical application of Harnack inequalities lies in the proof of the H¨older continuity of the solutions, but they have also many applications for deriving local or global estimates of solutions. 1.3.1

The Serrin’s results

The assumptions on the cj made by Serrin [Serrin (1964)], [Serrin (1965)] are as follows: N

N

(i) c3 , c4 ∈ L p−1 (Ω), c5 ∈ LN + (Ω), c1 , c2 , c6 , c7 ∈ L p + (Ω)

if p < N

N

(ii) c3 , c4 ∈ L N −1 + (Ω), c5 ∈ LN + (Ω), c1 , c2 , c6 , c7 ∈ L1+ (Ω) if p = N N

(iii) c3 , c4 ∈ L p−1 (Ω), c5 ∈ Lp (Ω), c1 , c2 , c6 , c7 ∈ L1 (Ω)

if p > N, (1.3.6) for some > 0. However we will concentrate on the case 1 < p ≤ N , since the weak solutions are automatically H¨older continuous if p > N . The local upper estimate for subsolutions is the following, Theorem 1.3.1 Assume 1 < p ≤ N and (1.3.6) holds. If u is a weak subsolution of (1.3.2) in a ball B3R := B3R (x0 ), it satisﬁes

N N R p u+ L∞ (BR ) + R ∇u+ Lp (BR ) ≤ m u+ Lp (B2R ) + kR p , (1.3.7) where m = m c, c , p, N, , c3

, R c5

N

L p−1 (B2R )

N

L 1− (B2R )

, R c6

,

N

L p− (B2R )

and k=

c4

N L p−1

(B2R )

+ R c7

N L p−

1 p−1

(B2R )

+ R c2

N L p−

p1 (B2R )

,

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Local and global aspects of quasilinear degenerate elliptic equations

if 1 < p < N , and where N m = m c, c , N, , c3 N −1− L

(B2R )

, R c5

N L 1−

(B2R )

, R c6

N L N −

(B2R )

and k = c4

+ R c7

N

L N −1− (B2R )

N

L N − (B2R )

N1−1 + R c2

N

L N − (B2R )

N1 ,

if p = N . Remark. By changing u into −u it is easy to see that the above inequality still holds for supersolution u as well, provided u+ is replaced by u− . The following weak Harnack inequality is fundamental for proving regularity of weak solutions. Theorem 1.3.2 Assume 1 < p ≤ N and (1.3.6) holds. If u is a positive weak supersolution of (1.3.2) in B3R and p − 1 ≤ q < NN(p−1) −p , it satisﬁes R

−N q

−1 −1

u q L (B

2R )

≤ m1 ess inf u + k1 , BR

(1.3.8)

where m1 and k1 depend on the same quantities as m and k in Theorem 1.3.1. Both Theorem 1.3.1 and Theorem 1.3.2 are proved by Moser’s iterative scheme which consists in estimating u Lτ a (Br ) in term of u La (Br ) for some τ > 1 and r < r by a judicious choice of cut-oﬀ functions. Combining estimates (1.3.6) and (1.3.8) thanks to John and Nirenberg theorem (see [Serrin (1964), Lemmas 6, 7]) we infer the Harnack inequality. Theorem 1.3.3 Assume 1 < p ≤ N and (1.3.6) holds. If u is a positive weak solution of (1.3.2) in B3R , it satisﬁes (1.3.9) ess sup u ≤ m2 ess inf u + k2 , BR

BR

where m2 and k2 depend on the same quantities as m and k in Theorem 1.3.1. It is important to notice that if the terms c2 , c4 and c7 in (1.3.6) are identically 0, then all the constants k are 0 and (1.3.9) inherits the following form,

,

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25

ess sup u ≤ m2 ess inf u.

(1.3.10)

BR

BR

When p > N the Harnack inequality takes another form. The requirements on the coeﬃcients cj are the following

c3 , c4 ∈ Lp (Ω) , c5 ∈ Lp (Ω) , c1 , c2 , c6 , c7 ∈ L1 (Ω).

(1.3.11)

Theorem 1.3.4 Assume p ≥ N , (1.3.1) and (1.3.11) hold. If u is a positive weak solution of (1.3.2) in B3R , it satisﬁes u(x) ≤ (u(y) + k)ec(

|x−y| R

)

1− N p

∀(x, y) ∈ BR ,

(1.3.12)

where

p−N p−N c = c p, N, c, c , R p c3 Lp , R p c5 Lp , Rp−N ( c1 L1 + c6 L1 ) ,

and k=

1 p−1 p−N 1 + Rp−N c2 L1 ) p . R p c4 Lp + Rp−N c7 L1

As a consequence it can be proved, [Serrin (1964), Th 8], that the solutions are H¨older continuous. Theorem 1.3.5 Assume 1 < p ≤ N and (1.3.6) holds. If u is a weak solution of (1.3.2) in a domain Ω, then it is essentially H¨ older continuous in Ω. More precisely, for any bounded subdomains D, D such that D ⊂ D ⊂ D ⊂ Ω and |u| ≤ M in D , there exists α ∈ (0, 1) subject to the same structural assumptions as the mj such that |u(x) − u(y)| ≤ C |x − y|

α

∀(x, y) ∈ D × D,

(1.3.13)

where C = C(m2 , k2 , M, D, D ). Proof. Let z ∈ Ω and r0 > 0 such that B 2r0 (z) ⊂ Ω. For 0 < r ≤ r0 we set m(r) = ess sup u and m(r) = ess inf u, Br (z)

Br (z)

and u = m(r) − u, u = u − m(r). Then u and u are nonnegative in Br (z). The function u is a weak solution of an equation of the form −div A(x, u, ∇u) + B(x, u, ∇u) = 0,

(1.3.14)

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in Br (z) where the functions A and B verify inequalities (1.3.1) in which c and c are unchanged, the cj for j = 1, 3, 5, 6, are replaced by ﬁxed multiples cj independent of r (and thus satisfying uniform bounds in their corresponding Lebesgue spaces), while for j = 2, 4, 7 they are multiplied by positive power of r. Applying Theorem 1.3.3 in Br (z) to u and u, we obtain m(r) − m( r3 ) = ess sup u ≤ m2 B r (z) 3

ess inf u + k 2 B r (z)

3 ≤ m2 m(r) − m( r3 ) + k 2 .

Proceeding similarly with u, we obtain m( r3 ) − m(r) ≤ m2 m( r3 ) − m(r) + k2 .

(1.3.15)

(1.3.16)

Notice that m2 , m2 are independent of r while there exists a > 0 depend older’s ing on the norms of the cj such that max{k 2 , k 2 } ≤ ar p−1 by H¨ inequality. By addition m( r3 ) − m( r3 ) ≤

m2 − 1 m2 (m(r) − m(r)) + k2 + k2 . m2 + 1 m2 + 1

(1.3.17)

2am2 2 −1 Put θ = m m2 +1 < 1, τ = m2 −1 and denote by ω(r) the oscillation of u in Br (z). Then (1.3.17) becomes ω( 3r ) ≤ θ ω(r) + τ r p−1 ∀0 < r ≤ r0 .

Since ω is nondecreasing, for any s ≥ 3 there holds ∀0 < r ≤ r0 . ω( rs ) ≤ θ ω(r) + τ r p−1 Starting with r = r0 and iterating this inequality, we obtain r0 p−1 r0 ∀n ∈ N∗ . ω( srn0 ) ≤ θ ω( sn−1 ) + τ sn−1 An easy induction yields −n

ω( srn0 ) ≤ θn ω(r0 ) + m3 max{θn , s p−1 }, where m3 depends in the structural constants of A and B. Choosing s such − that θ ≥ s p−1 we derive ω( srn0 ) ≤ θn (ω(r0 ) + m3 ). For ρ ∈ (0, s−1 ) we can ﬁnd n ∈ N such that r0 s−n−1 ≤ ρ < r0 s−n , thus α ω(ρ) ≤ ω( srn0 ) ≤ θn (ω(r0 ) + m3 ) ≤ ρα rs0 (ω(r0 ) + m3 ),

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27

if we choose α ∈ (0, 1) such that θ ≤ s−α . This implies the claim.

A more general form of Harnack inequality is due to [Trudinger (1967)]. The class of operators to which it applies is often called quasilinear equations with natural growth in the gradient. The growth assumption on B is |B(x, r, ξ)| ≤ b0 |ξ|p + b1 (x) |ξ|p−1 + b2 (x) |r|p−1 .

(1.3.18)

Furthermore, the integrability requirements on the bj and the cj are lowered provided the solution satisﬁes some local bound. If ρ → (ρ) is a positive decreasing function on (0, ∞) tending to 0 when ρ → 0 we set for k ≥ 1, Lk (Ω) = {φ ∈ Lk (Ω), φ Lk < ∞} where φ Lk =

φ Lk (Ω∩Bρ (z))

sup

(ρ)

z∈Ω,ρ>0

The assumptions on the bj , cj in (1.3.6)-(i)-(ii) and (1.3.18) are the following: N

N

c1 , b2 ∈ Lρpδ (Ω) , c3 ∈ L p−1 (Ω) , b1 ∈ LN ρδ (Ω),

(1.3.19)

for some δ > 0, assuming for simplicity 1 < p ≤ N . Theorem 1.3.6 Let 1 < p ≤ N and Ω ⊂ RN be a domain. Assume that A and B satisfy (1.3.1) (i)-(ii) with c2 = c7 = 0, and (1.3.18) respectively, and that the integrability conditions (1.3.19) hold. If R > 0 is such that B3R (x0 ) = B3R ⊂ Ω, let us deﬁne μ = μ(R) by 1

1

μ = R−1 c3 p−1N

L p−1 (B2R )

+ Rδ−1 c1 + b3 + bp1 p N L

p ρδ

, (B2R )

if 1 < p < N , and

1 μ = Rδ−1 c3 N −1N

L N −1 (B2R )

N1

+ c1 + b 3 + b N 1 L1

ρδ

(B2R )

,

if p = N . Then, if u is a weak solution of (1.3.2) in B3R which satisﬁes |u| ≤ M in B2R , there exists M2 > 0 depending on c, c , N , p, μ(R) and M such that ess sup ≤ M2 ess inf u. BR

BR

(1.3.20)

This inequality implies also the H¨older continuity of the solutions of (1.3.2). The next result characterizes the behaviour of solutions of weak subsolution of (1.3.2) with an isolated singularity.

.

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Theorem 1.3.7 Assume 1 < p ≤ N and (1.3.6) holds. If u is a weak solution of (1.3.2) in Ω \ {0} which is bounded from below near 0, then either u can be extended to whole Ω as weak solution of (1.3.2) in Ω, or there exist 0 > 0 and c > 0 such that the following estimates hold for 0 < |x| < 0 < 1, N −p

N −p

(i)

c|x|− p−1 ≤ u(x) ≤ c−1 |x|− p−1

if 1 < p < N

(ii)

c ln |x|−1 ≤ u(x) ≤ c−1 ln |x|−1

if p = N.

(1.3.21)

The next result due to [Reshetniak (1966)] extends Liouville theorem to weak solutions of −div A(x, u, ∇u) = 0

in RN ,

(1.3.22)

under the assumption that A : RN × R × R → RN is a Caratheodory vector ﬁeld such that there exist two positive constants such that for all (x, r, ξ) ∈ RN × R × R, (i)

A(x, r, ξ), ξ ≥ c|ξ|p

(ii)

|A(x, r, ξ)| ≤ c |ξ|p−1 .

(1.3.23)

Theorem 1.3.8 Assume p > 1. If u is a bounded weak solution of (1.3.22) in RN , it is a constant. Proof. We give the proof in the case 1 < p ≤ N , the case p > N being similar thanks to Theorem 1.3.4. For R > 0 we set m(R) = max{u(x) : |x| ≤ R} and m(R) = min{u(x) : |x| ≤ R}. Since k 2 = k 2 = 0, inequality (1.3.17) becomes m(R) − m(R) ≤

m2 − 1 m(3R) − m(3R), m2 + 1

(1.3.24)

where m2 > 1 is a constant independent of the solution as it is shown in Theorem 1.3.3. Iterating this inequality, we obtain m(R) − m(R) ≤

m2 − 1 m2 + 1

k m(3k+1 R) − m(3k+1 R)

∀k ∈ N.

(1.3.25) Letting k → ∞ and using the boundedness of u, we infer m(R) = m(R) for all R.

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Remark. The above proof shows actually that there exists a constant β > 0 such that any weak solution u of (1.3.22) in RN satisfying lim

|x|→∞

is a constant. If we set θ =

m2 −1 m2 +1 ,

u(x) = 0, |x|β

the exponent β is given by

β=− 1.3.2

(1.3.26)

ln θ . ln 3

(1.3.27)

Regularity

1.3.2.1

Local regularity

Let Ω be a domain in RN , and A : Ω×R×RN → RN and B : Ω×R×RN → R be two Caratheodory vector ﬁelds which satisfy (1.3.1) (i)-(ii) for A and (1.3.18) for B, subject to the integrability conditions (1.3.6) and/or (1.3.19). These assumptions imply the H¨ older continuity of the weak solutions of −div A(x, u, ∇u) + B(x, u, ∇u) = 0

in Ω.

(1.3.28)

In order to reach an higher regularity, it is assumed that A is diﬀerentiable. If g0 > 0, we introduce the class of positive C 1 functions on R subject to the inequality 0≤

tg (t) ≤ g0 g(t)

∀t ∈ R.

(1.3.29)

We set G(t) =

t

g(s)ds

∀t ∈ R.

0

It is easy to verify that G is a C 2 convex function and that there holds ∀t ≥ 0

(iii)

tg(t) ≤ G(t) ≤ tg(t) 1 + g0 G(a) a ≤ G(b) b g(t) ≤ g(2t) ≤ 2g0 g(t)

(iv)

ag(b) ≤ ag(a) + bg(a)

∀a, b ≥ 0.

(i) (ii)

∀b ≥ a ≥ 0 ∀t ≥ 0

(1.3.30)

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Concerning A and B, we assume that the following inequalities hold |Dξ A| ≤ Λ(|r|)

(i)

g(|ξ|) |ξ|

(iii)

∂Ai g(|ξ|) 2 |η| ∀η ∈ RN (x, r, ξ)ηi ηj ≥ ∂ξj |ξ| |A(x, r, ξ) − A(y, t, ξ)| ≤ Λ1 (|r|)(1 + g(|ξ|))(|x − y|γ + |r − t|γ )

(iv)

|B(x, r, ξ)| ≤ Λ1 (|r|)(|r|)(1 + g(ξ)),

(ii)

i,j

(1.3.31) where α ∈ (0, 1], Λ and Λ1 are positive increasing functions and g satisﬁes 1,1 (Ω) such (1.3.29). We denote by W 1,G (Ω) the class of functions u ∈ Wloc 1 that G(u) ∈ L (Ω). If we assume that A and B satisfy (1.3.31), then 1,1 (Ω) ∩ L∞ (Ω) the functions A(., u, ∇u) and B(., u, ∇u) for any u ∈ Wloc are integrable in Ω, thus this class is the natural framework for studying bounded weak solutions of (1.3.28). The following local regularity theorem is proved in [Lieberman (1991), Th 1.7]. Theorem 1.3.9 Assume p > 1 and A and B satisfy (1.3.31). Then for any M > 0, there exists α ∈ (0, 1), depending on γ, N , g0 , Λ1 (M ) and Λ(M ) such that any weak solution u ∈ W 1,G (Ω) ∩ L∞ (Ω) of (1.3.28) verifying u L∞ ≤ M , satisﬁes u C 1,γ (Ω ) ≤ c,

(1.3.32)

for any bounded domain Ω ⊂ Ω ⊂ Ω, where the constant c > 0 depends on γ, N , g0 , g(1), Λ1 (M ), Λ(M ) and dist (Ω , Ωc ).

Applications The conditions (1.3.31) are veriﬁed if B satisﬁes (1.3.18) and (i) (ii)

|Dξ A| (κ + |ξ|)2 + |Dr A| (κ + |ξ|) + |Dx A| ≤ Λ(|r|)(κ + |ξ|)p ∂Ai i,j

∂ξj

(x, r, ξ)ηi ηj ≥ λ(|r|)(κ + |ξ|)p−2 |η|2

∀η ∈ RN ,

(1.3.33)

for some p > 1, where Λ and λ are positive functions deﬁned on R+ , respectively increasing and decreasing, and where κ = 1 or κ = 0.

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1.3.2.2

31

Boundary regularity

In this section we consider solutions of the homogeneous Dirichlet problem, −div A(x, u, ∇u) + B(x, u, ∇u) = 0 u=0

in Ω (1.3.34) in ∂Ω.

We assume that there exist positive constants λ, γ and γ ∈ (0, 1] and three continuous functions μ : Ω × R → R and Λ : R+ → R+ , C increasing, such that the mappings (x, r, ξ) → A(x, r, ξ) = (A1 (x, r, ξ), ..., AN (x, r, ξ)) and (x, r, ξ) → B(x, r, ξ) satisfy a.e. in Ω and for all r, t ∈ R and ξ, η ∈ RN , (i) (ii)

∂Ai

p−2 (x, r, ξ)ηi ηj ≥ λ μ2 (x, r) + |ξ|2 2 |η|2

∂ξj i,j ∂Ai p−2 2 2 2 ∂ξj (x, r, ξ) ≤ Λ(|r|) μ (x, r) + |ξ|

(iii)

|A(x, r, ξ) − A(y, t, ξ)| ≤ β(1 + |ξ|p−2 + |ξ|p−1 )(|x − y|γ + |r − t|γ )

(iv)

|B(x, r, ξ)| ≤ C(|r|)(1 + |ξ|p ).

(1.3.35) The following result is proved in [Lieberman (1988), Th 1] with μ(x) ≡ κ where κ is either 0 or 1 and extended under the form stated here in [Porretta, V´eron (2009), Th A1]. Theorem 1.3.10 Let Ω be a bounded domain in RN with a C 1,δ boundary for some δ ∈ (0, 1]. Assume p > 1 and assumptions (1.3.35) hold. Then for any M > 0 there exist a positive constant c = c(λ, Λ, β, Ω, p, M ) and α ∈ (0, 1] such that if u is a bounded solution of (1.3.34) such that u L∞ ≤ M , then u ∈ C 1,γ (Ω) and more precisely u C 1,α (Ω) ≤ c. 1.3.3

(1.3.36)

Maximum principle

Two important tools in the study of second order elliptic equations are the strong maximum and the Hopf boundary lemma; they deal with the linear inequality in a domain Ω ⊂ RN . Lu = −

N i,j=1

∂2u ∂u + bi (x) + c(x)u ≥ 0. ∂xi ∂xj ∂x i i=1 N

aij (x)

(1.3.37)

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We assume that the functions aij , bj and c are measurable and locally bounded and that L is uniformly elliptic in the sense that there exists α > 0 such that for almost all x ∈ Ω, N

aij (x)ξi ξj ≥ α |ξ|2

∀ξ = (ξ1 , ..., ξN ) ∈ RN .

(1.3.38)

i,j=1

The strong maximum principle is well known in the case c(x) ≥ 0, see [Protter, Weinberger (1967), Th 7], but it is noticed in [Serrin (1971)] that it also holds with no sign assumption on c(x) (see also [Gidas, Ni, Nirenberg (1979), p. 212]). Proposition 1.3.11 Assume u ∈ C 2 (Ω) is a nonnegative solution of (1.3.38). If there exists x0 ∈ Ω such that u(x0 ) = 0, then u ≡ 0 in Ω. Hopf boundary lemma is the following, Proposition 1.3.12 Assume u ∈ C 2 (Ω) is a nonnegative nontrivial solution of (1.3.38). Let x0 ∈ ∂Ω be such that u ∈ C(Ω∪{x0 }) and u(x0 ) = 0, then if there exists a ball Br (a) ⊂ Ω such that x0 ∈ ∂Br (a), then for any unit vector n at x0 , pointing outward Ω, there holds lim sup t→0+

u(x0 + tn) − u(x0 ) < 0. t

(1.3.39)

Since Proposition 1.3.11 and Proposition 1.3.12 are of a local nature, they also hold in the equations are set in a subdomain S of a Riemannian manifold (M, g) and n is a unit vector in Tx0 M pointing outward S. These two results have been extended in the nonlinear framework [Vazquez (1984), Th 5]. The proof is an adaptation of the one in the linear case. Proposition 1.3.13 Let Ω ⊂ RN be a bounded domain. If A : Ω × R × RN → RN and B : Ω × R × RN → R are Caratheodory functions which satisfy (1.3.35) for some p > 1. Assume also that the function C in (1.3.35)-(iv) satisﬁes C(0) = 0 and either C(s) = 0 for 0 < s ≤ s0 for some s0 > 0, or C(s) > 0 for s > 0 and 1 ds

= ∞, (1.3.40) p J(s) 0 s C(τ )dτ . If u ∈ C 1 (Ω)∩W 1,∞ (Ω) is a nonnegative solution where J(s) = 0

of (1.3.28) such that A(., u, ∇u) ∈ L2 (Ω) which does not vanish identically in Ω, it is positive everywhere in Ω. If we assume moreover that u ∈

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C 1 (Ω ∪ {x0 }) for some x0 ∈ ∂Ω and that ∂Ω satisﬁes the interior sphere condition at x0 , then ∂u (x0 ) < 0, ∂n

(1.3.41)

where n is any outward unit vector at x0 to a sphere B ⊂ Ω such that x0 ∈ ∂B. 1.4

Existence result

In this section we present some useful existence results for solutions of quasilinear equations associated to monotone operators between Banach spaces and Leray-Lions operators. 1.4.1

Method of monotonicity

We assume here that V is a real separable and reﬂexive Banach space with norm . V and topological dual V with associated norm . V . A mapping A : V → V is bounded if it transforms bounded sets into bounded sets. Theorem 1.4.1 Let v → A(v) be a bounded operator from V to V , continuous from every ﬁnite dimensional subspace of V into V endowed with the weak topology. Assume A is coercive i.e. lim

v V →∞

A(v)(v) = ∞, v V

(1.4.1)

and monotone i.e. (A(v) − A(v ))(v − v ) ≥ 0

∀(v, v ) ∈ V × V.

(1.4.2)

Then A is onto. Proof. Step 1. Assume V is ﬁnite dimensional. Then, up to changing the norms, we can assume that V = V is endowed with an Euclidean structure and (1.4.1) remains valid. Thanks to (1.4.1) for any k > 0 there exists R = Rk > 0 such that A(v)(v) = A(v), v ≥ kR

∀v ∈ ∂BR .

(1.4.3)

Let θ ∈ [0, 1]. If v ∈ ∂BR , there holds

θA(v) + (1 − θ) k v, v ≥ kR =⇒ θA(v) + (1 − θ) k v ≥ k, (1.4.4)

R R V

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k which implies that there is no solution of θA(v) + (1 − θ) R v = f on ∂BR if f V < k. Because of homotopy invariance, the degree of the restriction of k θA + (1 − θ) Id to BR relative to f ∈ Bk is the same for all θ ∈ [0, 1] and R it is one. If θ = 1, it implies that there exists v ∈ BR such that A(v) = f .

Step 2. Approximate solutions. Since V is separable there exists a sequence {wm }m∈N∗ ⊂ V of linearly independent vectors with the following property: " if Vm is the subspace spanned by w1 , ..., wm , then m Vm is dense in V . Let f ∈ V . We claim that for any m ∈ N∗ there exists um ∈ Vm such that A(um )(w) = f (w)

∀w ∈ Vm .

(1.4.5)

Indeed, we introduce m elements θ1 , ..., θm of V such that θj (wi ) = δij , denote by Vm the space generated by the θj for j = 1, ..., m and deﬁne the linear operator Pm from V to V by Pm (v ) =

m

v (wj )θj

∀v ∈ V .

j=1

If we set Bm (v) = Pm (A(v)), then (1.4.5) is equivalent to Bm (um ) = Pm (f ). Since Bm (v)(v) = A(v)(v) the coercivity of Bm in Vm is ensured, therefore there exists um satisfying (1.4.5). Step 3. Convergence. By (1.4.5) |A(um )(um )| = |f (um )| ≤ um V f V , which implies that um V ≤ c for some c > 0 by the coercivity assumption. Since V is reﬂexive and A is bounded, there exist u ∈ V , φ ∈ V and a subsequence {mk } such that u mk u

and

A(umk ) φ

weakly.

Using (1.4.5) we obtain f (v) = φ(v) for all v ∈ Vm and all m ∈ N∗ . Since " the sequence of Vm is increasing and Vm is dense in V we infer f = φ. Let v ∈ Vmk0 , then for mk ≥ mk0 , we have from (1.4.2) (A(umk ) − A(v))(umk − v) = A(umk )(umk − v) − A(v)(umk − v) ≥ 0.

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Hence f (umk − v) − A(v)(umk − v) ≥ 0 by (1.4.5), therefore f (u − v) − A(v)(u − v) = (f − A(v)(u − v) ≥ 0

∀v ∈ Vmk0 .

(1.4.6)

" Since k0 is arbitrary, using again the density of Vm , we obtain that (1.4.6) is valid for any v ∈ V . Now we ﬁx w ∈ V , λ > 0 and take v = u − λw, then (f − A(u − λw)(w) ≥ 0. Since A(u−λw) A(u) weakly in V as λ → 0, we derive (f −A(u)(w) ≥ 0 for all w ∈ V , which implies f = A(u). The next variant has several applications to quasilinear equations. Theorem 1.4.2 Let v → A(v) be a bounded operator from V to V , continuous from every ﬁnite dimensional subspace of V into V endowed with the weak topology and coercive in the sense of (1.4.1). Assume also that ˜ u) = A(u) there exists a bounded mapping A˜ : V × V → V such that A(u, with the following properties: ˜ tv0 ) is continuous from (i) For all (u, v0 ) ∈ V × V the mapping t → A(u, R into V endowed with the weak topology, and A˜ is monotone in its second argument, i.e. ˜ u) − A(u, ˜ v))(u − v) ≥ 0 (A(u,

∀(u, v) ∈ V × V.

(1.4.7)

˜ n , un ) − A(u ˜ n , u))(un − (ii) If {un } ⊂ V converges weakly to u and (A(u ˜ v) weakly in V for all v ∈ V . ˜ n , v) A(u, u) → 0, then A(u ˜ n , v) v weakly in V , (iii) If {un } ⊂ V converges weakly to u and A(u then ˜ n , v)(un ) → v (u). A(u Then A is onto. Remark. A mapping which is continuous on any straight line is called hemi˜ v) is hemicontinuous from continuous . Therefore for any u ∈ V , v → A(u, V to V -weak. Application. Let 1 < p < ∞, Ω be a bounded domain in RN with a Lipschitz continuous boundary and V a subspace of W 1,p (Ω) closed for the norm . W 1,p , which contains W01,p (Ω). Let B : Ω × R → R and

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Aj : Ω × R × RN → R, j = 1, ..., N , be Caratheodory functions. We denote A = (A1 , ..., AN ) and we consider the following assumptions (i) |B(x, r)| ≤ c0 |r|p−1 + k(x) ∀(x, r) ∈ Ω × R ∀(x, r, ξ) ∈ Ω × R × RN , (ii) |A(x, r, ξ)| ≤ c |r|p−1 + |ξ|p−1 + k(x) (1.4.8) where c > 0, c0 ≥ 0 and k ∈ Lp (Ω). Thanks to these assumptions, for any v ∈ V the functions Aj (x, v, ∇v) belong to Lp (Ω). Then for any (u, v) ∈ V × V the integral (A(x, u, ∇u), ∇v + B(x, u)v) dx, (1.4.9) a(u, v) = Ω

is well deﬁned. Furthermore v → a(u, v) is a continuous linear mapping over V , therefore ∀v ∈ V, where A(u) ∈ V .

a(u, v) = A(u)(v)

(1.4.10)

For f ∈ V , the following equivalence holds: a(u, v) = f (v) ∀v ∈ V ⇐⇒ A(u) = f.

(1.4.11)

The next result is easy to prove by using Theorem 1.4.1 and Theorem 1.4.2 (see e.g. [Leray, Lions (1965), Th. 2]). Corollary 1.4.3 and all r ∈ R

Assume (1.4.8) holds. If A satisﬁes for almost all x ∈ Ω

A(x, r, ξ) − A(x, r, ξ ), ξ − ξ > 0

∀(ξ, ξ ) ∈ RN × RN , ξ = ξ , (1.4.12)

and A(x, r, ξ), ξ ≥ c |ξ|p − c2

∀ξ ∈ RN ,

(1.4.13)

where c > 0 and c2 are independent of x and r, then there exists a real number λ0 such that for any λ > λ0 and any f ∈ V one can ﬁnd u ∈ V such that ∀v ∈ V. (1.4.14) a(u, v) + λ |u|p−2 uv = f (v) Ω

An operator A such that (1.4.8)-(ii), (1.4.11) and (1.4.13) are satisﬁed is called a second order Leray-Lions operator. The number λ0 depends both on c0 , p and Ω.

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1.4.2

37

Method of super and sub solutions

In this section it is assumed that p > 1 and Ω is a domain in RN or on a Riemannian manifold, not necessarily bounded. We give existence results with precise two-sided estimates for some solution of equation (1.3.2) in presence of a supersolution and a subsolution. Concerning B, we assume the existence of an increasing function C from R+ to R+ such that for almost all x in Ω and all r ∈ R and ξ ∈ RN , there holds |B(x, r, ξ)| ≤ C(|r|)(1 + |ξ|p ).

(1.4.15)

By a solution of (1.3.1) we mean a function u ∈ W01,p (Ω) ∩ L∞ (Ω) which satisﬁes −div A(., u, ∇u) + B(., u, ∇u) = 0

in D (Ω).

(1.4.16)

Deﬁnition 1.5 A function ζ ∈ W 1,p (Ω) ∩ L∞ (Ω) verifying ζ ≤ 0 in a neighborhood of ∂Ω is a subsolution of problem (1.4.16) above if −div A(., ζ, ∇ζ) + B(., ζ, ∇ζ) ≤ 0

in D (Ω).

(1.4.17)

Similarly ζ ∈ W 1,p (Ω) ∩ L∞ (Ω) verifying ζ ≥ 0 in a neighborhood of ∂Ω is a supersolution of problem (1.4.16) above if −div A(., ζ, ∇ζ) + B(., ζ, ∇ζ) ≥ 0

in D (Ω).

(1.4.18)

The next result has many applications [Boccardo, Murat, Puel (1984)]. Theorem 1.4.4 Assume p > 1, A is a second order Leray-Lions operator, i.e. (1.4.8)-(ii), (1.4.12) and (1.4.13) hold, and B satisﬁes (1.4.15). If there exist a supersolution φ and a subsolution ψ of (1.4.16), belonging to W 1,∞ (Ω) and such that ψ ≤ φ, then there one can ﬁnd a solution u of of (1.4.16) satisfying ψ ≤ u ≤ φ.

Proof. Let us deﬁne the operator A from W01,p (Ω) into W −1,p (Ω) by A(v)(ζ) = A(x, v, ∇v).∇ζdx ∀(v, ζ) ∈ W01,p (Ω) × W01,p (Ω). Ω

(1.4.19) In [Boccardo, Murat, Puel (1984)], in (1.3.2)-(i) it is assumed that c1 = c2 = 0, however this assumption is unnecessary in order to ensure the

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coercivity of A i.e. A(v) = ∞. v W 1,p →∞ v W 1,p lim

(1.4.20)

Actually, we can even assume that c2 ∈ L1 (Ω). The proof is almost the same as the one of [Boccardo, Murat, Puel (1984), Th. 2.1], therefore we present a sketch of it and emphasize the modiﬁcations needed due to the fact that c2 = 0, and we simplify some points. Step 1. The truncated problem. The key idea is to truncate Aj and B by ψ and φ and deﬁne for almost all x ∈ Ω and all (r, ξ) ∈ R × RN ⎧ Aj (x, r, ξ) if ψ(x) < r < φ(x) ⎪ ⎪ ⎨ Hj (x, r, ξ) = Aj (x, ψ(x), ξ) (1.4.21) if r ≤ ψ(x) ⎪ ⎪ ⎩ if φ(x) ≤ r, Aj (x, ψ(x), ξ) and ⎧ B(x, r, ξ) ⎪ ⎪ ⎨ F (x, r, ξ) = B(x, ψ(x), ∇ψ(x)) ⎪ ⎪ ⎩ B(x, φ(x), ∇φ(x))

if ψ(x) < r < φ(x) if r ≤ ψ(x)

(1.4.22)

if φ(x) ≤ r.

Notice that F is no longer a Caratheodory function. If v ∈ W01,p (Ω) we set Hj (v, ∇v)(x) = Hj (x, v(x), ∇v(x)) and F (v, ∇v)(x) = F (x, v(x), ∇v(x)), for j = 1, ..., N and for almost all x ∈ Ω. We put H = (H1 , ..., HN ) and consider the truncated problem of ﬁnding u ∈ W01,p (Ω) solution of in D (Ω).

−div H(u, ∇u) + F (u, ∇u) = 0

(1.4.23)

It is clear that the Hj are Caratheodory functions and (1.4.8)-(ii), (1.4.12) and (1.4.13) holds, thus the operator H deﬁned by H(u) = −div H(u, ∇u) is a Leray-Lions operator. The operator F deﬁned by F (u) = F (u, ∇u) is deﬁned for all u ∈ W 1,p (Ω), it is continuous from W 1,p (Ω) into L1 (Ω), furthermore there exists C0 > 0 such that p

|F (u, ∇u)(x)| ≤ C0 (1 + |∇u(x)| ) a.e. in Ω,

(1.4.24)

(see [Boccardo, Murat, Puel (1984), Lemmas 3.1,3.2]). Step 2. Two-sided estimates of solutions of (1.4.23). If u is a solution of (1.4.23), the function (ψ − u)+ belongs to W01,p (Ω), thus for any m >

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(u − ψ)+,m = inf{m, (ψ − u)+ } it is an element of W01,p (Ω) ∩ L∞ (Ω). Since ψ is a subsolution of (1.4.16) and A(., ψ, ∇ψ) = H(., ψ, ∇ψ), we have (H(ψ, ∇ψ) − H(u, ∇u), ∇(ψ − u)+,m (1.4.25) Ω +(B(ψ, ∇ψ) − F (u, ∇u))(ψ − u)+,m ) dx ≤ 0. Deﬁning ω0 = {x ∈ Ω : ψ(x) − u(x) ≥ 0}, and ω0m = {x ∈ Ω : 0 ≤ ψ(x) − u(x) ≤ m} ⊂ ω0 , then (ψ − u)+,m = 0

a.e. on Ω \ ω0 ,

and ∇(ψ − u)+,m = 0

a.e. on Ω \ ω0m .

By the deﬁnition of H and F , there holds a.e. on ω0 (and thus on ω0m ) (i)

A(x, u(x), ∇u(x)) = A(x, ψ(x), ∇u(x))

(ii)

A(x, ψ(x), ∇ψ(x)) = A(x, ψ(x), ∇ψ(x))

(iii)

F (x, u(x), ∇u(x)) = B(x, ψ(x), ∇ψ(x)).

Therefore B(ψ, ∇ψ) − F (u, ∇u))(ψ − u)+,m dx Ω

B(ψ, ∇ψ) − F (u, ∇u))(ψ − u)+,m dx = 0,

= ω0

and (1.4.25) becomes H(ψ, ∇ψ) − H(u, ∇u), ∇(ψ − u)+,m dx ≤ 0. ω0m

By (1.4.12) ∇(ψ − u)+,m = 0 a.e. on ω0m , which implies (ψ − u)+,m = 0 for all m > 0 and ﬁnally ψ ≤ u a.e. in Ω. Similarly u ≤ φ. Moreover it is classical that ∇u = ∇ψ (resp. ∇u = ∇φ) a.e. on the set where u and ψ (rest. u and φ) coincide. This implies that u is a solution of (1.4.16).

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Step 3. Approximate solutions. For k > 0 and (x, r, ξ) ∈ Ω × R × RN , we set # F (x, r, ξ) if |F (x, r, ξ)| ≤ k Fk (x, r, ξ) = (1.4.26) k sign(F (x, r, ξ)) if |F (x, r, ξ)| > k, and we consider the problem of ﬁnding v = vk ∈ W01,p (Ω) −div H(v, ∇v) + Fk (v, ∇v) = 0

in D (Ω).

(1.4.27)

˜ on W 1,p (Ω) × W 1,p (Ω) by We introduce the operator H 0 0 ˜ w) = −div H(v, ∇w) + Fk (v, ∇v). H(v, ˜ v) = −div H(v, ∇v) + Fk (v, ∇v). The ˜ w) ∈ W −1,p (Ω) and H(v, Then H(v, ˜ mapping H is bounded. It is monotone and hemicontinuous in its second variable. If a sequence {un } ⊂ W01,p (Ω) converges weakly to u and satisﬁes ˜ n , u))(un − u) ˜ n , un ) − H(u (H(u = H(un , ∇un ) − H(un , ∇u), ∇(un , u)dx → 0, Ω

then for all w ∈ W01,p (Ω) ˜ ˜ v)(w). H(un , v)(w) = H(un , ∇v), ∇wdx → H(u, ∇v), ∇wdx = H(u, Ω

Ω

Finally, if a sequence {un } ⊂ W01,p (Ω) converges weakly to u and ˜ n , v) v weakly in W −1,p (Ω), equivalently H(u H(un , ∇v), ∇wdx → v (w) = − V , ∇wdx, Ω

Ω

where V = (v1 , ..., vN ) ∈ (Lp (Ω))N , then H(un , ∇v) → H(u, ∇v) in p L (Ω) because of (1.4.8)-(ii) and Rellich-Kondrachov’s compactness theorem. Therefore ˜ n , v)(un ) = H(un , ∇v), ∇un dx → − V , ∇udx = v (u). H(u Ω

Ω

˜ satisﬁes the assumptions of Theorem 1.4.2 and there exists vk ∈ Thus H 1,p W0 (Ω) solution of (1.4.27).

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Step 4. Two-sided estimates of approximate solutions. We assume k > k0 where k0 = sup { B(., φ, ∇φ) L∞ , B(., ψ, ∇ψ) L∞ } .

(1.4.28)

This is precisely here that where we use the assumption that φ and ψ belong to W 1,∞ (Ω). Then Fk (., φ, ∇φ) = F (., φ, ∇φ) = B(., φ, ∇φ), and Fk (., ψ, ∇ψ) = F (., ψ, ∇ψ) = B(., ψ, ∇ψ). If vk ∈ W01,p (Ω) satisﬁes (1.4.27), then −div H(., ψ, ∇ψ) + div H(., vk , ∇vk ) + F (., ψ, ∇ψ) − Fk (., vk , ∇vk ) ≤ 0. As in Step 2 we multiply by (ψ − vk )m + = inf{m, (ψ − vk )+ }. Since F (., vk , ∇vk ) = F (., ψ, ∇ψ) a.e. on the set of x ∈ Ω where vk (x) ≤ ψ(x) and since k > k0 , we get F (., vk , ∇vk ) = F (., ψ, ∇ψ) = B(., ψ, ∇ψ). The proof follows by the strict monotonicity. Letting m → 0 yields ψ ≤ vk . Similarly vk ≤ φ. Step 5. W01,p -bounds on approximate solutions. For k ≥ k0 (see (1.4.28)) and t > 0 we set 2

ηk = vk etvk .

(1.4.29)

Since ψ ≤ vk ≤ φ, ηk ∈ W01,p (Ω). Therefore 2 2 H(vk , ∇vk ), ∇vk etvk dx + 2t H(vk , ∇vk ), ∇vk vk2 etvk dx Ω

Ω

2

Fk (v, ∇v)vk etvk dx = 0.

+ Ω

Using (1.4.13) and (1.4.15) we get, with C ∗ = max{ ψ L∞ , φ L∞ }, 2 2 2 p p p c |∇vk | etvk dx + 2ct |∇vk | vk2 etvk dx − C ∗ |∇vk | vk etvk dx Ω

Ω

Ω

≤ Ω

(1 + 2tvk2 )c2 e

2 tvk

dx + C ∗

2

vk etvk dx. Ω

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42

The right-hand side of the above inequality is bounded if t is so. Because c + 2ctvk2 − C ∗ vk ≥ c −

c C∗ 2 ≥ 8ct 2

∗2

if we choose t = C4c2 , we obtain that the norm of vk in W01,p (Ω) is bounded independently of k. Step 6. Convergence when k → ∞. By Step 4 and Step 5, there exist a subsequence of {vk }, still denoted by {vk } and a function u ∈ W01,p (Ω) ∩ L∞ (Ω) such that in Lq (Ω) ∀1 ≤ q < ∞, vk → u ψ ≤ u ≤ φ a.e. in Ω ∇vk ∇u in Lp (Ω) − weak.

(i) (ii) (iii)

We have from (1.4.23), H(vk , ∇vk ), ∇(vk − u)dx + Fk (vk , ∇vk )(vk − u)dx = 0. Ω

(1.4.30)

(1.4.31)

Ω

This can be re-written as H(vk , ∇vk ) − H(vk , ∇u), ∇(vk − u)dx + H(vk , ∇u), ∇(vk − u)dx Ω

Ω

Fk (vk , ∇vk )(vk − u)dx = 0.

+ Ω

(1.4.32)

By (1.4.30) and (1.4.15) H(vk , ∇u), ∇(vk − u)dx → 0, Ω

and

Fk (vk , ∇vk )(vk − u)dx → 0, Ω

when k → ∞. This implies H(vk , ∇vk ) − H(vk , ∇u), ∇(vk − u)dx → 0. Ω

Because H(vk , ∇vk ) − H(vk , ∇u), ∇(vk − u) ≥ 0, there exists a subsequence {k } such that H(x, vk (x), ∇vk (x)) − H(x, vk (x), ∇u(x)), ∇(vk (x) − u(x)) → 0, (1.4.33)

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as k → ∞, for all x ∈ Ω \ E where E ⊂ Ω has zero Lebesgue measure. We can also assume that vk (x) → u(x) for all x ∈ Ω \ E. Let x ∈ Ω \ E, ξ ∗ be any cluster point of {∇vk (x)} and set ξ = ∇u(x). Notice that because of (1.4.8)-(ii), (1.4.13) there holds p

c |∇vk (x)| ≤ c2 (x) + k (x) + F (x, vk (x), ∇u(x)), ∇u(x) − F (x, vk (x), ∇u(x)), ∇vk (x) − F (x, vk (x), ∇vk (x)), ∇u(x) p−1 + k(x) (|∇vk (x)| + |∇u(x)|) ≤ c2 (x) + k (x) + 3c |vk (x)| p p−1 p−1 , |∇vk (x)| + |∇u(x)| |∇vk (x)| + c |∇u(x)| + |∇u(x)| (1.4.34) where k (x) denotes the left-hand side of (1.4.33). This yields c |ξ ∗ |p ≤ c2 (x) + 3c (sup{|φ(x)|, |ψ(x)|})p−1 + k(x) (|ξ ∗ | + |ξ|) + c |ξ|p + |ξ|p−1 |ξ ∗ | + |ξ| |ξ ∗ |p−1 . (1.4.35) Therefore |ξ ∗ | < ∞. Then, from (1.4.33) H(x, u(x), ξ ∗ ) − H(x, u(x), ξ), ξ ∗ − ξ) = 0.

(1.4.36)

This implies that ξ ∗ = ξ, thus ∇vk (x) → ∇u(x) for almost all x in Ω. Using Young’s inequality we derive from (1.4.34) p p M1 |∇vk (x)| ≤ c2 (x) + M2 (sup{|φ(x)|, |ψ(x)|}) + (k(x))p (1.4.37) p + M3 |∇u(x)| + k (x). Since k → 0 in L1 (Ω), there exist Φ ∈ L1 (Ω), Φ ≥ 0 and a subsequence p {k } such that |k | = k ≤ Φ. This implies that the sequence {|∇vk (x)| } is uniformly integrable and thus ∇vk → ∇u

in Lp (Ω),

(1.4.38)

by the Vitali convergence theorem. Therefore H(vk , ∇vk ) → H(u, ∇u) in Lp (Ω) and F (vk , ∇vk ) → F (u, ∇u) in L1 (Ω). By Step 2, u is a solution of (1.4.16). Remark. It is possible to prove (see [Boccardo, Murat, Puel (1984), Prop. 3.8]) that ∇vk , and thus ∇u remains bounded is some Lqloc (Ω) for some q > p.

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The previous theorem can be extended to non-homogeneous boundary value problems. Corollary 1.4.5 Assume p > 1, Ω ⊂ RN is a bounded domain, and the functions A and B satisfy the assumptions of Theorem 1.4.4. If there exist a supersolution φ and a subsolution ψ of (1.4.16), belonging to W 1,∞ (Ω) and such that ψ ≤ φ, then for any χ ∈ W 1,∞ (Ω) such that ψ ≤ χ ≤ φ there exists a function u ∈ W 1,p (Ω) of (1.4.16) satisfying ψ ≤ u ≤ φ and u ˜ = u − χ ∈ W01,p (Ω). Proof. We put ˜ r, ξ) = A(x, r + χ(x), ξ + ∇χ(x)), A(x, and ˜ r, ξ) = B(x, r + χ(x), ξ + ∇χ(x)). B(x, Then |B(x, r + χ(x), ξ + ∇χ(x))| ≤ C(|r + χ(x)|)(1 + |ξ + ∇χ(x)|p ) p

≤ 2p (1 + ∇χ L∞ )C (|r| + χ L∞ ) (1 + |ξ|p ) , hence (1.4.15) holds with C replaced by p ˜ C(r) = 2p (1 + ∇χ L∞ )C (|r| + χ L∞ ) .

Next ˜ r, ξ) − A(x, ˜ r, ξ ), ξ − ξ A(x, = A(x, r + χ(x), ξ + ∇χ(x) − A(x, r + χ(x), ξ + ∇χ(x), ξ − ξ >0

if ξ = ξ .

Furthermore ˜ A(x, r, ξ) ≤ c |r + χ(x)|p−1 + |ξ + ∇χ(x)|p−1 + k(x) ≤ 2p c |r|p−1 + |ξ|p−1 + k(x) + |χ(x)|p−1 + |∇χ(x)|p−1 . Finally p

˜ r, ξ), ξ ≥ c |ξ + ∇χ(x)| − c2 (x) A(x, ≥ 2−p c |ξ| − (c |∇χ(x)| + c2 (x)). p

p

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˜ there holds Set ψ˜ = ψ − χ and φ˜ = φ − χ. By the deﬁnition of A˜ and B, ˜ ψ, ˜ ∇ψ) ˜ + B( ˜ ψ, ˜ ∇ψ) ˜ ≤0 −div A(

in D (Ω),

(1.4.39)

˜ ∇φ) ˜ + B( ˜ ∇φ) ˜ ≥0 ˜ φ, ˜ φ, −div A(

in D (Ω).

(1.4.40)

and

˜ there exists u Since ψ ≤ u ≤ φ is equivalent to ψ˜ ≤ u ˜ ≤ φ, ˜ ∈ W01,p (Ω) solution of ˜ u, ∇˜ ˜ u, ∇˜ −div A(˜ u) + B(˜ u) = 0

in D (Ω),

(1.4.41)

˜ Hence u = u which satisﬁes ψ˜ ≤ u ˜ ≤ φ. ˜ + χ fulﬁlls the requirements of Theorem 1.4.4. Remark. If the function χ is only given on ∂Ω, then the function X(x) = max{χ(z) − ∇χ L∞ (∂Ω) |x − z| : z ∈ ∂Ω}

∀x ∈ RN , (1.4.42)

is a Lipschitz continuous extension of χ to RN which satisﬁes ∇X L∞ (RN ) = ∇χ L∞ (∂Ω) . Theorem 1.4.6 Let p > 1, Ω ⊂ RN be a domain non necessarily bounded and A : Ω × R × RN → RN and B : Ω × R × RN → R be Caratheodory functions. We assume that A is strictly monotone, i.e. (1.4.12) holds, and for any bounded subdomain G ⊂ G ⊂ Ω there exist positive constants cG , cG , an increasing function ΓG : R+ → R+ and c2,G ∈ L1 (G) and kG ∈ Lp (G) such that ∀(x, r, ξ) ∈ G × R × RN (i) |A(x, r, ξ)| ≤ cG |r|p−1 + |ξ|p−1 + kG (x) (ii)

A(x, r, ξ), ξ ≥ cG |ξ|p − c2,G (x)

∀(x, r, ξ) ∈ G × R × RN

∀(x, r, ξ) ∈ G × R × RN. (1.4.43) If there exist a supersolution φ and a subsolution ψ of (1.4.16), belonging 1,∞ 1,p (Ω) and such that ψ ≤ φ, then there exists a function u ∈ Wloc (Ω) to Wloc satisfying ψ ≤ u ≤ φ and solution of (1.4.16). (iii)

|B(x, r, ξ)| ≤ ΓG (|r|)(1 + |ξ|p )

Proof. Let {Ωn } be an increasing sequence of bounded smooth subdomains of Ω such that Ωn ⊂ Ωn+1 . Set χ = 12 (φ+ψ). It follows from Corollary 1.4.5 ∞ (Ωn ) to that for each n ∈ N∗ , there exists a solution un ∈ W 1,p (Ωn ) ∩ L −div A(., un , ∇un ) + B(., un , ∇un ) = 0

in D (Ωn ),

(1.4.44)

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46

satisfying ψ ≤ un ≤ φ in Ωn and such that un − χΩn ∈ W01,p (Ωn ). Step 1. A priori estimates. Let G ⊂ G ⊂ Ω be a bounded open domain and nG ∈ N such that G ⊂ Ωn for all n ≥ nG . Let θ ∈ C0∞ (RN ), 0 ≤ θ ≤ 1, 2 and θ = 1 on ω ⊂ G. Set ηn = un etun , t > 0. A(un , ∇un ), ∇(θηn )dx + B(un , ∇un )θηn dx = 0. G

G

Then 2 2 A(un , ∇un ), ∇un etnvn θp dx + 2t A(un , ∇un )), ∇un u2n etnn θp dx G

G

2

2

A(un , ∇un ), ∇θun etnvn θp−1 dx +

+p

B(un , ∇un )un etun θp dx = 0.

G

G

By (1.4.43)-(ii) the sum of ﬁrst two terms on the left is larger than 2 p p tu2n 2 cG (1 + 2tun ) |∇un | θ e dx − cG (1 + 2tu2n )c2,G θp etun dx. G

G

If mG = max{ φ L∞ (G) , ψ L∞ (G) }, there holds B(un , ∇un )un etu2n θp dx ≤ ΓG (mG ) (1 + |∇un |p ) un etu2n θp dx. G

G

By Young’s inequality, with > 0, 1 2 2 p tnvn p−1 p A(un , ∇un ), ∇θun e θ dx ≤ p |∇θ| un etun dx G G 2 p |A(un , ∇un )| un etun θp dx + (p − 1)p G

+ pp cG p p

1 ≤ p

p

2

|∇θ| un etun dx G

2 p p |∇un | + |un | + kGp un etun θp dx.

G

Combining the above inequalities, we obtain 2 p p cG 1 + 2tu2n − pp cG p + ΓG (mG ) un |∇un | θp etun dx G

≤

1 p p p + pp cG p |un | + kGp |∇θ| p G 2 1 + 2tu2n θp c2,G etun dx, + cG

G

2

etun dx

(1.4.45)

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which yields, using the fact that |un (x)| ≤ mG in G and 0 ≤ θ ≤ 1, 2 p cG 1 + 2tu2n − pp cG p + ΓG (mG ) un |∇un |p θp etun dx G

≤

∇θ L∞ 2 p p p etmG |G| + pp c m G G p 2 kGp (x) + cG (1 + tm2G )c2,G (x) dx. + etmG

(1.4.46)

G

Hence the right-hand side of (1.4.46) is dominated by a constant RG (t)

. which depends on t, G, mG (i.e. φG and ψG ), θ and kGp + c2,G L1 (G)

If we ﬁx

t := t0 =

2 pp cGp p + ΓG (mG ) 4c2G

,

we obtain that

2 p p pp c + Γ G (mG ) G cG p 2tcG u2n − pp cG p + ΓG (mG ) un +cG ≥ cG − . ≥ 8cG t0 2 This implies

p

|∇un | θp dx ≤ RG (t0 )

cG

∀n ≥ nG .

(1.4.47)

G

Step 2. Convergence. Using Cantor’s diagonal sequence construction, we infer from Step 1 that there exist and increasing function τ : N → N and 1,p u ∈ L∞ loc (Ω) ∩ Wloc (Ω) such that (i) (ii) (iii) (iv)

uτ (n) → u ψ≤u≤φ uτ (n) → u ∇uτ (n) ∇u

in Lqloc (Ω) ∀1 ≤ q < ∞ a.e. in Ω a.e. in Ω in Lploc (Ω) - weak.

(1.4.48)

Let G ⊂ G ⊂ Ω and θ ∈ C0∞ (Ω) be as in Step 2. Then, as in the proof of Theorem 1.4.4-Step 6, B(uτ (n) , ∇uτ (n) )θ(uτ (n) − u)dx + A(uτ (n) , ∇uτ (n) ), ∇(uτ (n) − u)θdx Ω

+ Ω

Ω

A(uτ (n) , ∇uτ (n) ), ∇θ(uτ (n) − u)dx = 0,

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48

which can be re-written as follows: A(uτ (n) , ∇uτ (n) ) − A(uτ (n) , ∇u), ∇(uτ (n) − u)θdx Ω

A(uτ (n) , ∇u), ∇(uτ (n) − u)θdx

+ Ω

A(uτ (n) , ∇uτ (n) ), ∇θ(uτ (n) − u)dx

+

Ω

+

Ω

B(uτ (n) , ∇uτ (n) )(uτ (n) − u)θdx = 0.

Let us denote by a1 (n), a2 (n), a3 (n) and a4 (n) the four terms, in this order, of the above inequality. Using (1.4.48), we see that a2 (n), a3 (n) and a4 (n) tend to 0 when n → ∞. Therefore, if we set 1,n (.) = A(., uτ (n) , ∇uτ (n) ) − A(., uτ (n) , ∇u), ∇(uτ (n) − u)θ(.), then 1,n (x)) > 0 on the set of x such that ∇(uτ (n) − u)(x)θ(x) = 0. Estimate (1.4.34) takes the form p cG ∇uτ (n) ≤ c2,G (x) + 1,n (x) p−1 + kG (x) ∇uτ (n) (x) + |∇u(x)| + 3cG uτ (n) (x) p−1 p p−1 ∇uτ (n) (x) + |∇u(x)| ∇uτ (n) (x) . + cG |∇u(x)| + |∇u(x)| (1.4.49) If x is a regular point for the convergence a.e. everywhere, ∇u(x) = ξ and ξ ∗ is any cluster point of the sequence {∇uτ (n)}, we obtain again that ξ ∗ is ﬁnite and A(x, u(x), ∇ξ ∗ ) − A(x, u(x), ξ), ξ ∗ − ξ)θ(x) = 0. This implies that ξ ∗ = ξ on the subset G of x ∈ G where θ(x) > 0. Since θ is arbitrary with support in G, we conclude that ∇uτ (n) (x) → ∇u(x) a.e. in G and hence a.e. in Ω. Finally, we use Young’s inequality and derive p from (1.4.49) the upper estimate of ∇uτ (n) , namely p M1,G ∇uτ (n) (x) ≤ c2,G (x) + M2,G (sup{|φ(x)|, |ψ(x)|})p + (kG (x))p + M3,G |∇u(x)|p + 1,n (x). (1.4.50) a subsequence {n Since there exist a function ΦG ∈ L1 (G) and } such p that |1,n | ≤ ΦG a.e. in G, we obtain that { ∇uτ (n ) (x) θ} is uniformly

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integrable in G and consequently ∇uτ (n ) − ∇up θdx → 0 as n → ∞.

(1.4.51)

G

This convergence holds for any function θ and the subsequence {n } may depend on θ. However, there exists a sequence {θn } ⊂ C0∞ (RN ) such that 0 ≤ θn ≤ 1 , supp θn ⊂ Ωn and θ ≡ 1 on Ωn−1 . Finally we can extract a diagonal sequence denoted by {˜ τ (n)} from {τ (n)} such that {uτ˜(n) } possesses the following properties (i) (ii) (iii) (iv)

uτ˜(n) → u ψ≤u≤φ uτ˜(n) → u ∇uτ˜(n) → ∇u

in Lqloc (Ω) ∀1 ≤ q < ∞, a.e. in Ω a.e. in Ω in Lploc (Ω).

(1.4.52)

This implies that u is a solution of (1.4.16) and it satisﬁes ψ ≤ u ≤ φ. The framework of second order Leray-Lions operators in RN can be adapted easily to a Riemannian manifold (M d , g). We assume that A : (x, r, ξ) ∈ M × R × T M → A(x, r, ξ) ∈ Tx M and B : (x, r, ξ) ∈ M × R × T M → B(x, r, ξ) ∈ R are Caratheodory vector ﬁelds and they satisfy (i) A(x, r, ξ), ξg ≥ c|ξ|p − c2 (x) (ii) |A(x, r, ξ)| ≤ c |ξ|p−1 + |r|p−1 + k(x)

∀(x, r, ξ) ∈ M × R × T M

(iii) |B(x, r, ξ)| ≤ Γ(|r|)(1 + |ξ|p )

∀(x, r, ξ) ∈ M × R × T M

∀(x, r, ξ) ∈ M × R × T M

ˆ ξ − ξ ˆg >0 (iv) A(x, r, ξ) − A(x, r, ξ), ˆ ∈ M × R × T M × T M s.t. ξ = ξ, ˆ ∀(x, r, ξ, ξ) (1.4.53) where c, c are positive constants, k ∈ Lp (M ), c2 ∈ L1 (M ) and Γ : R+ → R+ is an increasing function. The equation under consideration is the following −divg A(., u, ∇u) + B(., u, ∇u) = 0

in D (M ).

(1.4.54)

The next result is the exact analogue of Theorem 1.4.4 in a geometric setting. Theorem 1.4.7 Assume p > 1, (M d , g) is a complete Riemannian manifold and S ⊂ M is a bounded domain with a nonempty boundary ∂S and

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A and B satisfy (1.4.53). If there exist two functions ψ and φ in W 1,∞ (S)) such that ψ ≤ φ and (i) − divg A(., ψ, ∇ψ) + B(., ψ, ∇ψ) ≤ 0 in D (S), ψ ≤ 0 on ∂S (ii) − divg A(., φ, ∇φ) + B(., φ, ∇φ) ≥ 0 in D (S), φ ≥ 0 on ∂S, (1.4.55) then one can ﬁnd u ∈ W01,p (S) ∩ L∞ (S) such that ψ ≤ u ≤ φ in S, solution of (1.4.54) in D (S) (instead of D (M )). Another useful result is the following transposition of Theorem 1.4.6. Theorem 1.4.8 Assume p > 1, (M d , g) is a complete Riemannian manifold and S ⊂ M is a subdomain. Let A : M × R × T M → T M and B : M × R × T M → R be Caratheordory functions. We assume that A is strictly monotone, i.e. (1.4.53)-(iv) holds, and for any bounded subdomain G ⊂ G ⊂ S there exist positive constants cG , cG , an increasing function ΓG : R+ → R+ and c2,G ∈ L1 (G) and kG ∈ Lp (G) such that (i) |A(x, r, ξ)| ≤ cG |r|p−1 + |ξ|p−1 + kG (x) ∀(x, r, ξ) ∈ G × R ×T M (ii)

A(x, r, ξ), ξ ≥ cG |ξ|p − c2,G (x)

∀(x, r, ξ) ∈ G × R ×T M

(iii) |B(x, r, ξ)| ≤ ΓG (|r|)(1 + |ξ| )

∀(x, r, ξ) ∈ G × R ×T M. (1.4.56) 1,∞ If there exist two functions ψ and φ in Wloc (S)) such that ψ ≤ φ and p

(i) −divg A(., ψ, ∇ψ) + B(., ψ, ∇ψ) ≥ 0 in D (S) (ii) −divg A(., φ, ∇φ) + B(., φ, ∇φ) ≥ 0 in D (S),

(1.4.57)

1,p (S) ∩ L∞ then one can ﬁnd u ∈ Wloc loc (S) such that ψ ≤ u ≤ φ in S, solution of (1.4.54) in D (S).

If (M d , g) is a compact complete Riemannian manifold, then the dual of W 1,p (M ) is W −1,p (M ). The next statement is a variation of Theorem 1.4.4. Corollary 1.4.9 Assume p > 1, (M d , g) is a compact complete Riemannian manifold and the functions A and B satisfy (1.4.53). If there exist two functions ψ and φ in W 1,∞ (M ) such that ψ ≤ φ and (i)

p−1 ≤0 −divg A(., ψ, ∇ψ) + B(., ψ, ∇ψ) + δψ+

in D (M )

(ii)

−divg A(., φ, ∇φ) + B(., φ, ∇φ) − δφp−1 ≥0 −

in D (M ), (1.4.58)

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for some δ > 0, then one can ﬁnd u ∈ W 1,p (M ) ∩ L∞ (M ) solution of (1.4.54) such that ψ ≤ u ≤ φ. Proof. Since ∇u Lp is just a seminorm on W 1,p (M ), we indicate the modiﬁcations needed in order to follow the steps of the proof of Theorem 1.4.4. For 0 < σ ≤ δ, we consider the coercive problem −divg A(., u, ∇u) + B(., u, ∇u) + σ |u|

p−2

u=0

in D (M ).

(1.4.59)

and deﬁne Aσ from W 1,p (M ) into W −1,p (M ) by p−2 Aσ (v)(ζ) = A(., v, ∇v), ∇ζg + σ |u| uζ dvg ,

(1.4.60)

M

for (v, ζ) ∈ W 1,p (M ) × W 1,p (M ). The operator Aσ is coercive in the sense that it satisﬁes (1.4.20). The truncations of A and B by H and F are the same as in the proof of Theorem 1.4.4-Step 1 and the associated truncated problem is to ﬁnd u ∈ W 1,p (M ) such that −divg H(., u, ∇u) + F (., u, ∇u) + σ |u|

p−2

u=0

in D (M ).

(1.4.61)

Assuming that u is a solution of (1.4.60), we obtain, instead of (1.4.32), H(ψ, ∇ψ) − H(u, ∇u), ∇(ψ − u)+,m g M p−2

p−1 +σ(ψ+ − |u|

u)(ψ − u)+,m dvg

(1.4.62)

B(ψ, ∇ψ) − F (u, ∇u)(ψ − u)+,m dvg ≤ 0,

+ M

where (ψ − u)+,m = inf{m, (ψ − u)+ }. Then B(ψ, ∇ψ) − F (u, ∇u)(ψ − u)+,m dvg = 0. M

Denoting again ω0 = {x ∈ M : ψ(x) − u(x) ≥ 0} (resp. ω0m = {x ∈ M : m ≥ ψ(x) − u(x) ≥ 0}, we obtain H(ψ, ∇ψ) − H(u, ∇u), ∇(ψ − u)+,m g dvg ω0m

p−2

+σ ω0

p−1 (ψ+ − |u|

u)(ψ − u)+,m dvg ≤ 0,

which yields (ψ − u)+,m = 0 and thus ψ ≤ u. In the same way u ≤ φ and ﬁnally u is a solution of (1.4.59). Deﬁning Fk as in (1.4.26) for k > 0, we

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ﬁnd a solution v = vk,σ ∈ W 1,p (M ) of −div H(v, ∇v) + Fk (v, ∇v) + σ|v|p−2 v = 0

in D (M ).

(1.4.63)

Deﬁning k0 by (1.4.28) we infer that ψ ≤ vk,σ ≤ φ. In the same way as in Step 5 we derive that ∇vk,σ Lp is bounded independently of k and σ. Then, up to a subsequence, vk ,σ converges in any Lq (M ) for 1 ≤ q < ∞ and almost everywhere in M to some function uσ and ∇vk ,σ converges weakly to ∇uσ in Lp (M ). Since (1.4.33) is valid with vk replaced by vk ,σ , we deduce that ∇(vk ,σ − uσ ) Lp converges to 0. This implies that uσ is a solution of −div A(uσ , ∇uσ ) + B(uσ , ∇uσ ) + σ|uσ |p−2 uσ = 0

in D (M ),

(1.4.64)

and it satisﬁes ψ ≤ uσ ≤ φ. Furthermore ∇uσ Lp remains bounded independently of σ. Since uσ remains bounded, the same convergence method as the one used for letting k → ∞ applies. Therefore, up to a subsequence {σ } converging to 0, there exists u ∈ L∞ (M ) such that ∇u ∈ Lp (M ), ψ ≤ u ≤ φ with the property that uσ → u in any Lq (M ) (1 ≤ q < ∞) and a.e. in M , ∇uσ → ∇u in Lp (M ). At end u is a solution of (1.4.54). 1.5

The p-Laplace operator

Let Ω be a domain in RN . The p-Laplace is deﬁned by −Δp u = −div |∇u|

p−2

∇u,

(1.5.1)

where 1 < p < ∞. If f ∈ L1loc (Ω), solutions of −Δp u = f

in Ω,

(1.5.2)

1,p are usually understood in the weak sense, i.e. u ∈ Wloc (Ω) and (1.5.2) 1,p−1 holds in the sense of distribution in Ω. If p ≥ 2, u ∈ Wloc (Ω) is suﬃcient to give sense to this formulation in the sense of distributions. This is not the case if 1 < p < 2 and other deﬁnition will be given in Chapter 4.

An important property of the N -Laplace equation is its conformal invariance. Proposition 1.5.1

Assume Ω ⊂ RN is a domain which does not contain

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0 and u ∈ C 1 (Ω) satisﬁes (1.5.3) −ΔN u = f in Ω, x ˜ = {x : |x|−2 x ∈ Ω} deﬁned in Ω for some f , then v : x → v(x) = u |x| 2 satisﬁes x −2N ˜ in Ω. (1.5.4) −ΔN v = |x| f |x|2 This property will be very helpful to study quasilinear equations with N Laplacian in non-ﬂat domains and in particular to construct local supersolutions or subsolutions (see Chapter 3). 1.5.1

Comparison principles

In this particular case the results of Section I-3 are as follows: 1,p (Ω) is a weak supersolution (resp. Deﬁnition 1.6 A function u ∈ Wloc subsolution) of (1.5.2) if p−2 |∇u| ∇u.∇ζdx ≥ f ζdx ∀ζ ∈ Cc∞ (Ω), ζ ≥ 0, (1.5.5) Ω

Ω

resp.

|∇u|

Ω

p−2

∇u.∇ζdx ≤

f ζdx ∀ζ ∈ Cc∞ (Ω), ζ ≥ 0.

(1.5.6)

Ω

1,p Deﬁnition 1.7 A function u ∈ Wloc (Ω) ∩ C(Ω) is called p-harmonic if it is a weak solution of

−Δp u = 0

in Ω.

(1.5.7)

The requirement of continuity is not a restriction since any weak solution N 1,α . can be modiﬁed on a set with zero cR 1,p -capacity in order to become C 1,p (Ω) ∩ C(Ω) is called weakly pDeﬁnition 1.8 A function u ∈ Wloc superharmonic or a supersolution of (1.5.7) if it satisﬁes

−Δp u ≥ 0

in D (Ω).

(1.5.8)

It is weakly p-subharmonic, or a subsolution of (1.5.7), if −u is weakly p-superharmonic.

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An important property of the vector ﬁeld associated to the p-Laplace operator is the strict monotonicity inequality # p if p ≥ 2 22−p |X − Y | p−2 p−2 X − |Y | Y ).(X − Y ) ≥ (|X| 2 2 2 p−2 (p − 1) |X − Y | (1 + |X| + |Y | ) 2 , (1.5.9) for all X, Y ∈ RN (see [Lindqvist (2006)]). The following strict comparison principle will be used many times. Proposition 1.5.2 Let Ω ⊂ RN be a domain, p > 1 and c ∈ L∞ loc (Ω). Assume u and v belong to C 1 (Ω), satisfy −Δp u + cu ≤ 0

and − Δp v + cv ≥ 0

in D (Ω),

(1.5.10)

and ∇v never vanishes in Ω. If u ≤ v in Ω and if there exists x0 ∈ Ω such that u(x0 ) = v(x0 ), then u ≡ v in Ω. Proof. By the mean value theorem |∇u|

p−2

∂u ∂(u − v) p−2 ∂v − |∇v| = αij , ∂xi ∂xi ∂xi j

on the set of x ∈ Bδ (x0 ) where ∇u(x) = 0, with p−4 2 δij |ti ∇u + (1 − ti )∇v| αij = |ti ∇u + (1 − ti )∇v| ∂v ∂v ∂u ∂u ti . +(p − 2) ti + (1 − ti ) + (1 − ti ) ∂xi ∂xi ∂xj ∂xj At the tangency point of the graphs of u and v, ∇u(x0 ) = ∇v(x0 ) = 0, ∂v(x0 ) ∂v(x0 ) p−4 2 δij |∇v(x0 )| + (p − 2) . αij (x0 ) = |∇v(x0 )| ∂xi ∂xj The matrix A = (αij (x0 )) is symmetric and the inﬁmum of its spectrum is p−2 inf σ(A) = min{1, p− 2} |∇v(x0 )| . It remains positive deﬁnite in Bδ (x0 ) for δ small enough. If w = v − u, there holds ∂w ∂w αij + c+ w ≥ 0 in Bδ (x0 ). − ∂xj ∂xi i,j Thus w ≡ 0 in Bδ (x0 ). The completion of the proof follows by connectedness. A variant of this strong comparison principle is the following:

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Proposition 1.5.3 Let Ω ⊂ RN be a C 1 domain, p > 1 and c ∈ L∞ loc (Ω) Assume u and v belong to C 1 (Ω), satisfy (1.5.10), u ≤ v and ∇v never vanishes in Ω and there exists x1 ∈ ∂Ω such that u(x1 ) = v(x1 ). Then (i) either u ≡ v in Ω, ∂v ∂v (x1 ) < (x1 ). ∂n ∂n Proof. By Proposition 1.5.2, if there exists x0 ∈ Ω such that u(x0 ) = v(x0 ), then u and v coincide in Ω. If u < v in Ω, since the matrix A is uniformly positive deﬁnite in a neighborhood of x1 and v(x1 ) − u(x1 ) = 0, it follows by Hopf Lemma that (ii) or u < v in Ω and

∂v ∂v (x1 ) − (x1 ) < 0. ∂n ∂n Another approach of the comparison principle is to study the coincidence set between a supersolution and a subsolution [Guedda, V´eron (1988), Prop 2.1, 2.2]. Proposition 1.5.4 Let Ω ⊂ RN be a domain and p > 1. Assume u and v belong to C(Ω) ∩ W 1,p (Ω) and satisfy −Δp u = f

and − Δp v = g

in D (Ω)

(1.5.11)

where f, g ∈ Lp (Ω) satisfy f ≤ g. If u ≤ v in Ω, then either u < v in Ω or any connected component of the coincidence set C = {x ∈ Ω : u(x) = v(x)} has a nonempty intersection with ∂Ω. Proof. Assume K ⊂ Ω is a nonempty compact connected component of C and let G be a bounded domain such that K ⊂ G ⊂ G ⊂ Ω and G ∩ C = ∅. ¯ \ K and in particular v > uθ := u + θ for some θ > 0 on Then v > u in G ∂G. Since −Δp uθ + Δp v ≤ f − g and (uθ − v)+ has compact support in G, we derive p−2 p−2 |∇uθ | ∇uθ − |∇v| ∇v, ∇(uθ − v)dx ≤ 0, G∩uθ >v

which implies uθ ≤ v in G, thus K = ∅.

Proposition 1.5.5 Let Ω ⊂ RN be a bounded domain with a C 2 boundary and p > 1. Assume u and v belong to C 1 (Ω), vanish on ∂Ω and satisfy (1.5.11) where f, g ∈ Lp (Ω) satisfy 0 ≤ f ≤ g a.e. in Ω. If the set F = {x ∈ Ω : f (x) = g(x) a.e.} has zero measure, then we have the following:

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(i)

v(x) > u(x) ≥ 0

∀x ∈ Ω

(ii)

∂u ∂v (x) < (x) < 0 ∂n ∂n

∀x ∈ ∂Ω.

(1.5.12)

Proof. It follows from the assumptions that g is not zero a.e. zero thus v is not identically zero. By Vazquez maximum principle Proposition 1.3.13 there holds (i) v(x) > 0 ∀x ∈ Ω and (ii)

∂v (x) < 0 ∀x ∈ ∂Ω. ∂n

(1.5.13)

Moreover v ≥ u in Ω and the coincidence set C of u and v cannot have a connected component K ⊂ Ω. If we assume that C is not empty, there exists {xn } ⊂ K and x0 ∈ ∂Ω such that xn → x0 . Since u and v are C 1 in Ω, it follows from (1.5.13) (ii) that ∂v ∂u (x0 ) = (x0 ) = −a2 < 0. ∂n ∂n Set w = v − u, it satisﬁes, −Lw = g − f ≥ 0 where L is the linear strictly elliptic operator deﬁned in the proof of Proposition 1.5.3. Since w ≥ 0 is not identically zero in a neighborhood of x0 , ∂w ∂v ∂u (x0 ) = (x0 ) − (x0 ) < 0, ∂n ∂n ∂n contradiction. Therefore C is empty. 1.5.2

Isotropic singularities of p-harmonic functions

The fundamental solution μp of −Δp is deﬁned in RN \ {0} by ⎧ −p ⎨ γN,p |x|− Np−1 if p = N μp (x) = ⎩ γ ln |x|−1 if p = N, N

(1.5.14)

where the constants γN,p , γN are chosen so that −Δp μp = δ0

in D (RN ).

(1.5.15)

Since μp is rotationally invariant with respect to 0, we set μp (x) = μp (|x|).

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Let Ω be a domain in RN containing 0, Ω∗ = Ω \ {0}. The next result due to [Kichenassamy and V´eron (1988)] makes more precise the two-sided estimates in Theorem 1.3.7. Theorem 1.5.6 Let 1 < p ≤ N and u is a p-harmonic function in Ω∗ such that |u(x)| ≤ cμp (x) near 0. Then there exists a real number γ ∈ [−c, c] such that u − γμp ∈ L∞ loc (Ω).

(1.5.16)

Moreover, if γ = 0 the following relation holds N −p

lim |x| p−1 +|β| Dβ (u − γμp )(x) = 0,

x→0

(1.5.17)

for all β ∈ NN \ {0}. Furthermore |∇u| ∈ L1loc (Ω) and there holds p−2

−Δp u = |γ|

γδ0

in D (Ω).

(1.5.18)

Scaling techniques are among the standard methods to obtain pointwise estimates on the derivatives from similar estimates on functions itself. Lemma 1.5.7 Let 1 < p ≤ N and u is a p-harmonic function in B2∗ where it satisﬁes |u(x)| ≤ Aμp (x)+B. Then there exist constants α ∈ (0, 1) and c > 0 depending on N and p such that N −1

|∇u(x)| ≤ Ac|x|− p−1 N −1 α |∇u(x) − ∇u(x )| ≤ Ac|x|− p−1 −α |x − x | ,

(i) (ii)

(1.5.19)

for all x, x ∈ B1∗ satisfying |x| ≤ |x |. Proof. If 0 < a

0, solution of −Δp u = |γ|p−2 γδ0 u=g

in D (Ω) on ∂Ω.

(1.5.20)

Furthermore u is unique in the class of functions such that |u(x)| ≤ cμp (x) near 0. Another consequence of Theorem 1.5.6 is the following singular rigidity result. Corollary 1.5.9 Let 1 < p ≤ N . If u is a p-harmonic function in RN \ {0} such that |u(x)| ≤ a|μp (x)| + b for some nonnegative constants a, b, there exist two constants γ and λ such that |γ| ≤ a, |λ| ≤ b with the property that u(x) = γμp (x) + λ for all x ∈ RN \ {0}. Proof. The case p = N . We use the invariance of the equation under the x By Theorem 1.5.6 we see conformal transformation x → I0 (x) = |x| 2. that there exist constants γ and γ such that u − γμN ∈ L∞ (B1 ) and u − γ μN ∈ L∞ (B1c ). If |γ| + |γ | = 0, u is bounded and therefore it is constant by Theorem 1.3.8. Let us assume that |γ| + |γ | > 0, for example γ > 0, and put, for > 0 and R > 1, v = (1 − )γμN + (γ − (1 − )γ)μN (R) − K, where

K = max u − γμN L∞ (B1 ) , u − γ μN L∞ (B c ) . 1

By the comparison principle, u ≥ v in BR . Letting → 0 yields u ≥ γμN + (γ − γ)μN (R) − K

in BR .

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In the same way u ≤ γμN + (γ − γ)μN (R) + K

in BR .

Therefore the following two-sided inequality holds in BR , γμN + (γ − γ)μN (R) − K ≤ u ≤ γμN + (γ − γ)μN (R) + K.

(1.5.21)

Since μN (R) → −∞ when R → ∞, we infer that γ = γ . By Theorem 1.5.6 u−γμN admits a limit both at 0 and at inﬁnity; we set λ = lim|x|→∞ (u(x)− γμN (x)). For any > 0 there exists R() > 0 such that γμN (x) + λ − ≤ u(x) ≤ γμN (x) + λ +

∀x s.t. |x| ≥ R(). (1.5.22) The three functions u, γμN (x) + λ + and γμN (x) + λ − are N -harmonic in BR() \{0}, have the same singularity at x = 0 and are ordered on ∂BR() because of (1.5.22). Therefore (1.5.22) holds in whole RN . Letting → 0 yields u = γμN + λ. The case 1 < p < N . There exists a real number γ such that lim

x→0

u(x) =γ μp (x)

and u − γμp ∈ L∞ (RN ).

(1.5.23)

The following dichotomy occurs: (i) either the maximum of u−γμp is achieved at some point x0 ∈ RN \{0}, (ii) either the maximum of u − γμp is achieved at inﬁnity, (iii) or the maximum of u − γμp is achieved at 0. In case (i), it follows from Proposition 1.5.2 that u = γμN + (u − γμp )(x0 ). In case (ii), we set λ(r) = max{(u − γμp )(x) : |x| ≤ r}. Then the function λ is increasing, again by Proposition 1.5.2, and there exists some yr with norm 1 such that λ(r) = (u − γμp )(ryr ) and lim λ(r) = max{(u − γμp )(x) : 0 < |x| < ∞} := λ∗ .

r→∞

Set vr (y) = u(ry) − γμp (r). Then vr is p-harmonic in RN \ {0} and (i) (ii)

|∇vr (y)| = r |∇u(ry| ≤ c(a + b)|y|−1 |∇vr (y) − ∇vr (y )| ≤ c(a + b)|y|−1−α |y − y |α

∀|y| ≥ r−1 ∀|y | ≥ |y| ≥ r−1 .

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Hence there exist a sequence {rn } tending to ∞, a p-harmonic function v ∈ C 1 (RN \ {0}) and y0 with norm 1 such that {vrn } converges to v in the 1 -topology of RN \ {0} and {yrn } converges to y0 . Then Cloc v ≤ v(y0 ) = max{v(y) : y ∈ RN \ {0}}. This implies that v ≡ v(y0 ) = λ∗ is a constant function. Therefore, for any > 0, there holds λ∗ − ≤ u(rn y) − γμp (rn ) ≤ λ∗ + for rn large enough and |y| = 1. This implies lim (u − γμp )(x) = λ∗ ,

|x|→∞

by the maximum principle. Finally, comparing u and γμp )(x) + λ∗ in BR \ {0} for any R > 0 as in the previous case yields u = γμp )(x) + λ∗ . The proof in case (iii) is similar. 1.5.3

Rigidity theorems for p-harmonic functions

The following result (see [Kilpel¨ainen, Shahgholian, Zhong (2007)]) extends Theorem 1.3.8 and the remark hereafter to p-harmonic functions. It is noticeable that it proofs relies only on the H¨older regularity estimate of the gradient. Theorem 1.5.10 Let 1 < p < ∞. Then there exists a number α ∈ (0, 1) such that any p-harmonic function u in RN which satisﬁes |u(x)| = o(|x|1+α ) is of the form u(x) = 1, ..., N .

j

as |x| → ∞,

(1.5.24)

aj xj + b for some real numbers aj , b and j =

Proof. Let {Rk } be a sequence tending to inﬁnity and Sk = max{|u(x)| : |x| ≤ Rk }. The functions uk deﬁned by uk (x) = Sk−1 u(Rk x) are p-harmonic in RN and |uk (x)| ≤ 1 in B1 . By Theorem 1.3.9 the set of functions {uk } is uniformly bounded in C 1,α (Br ) for some α ∈ (0, 1) and any r < 1. Therefore the functions ck (x) =

R1+α |Du(Rk x) − Duk (0)| |Duk (x) − Duk (0)| = k , α |x| Sk |Rk x|α

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61

are uniformly bounded in Br . The growth condition implies Rk1+α = 0, k→∞ Sk lim

and therefore sup x∈Br

|Duk (x) − Duk (0)| |Du(y) − Du(0)| = sup → 0 as Rk → ∞. α |x| |y|α y∈BRk r

Thus Du(y) = Du(0) for all y ∈ RN , hence u is a linear functions of (x1 , ..., xN ). Conversely any linear function is p-harmonic. Remark. It is interesting to notice that the exponent α is the one which gives the H¨ older continuity of the gradient of p-harmonic functions. The next result is optimal since linear function are p-harmonic. Corollary 1.5.11 RN which satisﬁes

Let 1 < p < ∞. Then any p-harmonic function u in |u(x)| = o(|x|)

as |x| → ∞,

(1.5.25)

is a constant. Proof. By Theorem 1.5.10 we should have u(x) = aj are zero because of the sublinear estimate.

j

aj xj + b. But all the

The following theorem characterizes nonnegative p-harmonic functions in a half-space which vanishes on the boundary. Its proof is based upon a simple form of Carleson estimate. More elaborate statements will be stated in Chapter 2. Proposition 1.5.12 Assume 1 < p < ∞ and u is a nonnegative p harmonic function in RN + = {x := (x , xN ) : xN > 0} which vanishes on N ∂R+ . Then there exists a positive constant c1 = c1 (N, p) such that u(0, R) u(x , xN ) ≤ c1 xN R

+ ∀R > 0 and (x , xN ) ∈ B2R .

(1.5.26)

Proof. By the strong comparison principle we can assume that u > 0 in RN + . If R > 0, we set mpR = |u|p dx, B3R

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and deﬁne uR by uR (x) = R p m−1 R u(Rx). Then p |uR | dx = 1. N

B3

If we assume that the result does not hold there exists a sequence of points {ak } = {(ak , αk )} ⊂ B2+ such that uR (ak , αk ) ≥ kuR (0, 1). αk

(1.5.27)

Since uR Lp (B3 ) = 1, ∇uR L∞ (B2 ) ≤ m, where m = m(N, p) > 0, therefore |uR (ak , αk ) − uR (ak , 0)| = uR (ak , αk ) ≤ mαk , which implies uR (0, 1) ≤ mk −1 and uR (0, 1) = 0 which is a contradiction. Hence there exists c > 0 depending only on N and p such that uR (y , yN ) uR (0, 1)) ≤c yN 1

∀y = (y , yN ) ∈ B2+ .

By rescaling we obtain (1.5.26).

Theorem 1.5.13 Let 1 < p < ∞. Then any nonnegative p-harmonic N function in a half-space RN + which vanishes on its boundary ∂R+ is a linear function of the xN variable. Proof. We denote by eN the vector with coordinates (0, 0, .., 0, 1). Step 1. We claim that there exists c2 = c2 (N, p) such that u(ReN ) ≤ c2 Ru(eN )

∀R ≥ 2.

(1.5.28)

For r > 1 and x0 = 2reN = ReN , we deﬁne the positive p-harmonic function v in B2r (x0 ) \ Br (x0 ) by 2r 1−N t p−1 dt |x−x | v(x) = 2r0 . 1−N p−1 t dt r

This function is p-harmonic in B2r (x0 ) \ Br (x0 ), it vanishes on ∂B2r (x0 ) and has value 1 on ∂Br (x0 ). Since by Harnack inequality (see e.g. Theorem 1.3.3) there exists m2 = m2 (N, p) > 0 such that u(x0 ) ≤ m2 u(x)

∀x ∈ B |x0 | (x0 ), 2

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63

we obtain by comparison, m2 u(x) ≥ u(x0 )v(x)

∀x ∈ B2r (x0 ) \ Br (x0 ).

In particular m2 u(eN ) ≥ u(x0 )v(eN ) and a straightforward computation shows that v(eN ) ≥ (c2 r)−1 for some c3 = c3 (N, p) > 0. This implies u(x0 ) ≤ m2 c3 ru(eN ) and we get (1.5.28). Step 2. To complete the proof we use boundary Harnack inequality Proposition 1.5.12 which yields u(ReN ) u(x) ≤ c1 xN R

+ ∀x ∈ B2R

∀R > 0.

(1.5.29)

Combining (1.5.28) and (1.5.29) u(x) ≤ c1 xN

u(ReN ) ≤ c1 c2 xN Ru(eN ) R

+ ∀x ∈ B2R

∀R ≥ 2. (1.5.30)

This implies that u has a growth at most linear. We deﬁne now a function u ˜ in RN by $ if xN ≥ 0 u(x , xN ) u ˜(x , xN ) = (1.5.31) if xN < 0. −u(x , −xN ) Clearly u ˜ is p-harmonic in RN with an at most linear growth at inﬁnity. Thus it is a linear function by Theorem 1.5.10, which ends the proof. Remark. It is a consequence of Theorem 1.5.13 that the constant c which appears in Proposition 1.5.12 and could depend on the function u is actually equal to 1. The fact that c is independent of u is the real Carleson inequality and it is a key point in the proof of boundary Harnack inequality. 1.5.4

Isolated singularities of p-superharmonic functions

A remarkable property of positive weakly p-superharmonic functions with an isolated singularity lies in the presence of a Dirac mass at the singular point. This result which has many applications to equations with source reaction is proved in [Bidaut-V´eron (1989)]. 1,p (B1 \ {0}) ∩ C(B 1 \ {0}) Theorem 1.5.14 Let 1 < p ≤ N and u ∈ Wloc 1 such that Δp u ∈ Lloc (B1 \ {0}) in the sense of distributions in B1 \ {0} satisﬁes

−Δp u ≥ 0

in D (B1 \ {0}).

(1.5.32)

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64

Then N

,∞

p−1

N −p (B1 ), |∇u| (i) if 1 < p < N , up−1 ∈ Lloc 1 exist g ∈ Lloc (B1 ) and β ≥ 0 such that

−Δp u = g + βδ0

N

N −1 ∈ Lloc

in D (B1 ).

,∞

(B1 ) and there (1.5.33)

r

(ii) if p = N , ur ∈ L1loc (B1 ) for all r > 0, |∇u| ∈ L1loc (B1 ) for all r ∈ (0, N ). Furthermore, if lim u(x) = ∞ the conclusions of (i) holds. x→0

In the above theorem the function g is equal to Δp u a.e. The diﬃculty is to prove that g ∈ L1loc (B1 ). 1.6

Notes and open problems

1.6.1. The H¨older regularity of weak solutions of degenerate quasilinear equations associated to variational integral was initialy proved by Ladyzhenskaya and Ural’tseva. The C 1,α regularity was obtained for pharmonic type equations in [Uhlenbeck (1977)], [Evans (1982)] and [Lewis (1983)]. The extension to more general operators is due to [Tolksdorﬀ (1984)]. 1.6.2. It was conjectured by Serrin that the exponent α in (1.5.24) is equal 1 . There will be a discussion about optimality of this exponent at the to p−1 end of Chapter 2. 1.6.3. In Theorem 1.5.14-(ii) the question is still open wether the assumption lim u(x) = ∞ is necessary in order −ΔN u ∈ L1loc (B1 ) and (1.5.33) x→0

holds. A step in this direction would be the fact that ∇u ∈ LN,∞ loc (B1 ).

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Chapter 2

Separable solutions

2.1 2.1.1

Eigenvalue and eigenfunctions Dirichlet problem

In this section Ω is a bounded domain of RN and p > 1. The Rayleigh quotient Rp [v] is deﬁned for any v ∈ W01,p (Ω), v = 0, by |∇v|p dx Rp [v] = Ω . (2.1.1) |v|p dx Ω

If we assume that Ω ⊂ {x ∈ R

N

: 0 < xi < a1 }, then

Rp [v] ≥

p . apj

(2.1.2)

This is consequence of the fact that there exists a sequence {vn } ⊂ Cc∞ (Ω) which converges to v in W 1,p (Ω) and therefore Rp [v] = limn→∞ Rp [vn ]. By integration x1 p ∂vn p (t, x2 , ..., xN ))dt vn (x1 , x2 , ..., xN ) = ∂x1 0 p x1 p ∂vn ≤ x1p (t, x , ..., x ) 2 N dt. ∂x1 0

Since Rp [v] is invariant by orthogonal transformations in RN , and if R > 0 denotes the radius of the largest ball contained in the smallest rectangular parallelepiped box containing Ω, we obtain after a rotation, Rp [v] ≥ 65

p . Rp

(2.1.3)

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Local and global aspects of quasilinear elliptic equations

Since |Ω| ≤ ωN RN , where ωN is the volume of the Euclidean unit ball in RN , it follows p

pωNN Rp [v] ≥ p . |Ω| N Proposition 2.1.1

(2.1.4)

There exists φ1 ∈ W01,p (Ω) \ {0} such that

Rp [φ1 ] = inf Rp [v] : v ∈ W01,p (Ω) \ {0} := λ1 .

(2.1.5)

p p λ1 = inf ∇v Lp : v ∈ W01,p (Ω), v Lp = 1 .

(2.1.6)

Equivalently

Furthermore φ1 is C 1,α for some α ∈ (0, 1), it keeps a constant sign and it is unique up to an homothety, Proof. The existence of a minimizer, say φ1 , follows from the compactness of the imbedding of W01,p (Ω) into Lp (Ω). Because Rp [v] = Rp [|v|], we can suppose that φ1 ≥ 0. Since the mapping v → Rp [v] is C 1 on W01,p (Ω) \ {0}, d Rp [φ1 + tη]t=0 for any η ∈ W01,p (Ω), with η W 1,p < we can compute dt φ1 W 1,p and get

|∇φ1 |

p−2

∇φ1 .∇ηdx = λ1

Ω

|φ1 |

p−2

φ1 ηdx.

(2.1.7)

Ω

Thus φ1 is a nonnegative weak solution of −Δp φ = λ1 |φ|

p−2

φ.

(2.1.8)

We say that φ1 is an eigenfunction of −Δp with corresponding eigenvalue λ1 . By regularity φ1 is bounded and continuous in Ω up to a possible modiﬁcation on a set with zero measure, and by Harnack inequality, φ1 > 0. By Theorem 1.3.9, ∇φ1 is α-H¨older continuous for some α ∈ (0, 1). Actually any φ ∈ W01,p (Ω) which satisﬁes (2.1.8) in the weak sense (even with λ1 replaced by some λ > 0 if it exists) shares the same C 1,α -regularity. Assume that ψ1 is another minimizer of Rp . Since |ψ1 | satisﬁes (2.1.8), it satisﬁes Harnack inequality and thus it never vanishes in Ω. Therefore we can infer

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67

1

that ψ1 > 0 and we put w = ( 12 ψ1p + 12 φp1 ) p . Then w Lp = 1 and p

p 1 p (ψ1 + φp1 )1−p ψ1p−1 ∇ψ1 + φp−1 ∇φ 1 1 2 p 1 ψ1p−1 φp−1 1 1 = 1 ∇ψ + ∇φ 1 1 . 1 1 p p 1− p 2 p (ψ p + φp )1− p1 p 2 (ψ1 + φ1 ) 1 1

|∇w| =

ψ1p−1

p

Since X → |X| is convex and

1 2p

1− 1 p (ψ1p +φp 1)

+

set C := {x ∈ Ω : ψ1 (x) = φ1 (x)}, it follows |∇w|p

0, we obtain 1 1 p p |∇w| dx < |∇ψ1 | dx + |∇φ1 |p dx = λ1 . 2 Ω 2 Ω Ω p

Because w Lp = 1, ∇w Lp ≥ λ1 . Therefore |C| = 0 and ψ1 (x) = φ1 (x) a.e. in Ω. Deﬁnition 2.1.2 We say that a function φ ∈ W01,p (Ω) is an eigenfunction of −Δp in W01,p (Ω) if there exists λ, necessarily positive, such that −Δp φ = λ |φ|

p−2

φ

in Ω.

(2.1.9)

The set of eigenvalues is denoted by σ[−Δp ; W01,p (Ω)]. Remark. It is easy to see that σ[−Δp ; W01,p (Ω)] is a closed subset of R+ . Furthermore, if φ is an eigenfunction with eigenvalue λ and G is a connected component of {x ∈ Ω : φ(x) > 0}, φG is the ﬁrst eigenfunction of −Δp in W01,p (G) with corresponding eigenvalue λ. As a consequence of (2.1.4) p

p

λ|G| N ≥ pωNN .

(2.1.10)

Proposition 2.1.3 The eigenvalue λ1 is the smallest element of σ[−Δp ; W01,p (Ω)]. Furthermore φ1 is the only eigenfunction with constant sign and λ1 is an isolated eigenvalue.

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68

Proof. As already noticed in the proof of Proposition 2.1.1, any eigenfunction is bounded and, up to a modiﬁcation on a set with zero measure, it belongs to C 1,α for some α ∈ (0, 1). The fact that λ1 = min{λ : λ ∈ σ[−Δp ; W01,p (Ω)]} is clear from Proposition 2.1.1. If we assume that there exists λ ∈ σ[−Δp ; W01,p (Ω)], λ > λ1 , with a corresponding nonnegative eigenfunction φ, then φ > 0 by Harnack inequality. For > 0 we set φ1, = φ1 + , φ = φ + and η1 =

φp1, − φp φp−1 1,

and η =

φp − φp1, φp−1

.

Then ∇η1 =

p p−1 φ φ ∇φ1 − p 1 + (p − 1) ∇φ, φ1, φ1,

similarly for ∇η by exchanging φ1 and φ, thus η and η1 belong to W01,p (Ω),

|∇φ1 |

Ω

p−2

∇φ1 .∇η1 dx = λ1 Ω

|∇φ|

p−2

φp−1 η1 dx 1

∇φ.∇ηdx = λ φp−1 ηdx.

Ω

(2.1.11)

Ω

After some computations we obtain Ω

φp−1 φp−1 p 1 λ1 p−1 − λ p−1 φ1, − φp dx φ1, φ p p p φ1, − φp (|∇ ln φ1, | − |∇ ln φ | ) dx = Ω

−p

Ω

φp1, |∇ ln φ |

p−2

∇ ln φ . (∇ ln φ1, − ∇ ln φ ) dx

p−2

∇ ln φ1, . (∇ ln φ − ∇ ln φ1, ) dx.

φp |∇ ln φ1, |

−p Ω

Clearly p−1 p φp−1 φ p 1 λ1 p−1 − λ p−1 φ1, − φ dx = (λ1 − λ) (φp1 − φp ) dx. lim →0 Ω φ1, φ Ω

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Using the convexity of X → |X|p , p |∇ ln φ1, |

p−2

∇ ln φ1, .(∇ ln φ1, − ∇ ln φ ) p

≥ |∇ ln φ1, | − |∇ ln φ | ≥ p |∇ ln φ |

p−2

p

∇ ln φ .(∇ ln φ1, − ∇ ln φ ).

Put Ω = {x ∈ Ω : φ1, ≥ φ }, Ω = Ω \ Ω, then p φ1, − φp (|∇ ln φ1, |p − |∇ ln φ |p ) p−2 p−2 ∇ ln φ1, − |∇ ln φ | ∇ ln φ .∇ (ln φ1, − ln φ ) , ≥ pφp |∇ ln φ1, | on Ω and p p p φ1, − φp (|∇ ln φ1, | − |∇ ln φ | ) ≥ pφp1, |∇ ln φ1, |p−2 ∇ ln φ1, − |∇ ln φ |p−2 ∇ ln φ .∇ (ln φ1, − ln φ ) , on Ω . The right-hand side terms in the two above inequalities are nonnegative, thus (φp1 − φp ) dx ≥ 0. (λ1 − λ) Ω

Since we can replace φ by kφ for arbitrary positive k, the only possibility is λ1 = λ. Assume now that λ1 is not isolated. Then there exists a decreasing sequence {λn } of eigenvalues with corresponding eigenfunctions {φn } normalized by φn Lp = 1 converging to λ1 . By compactness we can assume that φn → φ in Lp (Ω) and ∇φn ∇φ weakly in Lp (Ω) for some φ ∈ W01,p (Ω) verifying φ Lp = 1. Furthermore the convergence of {φn } holds locally uniformly in Ω. Since . Lp is weakly lower semicontinuous, λ1 = lim λn = lim inf Rp [φn ] ≥ Rp [φ ], n→∞

n→∞

which implies φ = kφ for some nonzero k. Since λn > λ1 , φn has at least two nodal domains Ωn and Ωn . By the previous remark, p p p p −1 N pωNN (1 + ◦(1)). inf |Ωn | N , |Ωn | N ≥ λ−1 n pωN = λ By a selection lemma, lim sup inf {|Ωn |, |Ωn |} > 0, which contradicts φn → n→∞

φ in Lp (Ω) with φn Lp = 1.

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The question of the variation of the domain Ω gives rise to the same type of results as if p = 2 [Courant, Hilbert (1953)], with quite similar proofs. Proposition 2.1.4

Assume p > 1 and denote λ1 = λ1 (p, Ω). Then

(i) Ω1 Ω2 implies λ1 (p, Ω2 ) < λ1 (p, Ω1 ). (ii) If Ω1 ⊂ Ω2 ⊂ Ω3 ⊂ .... is an increasing sequence of domains in RN , then Ωj . lim λ1 (p, Ωj ) = λ1 (p, Ω) where Ω = (2.1.12) j→∞ j≥1

2.1.2

The case p → ∞ Let Ω be a bounded domain of RN . For any 1 < p < s

Proposition 2.1.5 there holds

p p λ1 (p, Ω) ≤ s s λ1 (s, Ω).

(2.1.13)

s

Proof. Let ψ ∈ Cc1 (Ω), ψ ≥ 0 and φ = ψ p , then φ ∈ Cc1 (Ω) and p1 p1 p s−p p |∇φ| dx ψ |∇ψ| dx

s Ω Ω p λ1 (p, Ω) ≤ = p1 p1 p φp dx ψ s dx Ω 1 Ω p1 − 1s 1s s s s ψ dx |∇| dx |∇|s dx s s Ω Ω ≤ = Ω p1 1s . p p s s ψ dx ψ dx Ω

Ω

we derive (2.1.13).

This implies that the function s → s s λ1 (s, Ω) is monotone nondecreasing. Therefore there exists

lim s s λ1 (s, Ω) ≤ p p λ1 (p, Ω ≤ lim s s λ1 (s, Ω). (2.1.14) s↑p s↓p If we take the inﬁmum over all the ψ ∈

Cc1 (Ω)

We denote by p− p− λ1 (p− , Ω) and p+ p+ λ1 (p+ , Ω) respectively the left-hand side and the right-side of the above limits. Corollary 2.1.6

Under the assumptions of Proposition 2.1.5, there holds lim λ1 (s, Ω) ≤ λ1 (p, Ω) ≤ lim λ1 (s, Ω). s↑p

s↓p

(2.1.15)

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Separable solutions

Theorem 2.1.7

71

For any bounded domain and p > 1, one has lim λ1 (s, Ω) = λ1 (p, Ω).

(2.1.16)

s↓p

Proof. For any φ ∈ Cc1 (Ω) there holds λ1 (s, Ω) ≤

Ω

|∇φ|s dx . |φ|s dx

Ω

Using Lebesgue convergence theorem and (2.1.15) yields |∇φ|s dx |∇φ|p dx Ω Ω lim λ1 (s, Ω) ≤ lim = , s↓p s↓p s p |φ| dx |φ| dx Ω

Ω

and if we take the inﬁmum over φ we obtain the claim.

Remark. The continuity to the left of s → λ1 (s, Ω) is not always veriﬁed and some delicate assumptions on the regularity of ∂Ω are needed (see [Lindqvist (1993)]). Let us denote

$ Λ∞ = inf

Proposition 2.1.8

% ∇φ L∞ 1 : φ ∈ Cc (Ω) . φ L∞

(2.1.17)

For any bounded domain Ω of RN , there holds Λ∞ =

∇ρ L∞ 1 , = ρ L∞ max ρ(x)

(2.1.18)

x∈Ω

where ρ(x) = dist (x, ∂Ω). Proof. By the triangle inequality the function ρ is a contraction. If φ ∈ Cc1 (Ω) there holds by the mean value theorem, |φ(x)| ≤ ρ(x) ∇φ L∞ . The claim follows since ∇ρ L∞ ∇φ L∞ 1 1 ≥ ≥ = . |φ(x)| ρ(x) ρ L∞ ρ L∞

(2.1.19)

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72

Proposition 2.1.9

There holds

lim p λ1 (p, Ω) = Λ∞ . p→∞

(2.1.20)

Proof. Taking ρ as test function in the minimization problem (2.1.1) yields ∇ρ Lp 1 p Rp [ρ] = = p1 . ρ Lp −1 p |Ω| (ρ(x)) dx

(2.1.21)

Ω

Hence lim sup p→∞

p Rp [ρ] ≤ lim sup p→∞

|Ω|−1

1

1 p1 = ρ ∞ = Λ∞ . L (ρ(x))p dx

Ω

(2.1.22) Let {φp } be the sequence of positive ﬁrst eigenfunctions of −Δp normalized 1 by |Ω|− p φp Lp = 1, then p1

−1 p |Ω| |∇φp | dx = p λ1 (p, Ω). Ω

Let N < m < p. By H¨older’s inequality m1

|Ω|−1 |∇φp |m dx ≤ p λ1 (p, Ω). Ω

Thus the set {φp }p≥m is uniformly bounded in W01,m (Ω). By Cantor diagonal process there exists a sequence {φpk } which converges to some function φ∞ , weakly in W01,m (Ω) for any m > N and strongly in any C α (Ω) for α =1− N m by Morrey inequality (1.2.35), and thus uniformly in Ω. Since 1 |Ω|− p φp Lp = 1, it follows supx∈Ω φ∞ (x) ≥ 1 by H¨older’s inequality and the uniform convergence. Using the weak lower semicontinuity of the norm in W 1,m and H¨older’s inequality, yields ∇φpk Lm φpk L∞ ∇φ∞ Lm ≤ lim inf ≤ lim inf Λpk . pk →∞ φpk m pk →∞ φ∞ Lm φpk Lm L This implies ∇φ∞ Lm ≤ φ∞ Lm

φ∞ L∞ lim inf Λpk . pk →∞ φ∞ Lm

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Therefore, for any m > N , ∇φ∞ Lm ≤ lim inf Λpk ; pk →∞ φ∞ L∞ henceforth this inequality holds also with m = ∞ and the claim follows using the minimizing property (2.1.17) and the reverse inequality (2.1.22). Remark. Using Harnack inequality, it is proved in [Lindqvist, Manfredi (1997)] that the function φ∞ is positive in Ω. This fact is important for application to eigenvalue problem (see e.g. Theorem 2.1.12). Contrary to the case p < ∞, the minimization problem (2.1.17) has too many solutions and no possibility to select the function ρ among them. If one expand the equation Δp φ + λ1 (p, Ω)φp−1 = 0, one ﬁnd that outside the set of critical points of φ, there holds |∇φ|2 Δφ λ1 (p, Ω) + Hφ (∇φ).∇φ + |∇φ|4−p φp−1 = 0. p−2 p−2

(2.1.23)

By deﬁnition the inﬁnite Laplacian is expressed by Δ∞ φ = Hφ (∇φ).∇φ =

j,k

∂ 2 φ ∂φ ∂φ . ∂xj ∂xk ∂xj ∂xk

(2.1.24)

It is proved in [Juutinen, Lindqvist, Manfredi (1999)] that the limit of (2.1.23) when p → ∞ exists, in the viscosity sense and the limit equation is % $ |∇φ(x)| , Δ∞ φ(x) = 0. (2.1.25) max Λ∞ − |φ(x)| Deﬁnition 2.1.10 tion of (2.1.25) if

A nonnegative function φ ∈ C(Ω) is a viscosity solu-

(i) For every x0 ∈ Ω and η ∈ C 2 (Ω) such that η(x0 ) = φ(x0 ) and φ(x) < η(x) if x = x0 in a neighborhood of x0 , there holds Λ∞ −

|∇η(x0 )| ≥0 |η(x0 )|

or Δ∞ η(x0 ) ≥ 0.

(2.1.26)

(ii) For every x0 ∈ Ω and η ∈ C 2 (Ω) such that η(x0 ) = φ(x0 ) and φ(x) > η(x) if x = x0 in a neighborhood of x0 , there holds Λ∞ −

|∇η(x0 )| ≤ 0 and Δ∞ η(x0 ) ≤ 0. |η(x0 )|

(2.1.27)

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It is shown in [Juutinen, Lindqvist, Manfredi (1999)] that if Ω = | √ 2 is {(x1 , x2 ) ∈ RN : |x1 | + |x2 | < 1}, the function d(x) = 1 − |x1 |+|x 2 not a viscosity solution of (2.1.25). Actually, the viscosity solutions have to be found as limit of the normalized eigenfunctions φp used in the proof of Proposition 2.1.9 and the following deep result is proved in [Juutinen, Lindqvist, Manfredi (1999), Th 1.21]. Theorem 2.1.11 For any sequence of positive normalized eigenfunctions {φpk }, converging weakly in W01,m (Ω) for any m > N , the limit function φ∞ = limpk →∞ φpk is a viscosity solution of (2.1.25). It is noticeable that even if the uniqueness of the normalized viscosity solution of (2.1.25) is not known, the parameter Λ∞ is uniquely determined. Theorem 2.1.12 Assume Ω is a bounded domain such that ∂Ω = ∂Ω and there exists a continuous positive solution φ ∈ C(Ω) vanishing on ∂Ω which is a viscosity solution of % $ |∇φ(x)| , Δ∞ φ(x) = 0. (2.1.28) max Λ − |φ(x)| Then Λ = Λ∞ . 2.1.3

Higher order eigenvalues

It is well-known that the spectrum of the Laplacian is completely obtained by the theory of compact (linear) operators in Hilbert spaces. It can also be constructed by induction with the following minimax formulation ⊥ } λk = min{R2 [v] : v ∈ Hk−1

for k = 1, ...

(2.1.29)

where H1 = H1 = N (Δ + λ1 I) and Hj = ⊕j=1 H where H = N (Δ + λ I), noticing that the spaces H are ﬁnite dimensional and mutually orthogonal in L2 (Ω). This construction can be adapted to construct a variational spectrum of −Δp for p = 2 in the following way. Let A be a closed subset of a Banach space X such that A = −A. The Krasnoselskii genus (see e.g. [Struwe (1996)]) of A, γ(A), is the smallest integer k such that there exists a continuous and odd mapping from A into Rk \ {0}. Let Sk be the collection of all closed symmetric subsets of W01,p (Ω such that γ(A) ≥ k and A ∩ S is compact where S = {u ∈ W 1,p : u Lp = 1}. The number λk = inf max Rp [v] A∈Sk v∈A

(2.1.30)

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75

is an eigenvalue of −Δp . The set of all such λk is the variational spectrum of −Δp denoted by σvar [−Δp ; W01,p (Ω)]. It is a deep open problem whether σ[−Δp ; W01,p (Ω)] and σvar [−Δp ; W01,p (Ω)] coincide. The asymptotic distribution of the λk is described by the following result [Garcia Azorero, Peral (1988)], [Friedlander (1989)]. Proposition 2.1.13 for all integer k, c1

k |Ω|

Np

There exist two positive constants, c1 , c2 such that,

≤ λk ≤ c2

k |Ω|

Np

∀λk ∈ σvar [−Δp ; W01,p (Ω)].

(2.1.31)

The proof is quite involved, but we have the next easy-to-prove result. Proposition 2.1.14 If an eigenfunction φ ∈ W01,p (Ω) has k nodal domains, i.e. connected components of {x ∈ Ω : φ(x) = 0}, and corresponding eigenvalue λ ∈ σ[−Δp ; W01,p (Ω)], the following inequality holds λ≤p

kωN |Ω|

Np .

(2.1.32)

Proof. Let Gj , j = 1, ..., k be the nodal domains of φ. Then, by (2.1.10) |Gj | ≥ This implies the claim.

p Np λ

ωN =⇒ |Ω| ≥ k

p Np λ

ωN .

Remark. It is a classical result due to Courant that an eigenfunction of −Δ in W01,2 (Ω) corresponding to the k-th eigenvalue λk has at most k nodal domains. This result admits an extension to the p-Laplacian [Dr` abek, Robinson (2004)] Proposition 2.1.15 Any eigenfunction φ ∈ W01,p (Ω) corresponding to the eigenvalue λk ∈ σvar [−Δp ; W01,p (Ω)] has at most 2k − 2 nodal domains. Furthermore if there exists an eigenfunction with k + nodal domains for some ∈ N, there exists an eigenfunction corresponding to the same eigenvalue λk with k − nodal domains. Eigenfunctions associated to a nodal domain G satisﬁes a universal estimate [Lindqvist (1995)]. Proposition 2.1.16 If G ⊂ Ω is a nodal domain of an eigenfunction φ ∈ W01,p (Ω) with corresponding eigenvalue λ ∈ σ[−Δp ; W01,p (Ω)], there

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76

holds if φ ≥ 0 in G,

N

2N +1 λ p sup ess φ ≤ N SN,p G

φdx,

(2.1.33)

G

where SN,p is the Sobolev constant. Proof. Let k > 0 and φk = max{φ − k, 0}. Since vk ∈ W01,p (G), |∇φ|p dx = λ φp−1 φk dx, Gk

Gk

where Gk = {x ∈ G : φ > k}. Since ap−1 ≤ 2p−1 (a − k)p−1 + 2p−1 k p−1 , + φp−1 φk dx ≤ 2p−1 φpk dx + 2p−1 k p−1 φk dx. Gk

Gk

Gk

By Sobolev inequality p SN,p

Gk

φpk dx

≤ |Gk |

p N

|∇φ|p dx, Gk

where SN,p is deﬁned in (1.2.33). Because −1 p p−1 p−1 λ |∇φ| dx = φ φk dx ≤ 2 Gk

Gk

Gk

we obtain −1 1−p

λ

2

|Gk |

p −N

p SN,p

φpk dx, +2p−1 k p−1

Gk

φpk dx

≤ Gk

φpk dx

+k

−1 1−p

≥ λ

2

|Gk |

p −N

p SN,p

p−1

φk dx, Gk

and ﬁnally −p p k p−1 φk dx ≥ λ−1 21−p |Gk | N SN,p −1 Gk

φk dx, Gk

Gk

φpk dx

− 1 |Gk |

1−p

p φk dx .

Gk

N , which holds true if k ≥ k1 , where If k is such that |Gk | ≤ 2−N λ− p SN,p N

−N φ L1 , there holds k1 = λ p 2N SN,p N

k p−1 |Gk |

N +1 N p−1

p ≥ λ−1 2−p SN,p

Gk

p−1 φk dx .

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Set f (k) =

∞

φk dx = Gk

77

|Gs | ds. Then f (k) = −|Gk | and

k

p

p

(f (k1 )) (N +1)p−N ≥ (f (k)) (N +1)p−N (2.1.34) N S p (N +1)p−N p p (N +1)p−N N,p (N +1)p−N k , + 2p λ − k1 on the set of the k larger or equal to k1 such that f (k) > 0. Therefore, if we set N

k2 = k1 +

2N λ p f (k1 ), N SN,p

inequality (2.1.34) cannot hold. This implies that f (k) = 0 for k ≥ k2 and (2.1.33) follows. If N = 1 and Ω = (a, b), the eigenvalue problem (|φ |

p−2

φ ) + λ |φ|p−2 φ = 0,

φ(a) = φ(b) = 0,

(2.1.35)

is thouroughly studied in [Otani (1984)]. This equation is integrable since |φ | + p

λ |φ|p = C. p−1

The explicit value of λ1 := λ1 (p, b − a) is

2 p (p − 1) 1 ds 2π p (p − 1) p √ λ1 = = . p b−a (b − a)p sin πp 1 − sp 0

(2.1.36)

The spectrum σ(−Δp ; W01,p ((a, b))) is the set of λk = k p λ1 for k ∈ N∗ . It coincides with the variational spectrum σvar (−Δp ; W01,p ((a, b))). The ﬁrst eigenvalue depends heavily

on p, but the remarkable and mysterious duality

p p λ1 (p , b − a) holds if 1p + p1 = 1. identity λ1 (p, b − a) = 2.1.4

Eigenvalues on a compact manifold

If (M d , g) is a compact Riemannian manifold with empty boundary, the divergence operator on X ∈ T M is denoted by divg . The p-Laplacian is deﬁned by −Δp,g u = −divg (|∇g u| p−2

= − |∇g u|

p−2

∇g u)

Δg u − (p − 2) |∇g u|

p−2

D2 u∇g u, ∇g ug ,

(2.1.37)

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whenever D2 u exists, thanks to the identiﬁcation of δu with the tangential gradient of u (it is one of the consequences of Nash imbedding theorems). When there is no ambiguity we denote Δp,g = Δp and Δ2,g = Δg . Note also that 2D2 u∇u, ∇ug = ∇|∇u|2 , ∇ug . Since ∂M = ∅, the spectrum of −Δp is considered for W 1,p (M ) functions. It is denoted by σ(Δp ; W 1,p (M ). Clearly min σ(Δp ; W 1,p (M ) = 0 and the corresponding eigenfunction are constant. If λ is a nonzero eigenvalue, necessarily positive, and φ an associated eigenfunction, then φ has zero average. The following characterization of the ﬁrst nonzero eigenvalue λ1 holds [V´eron (1991)]. Theorem 2.1.17

One has $ λ1 = inf

% p |∇g u| dvg : u ∈ Σ ,

(2.1.38)

M

where Σ=

$

u∈W

1,p

%

p

|u| dvg = 1,

(M ) : M

|u|

p−2

udvg = 0 .

(2.1.39)

M

Proof. Relation (2.1.38) deﬁnes λ1 as the minimum of the p-energy functional under two constraints. The set Σ is a C 0 submanifold of W 1,p (M ) which happens to be C 1 only if p ≥ 2. If Bk is the closed ball of W 1,p (M ) of radius k, Σ ∩ Bk is compact. Thus the inﬁmum is achieved by some φ ∈ Σ. If p ≥ 2 there exist two Lagrange multipliers α and β such that −Δp φ = α |φ|

p−2

p−2

u + β |φ|

,

(2.1.40)

in the weak sense. Integrating (2.1.40) yields β = 0 and α = λ1 . When 1 < p < 2, we set for > 0 small enough, $ % p p−2 (u2 + 2 ) 2 dvg = 1, (u2 + 2 ) 2 udvg = 0 , Σ = u ∈ W 1,p (M ) : M

M

(2.1.41) and

% p |∇g u| dvg : u ∈ Σ .

$ λ1, = inf

(2.1.42)

M

There exists a Lagrange multiplier μ (the other multiplier is zero) and φ ∈ Σ such that −Δp φ = μ (φ2 + 2 )

p−2 2

φ ,

(2.1.43)

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which veriﬁes

79

p

|∇g φ | dvg = μ

λ1, =

(φ2 + 2 )

M

p−2 2

φ2 dvg .

M

Hereafter we assume 1 < p < 2. The following inequality holds p

(2 + r2 ) 2 − p ≤ r2 (2 + r2 )

p−2 2

p

≤ (2 + r2 ) 2 ,

therefore (1 − p V ol(M ))μ ≤ λ1, ≤ μ . ˜1 = lim inf →0 λ1, = limn→∞ λ1, = limn→∞ μ for some sequence Set λ n n {n } converging to 0. Identity (2.1.43) implies that the set of functions φ remains bounded in W 1,p (M ), and thus in L∞ (M ) and ﬁnally in C 1,α (M ) for some α ∈ (0, 1) by Theorem 1.3.9. Then there exists a subsequence, extracted from {n } and still denoted by the same designation and function φ˜ ∈ W 1,p (M ) such that {φn } converges to φ˜ in C 1,β (M ) for any β ∈ (0, α). Clearly φ˜ ∈ Σ and ˜ p−2 φ, ˜ 1 |φ| ˜ −Δp φ˜ = λ

(2.1.44)

˜ 1 ∈ σ(−Δp ; W 1,p (M )). Let us assume that λ1 < λ ˜1 which indicates that λ and denote by φ an element of Σ for which the inﬁmum in (2.1.42) is achieved, thus |∇g φ|p dvg = λ1 . M

We write φ = φ+ − φ− . These two functions belong to W 1,p (M ) and,

(φ+ )p−1 dvg = M

(φ− )p−1 dvg ,

M

(φ+ )p + (φ− )p dvg = 1.

M

Using the mean value theorem we see that for any > 0 there is a unique σ() = σ > 1 such that p−2 σφ+ (2 + (σφ+ )2 ) 2 dvg = (φ− )p−1 dvg = (φ+ )p−1 dvg . (2.1.45) M

M

M

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There exists c1 > 0 such that, for r ≥ 0, 0 ≤ rp−1 − r(2 + r2 ) therefore

p−2 2

≤ c1 p−1 ,

p−2 σφ+ 2 + (σφ+ )2 2 dvg −

−c1 p−1 ≤ M

(σφ+ )p−1 dvg ≤ 0, M

where c2 = c1 V ol(M ) and

0 ≤ (σ p−1 − 1) (φ+ )p−1 dvg ≤ c2 p−1 .

(2.1.46)

M

Hence there exists 0 such that for 0 < ≤ 0 one has 0 < σ − 1 ≤ 12 , thus 0 ≤ σ − 1 ≤ c3 p−1 ,

(2.1.47)

for some c3 = c3 (p, φ, 0 ) > 0. By the mean value theorem and (2.1.47), we infer p (2 + (φ− )2 ) 2 dvg − (φ− )p dvg ≤ ( + φ− )p dvg − (φ− )p dvg ≤ c4 , M

M

and ﬁnally

M

M

2 + 2 p (φ+ )p dvg − 2 ( + (σφ ) ) dvg ≤ c5 . M

M

Denote M = {x ∈ M : φ (x) > 0} and M − = {x ∈ M : φ− (x) > 0}, hence 2 p 2 p + − 2 2 + (σφ − φ ) + (σφ− )2 2 dvg dvg = +

+

M

+ M−

M+

2 + (φ− )2

p2

dvg + p V ol(M \ (M + ∪ M − )).

We deduce that there exists c6 = c6 (p, M, φ) > 0 such that 2 p + − 2 2 1 − + (σφ − φ ) dvg ≤ c6 p−1 ,

(2.1.48)

and therefore one can ﬁnd η > 0 satisfying p (η)2 + (σηφ+ − ηφ− )2 2 dvg = 1,

(2.1.49)

M

M

and |1 − η| ≤ c7 p−1 .

(2.1.50)

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By combining (2.1.45) and (2.1.49) we derive σηφ+ − ηφ− ∈ Ση . Moreover p ∇g φ+ p dvg + ∇g φ− p dvg , (2.1.51) |∇g φ| dvg = λ1 = M−

M+

M

and ∇g (σηφ+ − ηφ− )p dvg = (ση)p

∇g φ+ p dvg +η p

M+

M

M−

∇g φ− p dvg . (2.1.52)

We ﬁnally infer from (2.1.51), (2.1.52) that ∇g (σηφ+ − ηφ− )p dvg ≤ c8 p−1 , λ1 −

(2.1.53)

M

which yields λ1,η ≤ λ1 + c8 p−1 ,

(2.1.54)

˜ 1 ≤ λ1 , a contradiction. Therefore λ ˜1 = λ1 and (2.1.38) holds. and λ Since an eigenfunction φ is continuous, M + = φ−1 ((0, ∞)) and M φ ((−∞, 0)) are open subsets of M and any connected component of M \ φ−1 (0) is a nodal domain of φ. If G is a connected component of M \ φ−1 (0), μ(G) denotes the ﬁrst eigenvalue of −Δp in W 1,p (G). = −1

Theorem 2.1.18 An eigenfunction corresponding to the ﬁrst nonzero eigenvalue λ1 has two nodal domains. Furthermore λ1 = min max{μ(ω), μ(˜ ω )}

(2.1.55)

(ω,˜ ω)∈S

where S is the set of couples of disjoint nonempty open subsets of M . Proof. Let ω and ω ˜ be two disjoint subdomains of M and ψ and ψ˜ the ω ) with two ﬁrst eigenfunctions (positive) of −Δp in W01,p (ω) and W01,p (˜ corresponding eigenvalues μ(ω) and μ(˜ ω ) respectively, normalized by max ψ = max ψ˜ = 1. ω

ω ˜

These functions are extended by 0 to become elements of W 1,p (M ). We can ﬁnd τ and τ˜ such that τ ψ + τ˜ψ˜ ∈ Σ. Since p p p ˜ τ ∇g ψ˜ dvg , (τ ψ + τ ˜ ψ) dv = μ(ω) |τ ∇ ψ| dv + μ(˜ ω ) ∇g ˜ g g g M

ω

ω ˜

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we obtain by convexity λ1 ≤ max{μ(ω), μ(˜ ω)} =⇒ λ1 ≤ min max{μ(ω), μ(˜ ω)}.

(2.1.56)

(ω,˜ ω )∈S

Let φ ∈ Σ be an eigenfunction of −Δp corresponding to λ1 . Assume M \{0} has at least three connected components ωj , j = 1, 2, 3, then λ1 ≤ min max{μ(ωj ), μ(ωj+1 )}, j∈Z/3Z

where the indices are numbered modulo 3. The restriction of ψ to ωj belongs to W01,p (ωj ), thus p p p |∇g φ| dvg = λ1 |φ| dvg ≥ μ(ωj ) |φ| dvg . ωj

ωj

ωj

This implies that, at least for two indices j, say 1, 2, λ1 = μ(ω1 ) = μ(ω2 ). By Hopf boundary lemma, at any point x ∈ ∂ωj where the interior sphere ∂ψ < 0 where nx is the unit outward normal vector to to ωj is satisﬁed, ∂n x any sphere contained in ωj and tangent to ∂ωj at x. Notice the set Rωj of regular points form a dense subset of ∂ωj since any point z in ωj admits proximum points in ∂ωj which are the points where the distance from z to ∂ωj is achieved; clearly ∂ωj satisﬁes the interior sphere condition at each of these proximum points . Therefore, if φ is positive in ω1 and ω2 , there exist at least two points y1 ∈ ∂ω1 \ ∂ω2 and y2 ∈ ∂ω2 \ ∂ω1 and we can replace ˜ 1 and ω ˜ 2 , for which μ(˜ ω1 ) < ω1 and ω2 by two larger disjoint domains ω ω2 ) < μ(ω2 ), which would lead to λ1 > max{μ(˜ ω1 ), μ(˜ ω2 )}, μ(ω1 ) and μ(˜ a contradiction with (2.1.56). If φ is positive in ω1 and ω3 and negative in ω2 and if μ(ω3 ) = μ(ω1 ), we proceed similarly, replacing only ω2 by ω3 . Finally, if μ(ω1 ) = μ(ω2 ) > μ(ω3 ), we have two possibilities: (i) either Rω1 = Rω2 , which implies ∂ω1 = ∂ω2 , in which case M = ω 1 ∪ω 2 , which is impossible. (ii) or there exists z ∈ Rω1 \ Rω2 (or z ∈ Rω2 \ Rω1 in the same way). Then ˜ 1 such that ω ˜ 1 ∩ ω2 = ∅ and μ(ω3 \ we can replace ω1 by a larger domain ω ω ˜ 1 ) < μ(˜ ω1 ) < μ(ω1 ), and then we can replace ω2 by a larger domain ω ˜2 such that ω ˜2 ∩ ω ˜ 1 = ∅ and μ(ω3 \ (˜ ω1 ∪ ω ˜ 1 ) < μ(˜ ω2 ) < μ(ω2 ). Therefore ω ˜1, ω1 ∪ ω ˜ 1 are three disjoint domains and max{μ(˜ ω1 ), μ(˜ ω2 ), μ(ω3 \ ω ˜ 2 and ω3 \(˜ ω1 ∪ ω ˜ 1 )} < μ(ω1 ) = λ1 which is a contradiction. (˜ When p = 2 there are many estimates of the ﬁrst nonzero eigenvalue of the Laplacian due to Lichnerowitz, Obata, Cheeger, Buser (see [Berger, Gauduchon, Mazet (1971) ] and the references therein). Many of these

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83

estimates have been extended to the p-Laplacian by Matei [Matei (2000)] and Valtorta [Valtorta (2014)]. Theorem 2.1.19 Suppose Riccg ≥ (d − 1)k > 0 and p ≥ 2. If λ1 = λ1 (p, M ) is the ﬁrst nonzero eigenvalue of −Δp , it satisﬁes λ1 ≥

p2

(d − 1)k p−1

.

(2.1.57)

Proof. Let u ∈ C 3 (M ). By Wiestenb¨ ock’s formula (1.1.14), p−2 Δp uΔg udvg = − ∇g u, ∇g Δg ug |∇g u| dvg M

M

2 2

D u + Riccg (∇g u, ∇g u) − 1 Δg |∇g u|2 |∇g u|p−2 dvg . = 2 M

Since

p−2

|∇g u|

2

p−2

Δg |∇g u| dvg = −

M

∇g |∇g u|

2

, ∇g |∇g u| g dvg

M

2−p 2 p−4 |∇g |∇g u| |2 |∇g u| dvg 2 M 2

2 2 2 and 4 D2 u |∇g u| ≥ ∇g |∇g u| , it follows an inequality valid for any =

C 3 function (and by approximation C 3 can be replaced by C 2 ), 2 Riccg (∇g u, ∇g u) |∇g u| Δp uΔg udvg ≥ M

M

2 p−4 +(p − 1)|∇g |∇g u| |2 |∇g u| dvg

≥

(2.1.58) Riccg (∇g u, ∇g u) |∇g u|

M

p−2

dvg

p

≥ (d − 1)k

|∇g u| dvg . M

If u = φ is a minimizer for problem (2.1.42), then −Δp u = μ (u2 +2 ) p−2 2 Δp uΔg udvg = (p − 1)μ (u2 + 2 ) 2 |∇g u| dvg M

M

− 2 (p − 1)μ

(u2 + 2 ) M

p−4 2

2

|∇g u| dvg ,

p−2 2

u,

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thus (p − 1)μ

(u2 + 2 )

p−2 2

2

|∇g u| dvg ≥ (d − 1)k

p

|∇g u| dvg .

(2.1.59)

M

Letting → 0 we obtain from the previous convergence, with μ → λ1 and u = φ → φ,

|φ|

(p − 1)λ1

p−2

2

|∇g φ| dvg ≥ (d − 1)k

|∇g φ|p dvg .

(2.1.60)

M

We derive (2.1.55) by H¨older’s inequality.

Other estimates involve either the diameter of M or the Cheeger’s isometric constant hM deﬁned by $ hM = inf

% P er(S) :S∈S , inf{V ol(M ), V ol(M )}

(2.1.61)

where S is the set of all closed hypersurfaces S (or imbedded (d − 1)-dim submanifolds) which divide M into two open submanifolds M and M and P er(S) is the (d − 1)-volume for the induced volume form vgS . Let p > 1, then λ1 = λ1 (p, M ) veriﬁes

Theorem 2.1.20

λ1 ≥

hM p

p .

(2.1.62)

Proof. Let φ be a ﬁrst eigenfunction. It is easy to construct a sequence {φn } ⊂ C 2 (M ) converging to φ in W 1,p (M ) such that, for each n ∈ N, |∇g φn | + |det(Hφn )| > 0 and ∇g φn = 0 on the set S = φ−1 n (0). We say that φn is a Morse function and 0 a regular value of φn . Notice that, since M is compact, the set of critical points of φn is ﬁnite. We put Mn = φ−1 n ((0, ∞)) −1 and Mn = φn ((−∞, 0)). Since λ1 = μ(M ) = μ(M ), p

lim

n→∞

Mn

|∇g φn | dvg

p

M

=

|∇g φ| dvg

p

Mn

|φn | dvg

p

M

|φ, | dvg

= λ1 .

(2.1.63)

The same relation holds if Mn is replaced by Mn , and without loss of

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generality we can assume V ol(Mn ) ≤ V ol(Mn ). By H¨ older’s inequality |∇g φpn | dvg = p |∇g φn | φp−1 n dvg Mn

Mn

1− p1

≤p

Mn

which yields Mn

|∇g φn | dvg

p

Mn

|φn | dvg

φpn dvg

⎛

p

≥

1 ⎜ ⎜ Mn pp ⎝

p1 p

|∇g φn | dvg

Mn

|∇g φpn | dvg

Mn

p

|φn | dvg

,

⎞p ⎟ ⎟ . ⎠

(2.1.64)

Set Sφpn (t) = {x ∈ Mn : φpn (x) > t}, sφpn (t) = {x ∈ Mn : φn (x) = tp }. By the co-area formula (1.2.39) sup φpn sup φpn p |∇g φn | dvg = dHN −1 dt = P er(sφn (t))dt. Mn

(t) ∂Mn

0

0

Since V ol(Mn ) ≤ V ol(Mn ), for every t > 0, V ol(Sφn (t)) ≤ V ol(Mn (t)) where Mn (t) = {x ∈ M : (φn |φn |p−1 )(x) < tp }, and because φn has only a ﬁnite number of critical points, there holds by the deﬁnition of hM , dHN −1 ≤ hM V ol(Sφpn (t)) = hM |Sφpn (t)| := λφpn (t), (t) ∂Mn

except at most for a ﬁnite set of values of t (see Section 1.2.1). This implies sup φpn p |∇g φn | dvg ≤ hM λφpn (t)dt = hM φpn dvg . (2.1.65) Mn

Mn

0

Inequality (2.1.62) follows from (2.1.63), (2.1.64) and (2.1.65).

Remark. If the Ricci curvature of M is nonnegative, it is proved in [Gallot (1983)] that hM ≥

2 , diam(M )

(2.1.66)

where diam(M ) is the maximal geodesic distance between two points of M . Therefore (2.1.62) implies the more explicit estimate λ1 (p, M ) ≥

2p pp (diam(M ))p

.

(2.1.67)

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When p = 2 Zhong and Yang [Yang, Zhong (1984)] proved the following result: Theorem 2.1.21

Assume Riccg ≥ 0, then λ1 (2, M ) ≥

π diam(M )

2 .

(2.1.68)

This estimate improves a previous one due to Li and Yau [Li, Yau (1980)]. Their proof is based upon an estimate of the function F (φ) = |∇g φ|2 ock formula. Recently 1−φ2 , assuming that |φ| < 1, and on the Weitzenb¨ [Naber, Valtorta (2014)], [Valtorta (2014)] gave a remarkable extension of this inequality to the p-Laplacian. Theorem 2.1.22

Assume Riccg ≥ 0 and p > 1, then p πp λ1 (p, M ) ≥ , p−1 diam(M )

(2.1.69)

where πp =

1

ds 2π √ = π . p p p sin( 1−s −1 p)

(2.1.70)

It is noticeable that this lower bound is much larger that the one obtained in (2.1.67). The proof is long and involves a delicate gradient comparison theorem with one-dimensional model. One of the intermediate result which is of intrinsic interest is the so-called p-Bochner formula. To state it, for any u ∈ C 2 (M ) we deﬁne the operator PuII by PuII (η) = |∇g u|p−2 Δg η + (p − 2)|∇g u|p−2 D2 η(∇g u), ∇g ug

∀η ∈ C 2 (M ), (2.1.71)

and set Au = Proposition 2.1.23 ∇g u = 0 there holds

D2 u∇g u, ∇g ug . |∇g u|2

(2.1.72)

Let p > 1 and u ∈ C 3 (M ). At any point where

1 II P (|∇g u|p ) = |∇g u|p−2 (∇g Δp u, ∇g ug − (p − 2)Au ) p u (2.1.73)

2 + |∇g u|2p−4 D2 u + p(p − 2)A2u + Riccg (∇g u, ∇g u) .

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Spherical p-harmonic functions

2.2 2.2.1

The spherical p-harmonic eigenvalue problem

Let S N −1 := (S N −1 , g0 ) be the unit sphere in RN endowed with the canonical metric g0 induced by the isometric imbedding into RN . We denote by (r, σ) the spherical coordinates in RN \ {0}, r > 0 and σ ∈ S N −1 , by ∇ ω the covariant gradient of a function ω deﬁned on S N −1 , by div = divg0 the divergence of vector ﬁelds on S N −1 and by Δ the Laplacian (or LaplaceBeltrami operator). If one looks for separable p-harmonic C 2 functions (p > 1) under the form u(x) = u(r, σ) = r−β ω(σ),

(2.2.1)

then ω satisﬁes 2 2 p−2 p−2 2 2 div β ω + |∇ ω| ∇ ω + βΛp (β) β 2 ω 2 + |∇ ω|2 2 ω = 0, (2.2.2) on S

N −1

with Λp (β) = β(p − 1) + p − N.

(2.2.3)

The spherical p-harmonic eigenvalue problem is to ﬁnd the couples (β, ω) which solve (2.2.2)-(2.2.3). Clearly βΛp (β) must be nonnegative. Constant −p , in which case the functions ω satisfy (2.2.2) if and only if β = 0 or β = Np−1 function u in (2.2.1) is called a fundamental solution. A necessary condition for the existence of non-constant solutions ω is (i)

−p , ∞) β ∈ (−∞, 0) ∪ ( Np−1

if 1 < p < N

(ii)

β ∈ (−∞, − p−N p−1 ) ∪ (0, ∞)

if p ≥ N.

(2.2.4)

If p = 2, (2.2.2) becomes −Δ ω = βΛ2 (β)ω.

(2.2.5)

The set of {βΛ2 (β)} is the set of integers {k(N + k − 2)}k∈N and the ω are the restriction to S N −1 of the k-homogeneous harmonic polynomials. Theorem 2.2.1 Let S S N −1 be a C 2 domain and p > 1. Then there exist two couples (β, ω) = (βS , ωS ) and (β, ω) = (βS , ωS ) where βS > 0,

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βS < 0, ωS , ωS ∈ W01,p (S) are positive solutions of p−2 div β 2 ω 2 + |∇ ω|2 2 ∇ ω p−2 + βΛp (β) β 2 ω 2 + |∇ ω|2 2 ω = 0 ω=0

(2.2.6)

in S in ∂S.

Furthermore ωS and ωS are C ∞ in S and unique up to a multiplication by a constant. If ω ∈ W01,p (S) is a positive solution of (2.2.6) with β > 0, v = − β1 ln ω satisﬁes p−2 p−2 −div 1 + |∇ v|2 2 ∇ v + β(p − 1) 1 + |∇ v|2 2 |∇ v|2 p−2 + Λp (β) 1 + |∇ v|2 2 |∇ v|2 = 0 in S lim v(σ) = ∞

ρ(σ)→0

in ∂S.

(2.2.7) where ρ(σ) is the geodesic distance from σ to ∂S. Note that this equation is never degenerate, therefore the solutions are C ∞ in S. The proof of Theorem 2.2.1 is a consequence of a more general result Theorem 2.2.2 Let (M d , g) be a d-dimensional Riemannian manifold with covariant gradient ∇ and divergence operator divg and let p > 1. For any relatively compact subdomain S ⊂ M with a nonempty boundary ∂S and β > 0, there exists a unique λ(β) > 0 such that the problem p−2 p−2 2 2 −divg 1 + |∇v| ∇v + β(p − 1) 1 + |∇v|2 2 |∇v|2 p−2 + λ(β) 1 + |∇v|2 2 |∇v|2 = 0 in S lim v(σ) = ∞

in ∂S,

ρ(x)→0

(2.2.8) where ρ(x) is the geodesic distance from x ∈ S to ∂S, admits a positive solution v ∈ C 2 (S) ∩ C(S). Furthermore v is unique up to the addition of a constant. When p = 2 this statement is customary in ergodic problems and the constant λ(β) is called an ergodic constant . By analogy this name remains in the quasilinear case even if no link with probability theory has been found so far. If we expand the equation the previous statement becomes

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Theorem 2.2.3 Let p > 1 and Δg be the Laplacian on (M d , g). For any relatively compact subdomain S ⊂ M with a nonempty boundary ∂S and β > 0, there exists a unique λ(β) > 0 such that the problem −Δg v −

2

p − 2 ∇ |∇v| , ∇vg + β(p − 1)|∇v|2 = −λ(β) 2 1 + |∇v|2 lim v(x) = ∞

in S in ∂S,

ρ(x)→0

(2.2.9) admits a positive solution v ∈ C 2 (S) ∩ C(S), unique up to the addition of a constant. Proof of Theorem 2.2.3. Existence. For > 0 consider the problem −Δg v −

2

p − 2 ∇ |∇v| , ∇vg + β(p − 1)|∇v|2 + v = 0 2 1 + |∇v|2 lim v(x) = ∞

in S in ∂S.

ρ(x)→0

(2.2.10) Construction of supersolutions and subsolutions. We denote by ρ(x) ˙ the signed geodesic distance from x ∈ M to ∂S which is equal to ρ(x) if x ∈ S and to −ρ(x) when x ∈ M \ S. Since ∂S is compact and C 2 , there exists δ > 0 such that for any x such that |ρ(x)| ˙ ≥ δ, there exists a unique σx ∈ ∂S ˙ < δ}. We extend such that |x − σx | = ρ(x). We set Nδ = {x ∈ M : |ρ(x)| ρ˙ to M \ Nδ as a smooth function ρ˜ and put u ˜(x) = −

1 M1 ln ρ˜(x) − M0 ρ˜(x) + , β

(2.2.11)

where the Mj will be chosen later on. Since |∇˜ ρ| = 1 in Nδ , |∇˜ u(x)|2 =

1 + 2βM0 ρ(x) + O(ρ2 (x)) β 2 ρ2 (x)

as ρ(x) → 0,

and Δg ρ˜(x) is bounded in M . Therefore 2

ug u| , ∇˜ p − 2 ∇ |∇˜ + β(p − 1)|∇˜ u|2 + ˜ u 2 1 + |∇˜ u|2 1 Δg − ρ˜ ln ρ˜ + 2(p − 1)M0 |∇˜ = ρ|2 + ψβ + M1 , ρ˜ β β

−Δg u ˜−

(2.2.12)

where ψβ is a bounded function depending on β and M0 . Thus we can choose M0 and M1 to ensure the positivity of the right-hand side term in

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(2.2.12). Notice also that the Mj and the bound for ψβ can be chosen independent of β when it remains in a compact subset of (0, ∞). Similarly the function u deﬁned by u(x) = −

1 M1 ln ρ˜(x) + M0 ρ˜(x) − , β

(2.2.13)

can be constructed as being a subsolution for suitable Mj compatible with the ones in u ˜. For 0 < h < δ we approximate u ˜ and u in the following way 1 ln(˜ ρ(x) − h) − M0 (˜ ρ(x) − h) + β 1 ρ(x) + h) − M0 (˜ ρ(x) + h) + uh (x) = − ln(˜ β u ˜h (x) = −

(i) (ii)

M1 M1 ,

(2.2.14)

and they are respectively supersolution in S \ Nδ and subsolution in S. The functions g and h are increasing functions of the nonnegative variable z. Consequently we can apply the comparison principle between supersolutions and subsolutions. Estimates. In this paragraph, we use the equation satisﬁed by the function 2 z = |∇u| , where u is a solution of (2.2.12), combined with Theorem 1.1.2. Set m = inf {Riccg (ξ, ξ) : |ξ| ≤ 1}. Any C 2 function deﬁned on M satisﬁes S

1 1 Δg |∇u|2 ≥ (Δg u)2 + ∇Δg u, ∇ug + m|∇u|2 , 2 d

(2.2.15)

by the Weitzenb¨ ock formula Theorem 1.1.2 and since |D2 u|2 ≥ d1 (Δg u)2 by Schwarz inequality. Identity (2.2.12) becomes −Δg z = −

ug p − 2 ∇z, ∇˜ + β(p − 1)z + ˜ u = 0. 2 1 + |∇u|2

(2.2.16)

Furthermore ∇Δg u ˜, ∇˜ ug = −

u, ∇˜ ug p−2 |∇z|2 ug )2 p−2 D2 z∇˜ p−2 (∇z, ∇˜ − + 2 2 2 4 4 1+|∇˜ 2 (1+|∇˜ 1+|∇u| u| u| )2

+β(p−1)∇z, ∇˜ ug +z. From (2.2.16) there exist two positive constants c0 , c1 , independent of and bounded with β such that 2 ) (∇z, ∇˜ u g u− )2 + . (Δg u)2 ≥ c0 z 2 − c1 (˜ 2 (1 + |∇˜ u| )2

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Using (2.2.16) we obtain Δg z + (p − 2) 2c1 − d −

u, ∇˜ ug Dz ∇˜

2

1 + |∇˜ u| (˜ u− )2 +

2c0 z 2 d

≥

(∇z, ∇˜ ug )2

(1 + |∇˜ u|2 )2

+ 2(m + )z

p − 2 |∇z|2 (∇z, ∇˜ ug )2 + (p − 2) + 2β(p − 1)∇z, ∇˜ ug . 2 2 2 1 + |∇˜ u| (1 + |∇˜ u| )2

2

2

Since 2c0dz + 2(m + )z ≥ c0dz − c2 by Young’s inequality, we derived that there exist positive constants Cj , j = 0, 1, 2, such that −Δg z − (p − 2)

u, ∇˜ ug Dz ∇˜ 2

1 + |∇˜ u|

+ C0 z 2 ≤ C1

2

|∇z|

2

1 + |∇˜ u|

+ C2 .

(2.2.17)

Let us deﬁne the operator A by Az = −Δg z − (p − 2)

u, ∇˜ ug Dz ∇˜

(2.2.18) 2 . 1 + |∇˜ u| In local coordinates Az = − i,j ai,j zxi xj + i,j bi zxi where the coeﬃcients ai,j and bi are bounded. The ellipticity condition is satisﬁed since ai,j ξi ξj ≤ max{1, p − 1}ξ, ξg ∀ξ = (ξ1 , ..., ξd ). min{1, p − 1}ξ, ξg ≤ i,j

It follows from (2.2.17) that z is a positive subsolution of an equation of the type 2

Az + h(z) + g(z) |∇z| = f,

(2.2.19)

where g(z) = −C1 /(1 + z), h(z) = C0 z 2 and f = C2 . The two functions g and h are increasing with respect to z ≥ 0, therefore the comparison principle applies for a subsolution and a solution of −Δg z − (p − 2)

u, ∇˜ ug Dz ∇˜ 2

1 + |∇˜ u|

+ C0 z 2 − C1

2

|∇z| = C2 . 1+z

(2.2.20)

It is easy to verify that there exist two large enough positive constants λ and μ such that the function z¯(x) =

λ + μ, ρ˜2 (x)

(2.2.21)

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is a supersolution of (2.2.20) which blows-up on the boundary. Henceforth, any bounded subsolution of the same equation is bounded from above by z¯. For 0 < h < δ we replace z¯ by z¯h deﬁned in S \ Nh by z¯h (x) =

λ + μ, (˜ ρ(x) − h)2

(2.2.22)

˙ < h}. The where, accordingly to the deﬁnition of Nδ , Nh = {x ∈ M : ρ(x) function z¯h is still a supersolution of (2.2.20) in S \ Nh which blows-up on the boundary. Therefore it dominates in this domain the restriction of any subsolution in S. Letting h → 0 implies that z¯ is an absolute dominant for any subsolution in S. As a consequence if u ∈ C 2 (S) satisﬁes (2.2.15) there holds |∇˜ u(x)| ≤

L0 + L1 , ρ˜(x)

(2.2.23) ∞

in S, for some positive constants L0 , L1 depending on u− L . Furthermore L0 and L1 can be chosen independent of β provided it remains in a bounded interval. Once uniform Lipschitz estimates hold, the regularity theory Theorem 1.3.9 and Theorem 1.3.10 presented in Chapter 1 yields local C 2,α estimates. Approximate solutions. By using Schauder’s ﬁxed point theorem it is easy to prove that for any n ∈ N∗ there exists a solution v := vn, to −Δg v −

2

p − 2 ∇ |∇v| , ∇vg + β(p − 1)|∇v|2 + v = 0 2 1 + |∇v|2 v=n

in S in ∂S.

(2.2.24) By comparison this solution is bounded from above by u ˜h for any 0 < h < δ, this leads to vn, (x) ≤ −

1 M1 ln ρ˜(x) − M0 ρ˜(x) + . β

(2.2.25)

Notice also that n → vn, is increasing by the maximum principle and it converges to some function v which satisﬁes also (2.2.25). Since {vn, } remains locally bounded in C 2,α (S), it converges in the local C 2 topology to v which is a C 2,α solution of (2.2.10). Furthermore v satisﬁes the same estimate (2.2.23) as u.

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The ergodic limit. Comparing v with uh and then to u we obtain the lower bound. Combining it with the upper bound, we see that v satisﬁes M1 1 M1 1 ln ρ˜(x) + M0 ρ˜(x) − ≤ v (x) ≤ − ln ρ˜(x) − M0 ρ˜(x) + . β β (2.2.26) This implies that v is locally bounded in S. Since ∇v is also locally bounded, it implies that there exist a subsequence {n } and a nonnegative constant λ0 such that n vn converges to λ0 locally uniformly in S. We ﬁx X0 ∈ S, set w = v − v (x0 ). Since w (x0 ) = 0, ∇w is locally bounded and there holds −

−Δg w −

2

p − 2 ∇ |∇w | , ∇w g + β(p − 1)|∇w |2 + w = −v (x0 ), 2 1 + |∇w |2

there exist a subsequence {nk } and a C 2+α function w such that {wnk } converges to w locally in the C 2 (S) topology. Thus w satisﬁes −Δg w −

2

p − 2 ∇ |∇w| , ∇wg + β(p − 1)|∇w|2 = −λ0 2 1 + |∇w|2

in S.

(2.2.27) At end we prove that w(x) tends to inﬁnity on the boundary. We set ψ(x) = −

1 ln ρ˜(x) + M0 ρ˜(x). β

Then 2 p − 2 ∇ ∇ψ , ∇ψg + β(p − 1)|∇ψ|2 + ψ −Δg ψ − 2 1 + |∇ψ|2 ρ˜ ln ρ˜ 1 Δg ρ˜ 2 − − 2(p − 1)M0 |∇˜ = ρ| + a β , ρ˜ β β where aβ is a bounded function. We can choose M0 > 0 and ρ0 ∈ (0, δ) such that ψ is a subsolution of (2.2.10) in Ω ∩ Nρ0 . Next we have that w (x) ≥ −c0 for some constant c0 > 0 on the set of x ∈ S such that ρ(x) = ρ0 since ∇w is locally bounded. Because ψ − c is still a subsolution for c ≥ 0 and independent of , we derive w (x) ≥ −

1 ln ρ˜(x) + M0 ρ˜(x) − c β

in Ω ∩ Nρ0 .

(2.2.28)

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Letting → 0 implies that w(x) → ∞ as ρ(x) → 0. In the same way the function ψ(x) = −

1 ln ρ˜(x) − M0 ρ˜(x) + c, β

is a supersolution and w, as any solution of (2.2.27) which blows-up on the boundary, satisﬁes −

1 1 ln ρ˜(x) + M0 ρ˜(x) − c ≤ w(x) ≤ − ln ρ˜(x) − M0 ρ˜(x) + c β β

in Ω ∩ Nρ0 . (2.2.29)

Proof of Theorem 2.2.3. Uniqueness. It will follow from the next statement. There exists a unique positive constant λ0 > 0 such that there is a function v ∈ C 2 (S) solution of −Δg v −

2

p − 2 ∇ |∇v| , ∇vg + β(p − 1)|∇v|2 + λ0 = 0 2 1 + |∇v|2

in S (2.2.30)

lim v(x) = ∞.

ρ(x)→0

The role of the ergodic constant λ0 is enlighted by the following equivalence result. Lemma 2.2.4 A function v ∈ C 2 (S) is a solution of (2.2.30) if and only if ω := e−βv ∈ C 2 (S) ∩ C(S) satisﬁes p−2 p−2 2 2 2 2 2 2 2 2 −divg β ω + |∇ω| ∇ω = βλ0 β ω + |∇ω| ω in S w=0

in ∂S.

(2.2.31) Moreover ω ∈ C (S) and ∇ω, ng < 0 on ∂S where n ∈ T S is the outward normal unit vector to ∂S. 1+α

Proof. A straightforward computation shows that the two problems (2.2.30) and (2.2.31) are exchanged by the setting ω := e−βv : if v ∈ C 2 (S) is a solution of (2.2.30), ω ∈ C(S) ∩ W 1,∞ (S) is a weak solution of (2.2.31). By Theorem 1.3.10, ω ∈ C 1 (S) and since (2.2.29) holds, v(x) + 1 ln ρ˜(x) ≤ c in Ω ∩ Nρ0 , (2.2.32) β

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which implies ρ˜(x)e−βc ≤ ω(x) ≤ ρ˜(x)eβc

in Ω ∩ Nρ0 .

(2.2.33)

Because w satisﬁes (2.2.23), β(L0 + ρ˜(x)L1 )e−βc ≤ |∇ω(x)| ≤ β(L0 + ρ˜(x)L1 )eβc

(2.2.34)

near the boundary, and thus ∇ω, ng < 0 on ∂S. Finally, since 2 β 2 ω 2 + |∇ω| > 0, the equation is not degenerate and thus the solutions are smooth. Conversely, any solution of (2.2.31) with the above indicated regularity is transformed into a C 2 (S) solution of (2.2.30). End of the proof of Theorem 2.2.3. Uniqueness of the ergodic limit. Assume λj , j = 0, 1, are two ergodic constants associated to solutions vj = e−βωj of (2.2.30) and (2.2.31). By energy estimates λj > 0. Assume λ0 > λ1 . By (2.2.33), there exists θ > 0 such that ω1 ≤ θω0 , and since the equation is homogeneous we can assume that θ = 1. The tangency conditions of the two graphs of the functions ω0 = e−βv0 and ω1 = e−βv1 implies that either there exists some x0 ∈ S such that ω1 (x0 ) = ω0 (x0 ) and ∇ω1 (x0 ) = ∇ω1 (x0 ), or ω1 < ω0 in S and there exists x0 ∈ ∂S such that ∇ω0 (x0 ), ng = ∇ω1 (x0 ), ng < 0. In the ﬁrst case the function z = v1 − v0 is nonnegative in S, achieves its minimal value 0 at x0 and satisﬁes at this point −Δg z(x0 ) − (p − 2)

D2 z(x0 )∇v0 (x0 ), ∇v0 (x0 )g 1 + |∇v0 (x0 )|2

= λ0 − λ1 > 0,

which is impossible because of ellipticity. In the second case the tangency condition at x0 yields ∇(ω0 (x0 ) − ω1 (x0 )), ng = 0, which implies |∇ω0 (x0 )|2 = |∇ω0 (x0 )|2 , whereas the function ζ = ω1 − ω0 is negative in S and vanishes at x0 . Since β 2 ωj2 + |∇ωj |2 > 0, the problem (2.2.31) is uniformly elliptic. Since ω0 (x0 ) = ω1 (x0 ), this is a contradiction with Hopf maximum principle. Because the ergodic limit λ0 is uniquely determined by β, we denote it by λ(β). Uniqueness of the eigenfunction up to homothety. Similarly to the previous argument, we consider two tangent solutions positive ω0 and ω1 (by homothety). We set ζ = ω1 − ω0 ≥ 0 and assume that ζ is not identically zero. We apply the strong maximum principle to ζ at x0 and derive that ζ has to vanish everywhere. Similarly Hopf maximum principle yields ζ ≡ 0, a contradiction. Note that this uniqueness implies that the whole sequence {v } converges to v as → 0.

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Next we study the properties of the ergodic constant. Proposition 2.2.5 Under the assumptions of Theorem 2.2.3 the mapping β → λ(β) is continuous and decreasing on (0, ∞). Furthermore lim λ(β) = ∞.

(2.2.35)

β→0

Proof. Monotonicity Assume 0 < β1 < β2 and let v,1 and v,2 be the corresponding solutions of (2.2.10) with β = β1 and β2 respectively. Using the maximum principle and letting n → ∞ in (2.2.24) yields v,1 > v,2 and thus λ(β1 ) = lim v,1 ≥ lim v,2 = λ(β2 ). →0

→0

If we assume that there exists 0 < β1 < β2 such that λ(β1 ) = λ(β2 ) = λ and denote by ω1 and ω2 the corresponding solutions of (2.2.31) with β = β1 and β2 and λ0 = λ(β1 ) and λ(β2 ) respectively, then ω1 ≤ ω2 and by (2.2.33), ρ(x) m−1 ρ˜(x) ≤ ω1 (x) ≤ ω2 (x) ≤ m˜

∀x ∈ S.

(2.2.36)

β2 β1

Put ω ˜ = ω . Then p−2 p−2 −divg β22 ω ˜ 2 + |∇˜ ω |2 2 ∇˜ ω − β2 λ β22 ω ˜ 2 + |∇˜ ω |2 2 ω ˜ p−1 β p−2 |∇ω1 |2 (p−1) β2 −1 β2 β2 1 β12 ω12 + |∇ω1 |2 2 ω1 < 0. = (p − 1) 1 − β1 β1 ω1 (2.2.37) Equivalently w ˜ is a strict supersolution of the equation satisﬁed by w2 . By homogeneity, we can assume that ω ˜ ≤ ω2 and that the coincidence set S of ω ˜ and ω2 is nonempty. Thus it must be a compact subset of S because ∇˜ ω = 0 on ∂S. Put 1 ˜ ) = w2 − w. ˜ z = − (ln ω2 − ln ω β2 Clearly z ≤ 0 is not identically zero and there exists x0 ∈ S where z(x0 ) = 0. Because of (2.2.29), z(x) → −∞ as ρ˜(x) → 0. If we expand (2.2.37), we obtain at x = x0 , −Δg z − (p − 2)

D2 z∇v2 , ∇v2 g + β2 (p − 1)(|∇v2 |2 − |∇˜ v |2 ) 1 + |∇v2 |2 v ), ∇˜ v g Hv˜ (∇˜ Hv˜ (∇v2 ), ∇v2 g ≤ 0. + (p − 2) − 1 + |∇˜ v |2 1 + |∇v2 |2

This contradicts the strong maximum principle. Thus λ(β1 ) > λ(β2 ).

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Continuity. Let {βn } ⊂ R+ be a sequence converging to β > 0, and vβn and v,βn be the corresponding solutions to (2.2.9) and (2.2.10) respectively with β = βn . Since v,βn remains locally bounded in S and converges locally in S to λ(βn ) as → 0, the set {λ(βn )} remains bounded. Up to a subsequence, still denoted by {βn }, {λ(βn )} converges to some λ ≥ 0 as n → ∞. Estimates (2.2.29) and (2.2.12) are valid with vβn and the constants therein uniform with respect to βn . Therefore vβn remains locally bounded in W 1,∞ (S) and thus in C 2,α (S). Up to a subsequence, not relabeled, {vβn } converges to v¯ as n → ∞, and v¯ satisﬁes −Δg v¯ − (p − 2)

Hv¯ (∇¯ v ), ∇¯ v g ¯ + β0 (p − 1)|∇¯ v |2 = −λ. 2 1 + |∇¯ v|

(2.2.38)

¯ = λ(β0 ) and the whole Because of the uniqueness of the ergodic constant λ sequence {λ(βn )} converges to λ(β0 ). Limit. By (2.2.31), 2 2 p 2 2 p−2 β ω + |∇ω|2 dvg = β(λ(β) + β) β ω + |∇ω|2 2 ω 2 dvg . S

S

Let λ1 (p, S) be the ﬁrst eigenvalue of −Δp in W01,p (S) and normalize ω by |∇ω|p dvg = 1. S

Then there holds

|ω|p dvg ≤ S

1 . λ1 (p, S)

Clearly

β 2 ω 2 + |∇ω|2

p

dvg ≥

S

|∇ω|p dvg = 1. S

If p ≥ 2, 2 2 p−2 p β ω + |∇ω|2 2 ω 2 dvg ≤ 2 2 (β p−2 |ω|p + ω 2 |∇ω|p−2 )dvg S

S

p p 2 2 p−2 p 2 2 β 1− ≤2 + |ω| dvg + 2 p p S p 22 2 p β p−2 + + 22. ≤ λ1 (p, S) p

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We derive p

22 β + λ(β) ≥ β

1 λ1 (p, S)

2 p−2 β +1 . + p

(2.2.39)

If 1 < p < 2, then S (β 2 ω 2

ω2 + |∇ω|2 )

2−p 2

dvg ≤ β p−2

|ω|p dvg ≤ S

β p−2 . λ1 (p, S)

Since β p−1 (λ(β) + β) ≥ λ1 (p, S) λ(β) ≥

λ1 (p, S) − β. β p−1

(2.2.40)

In the two cases, (2.2.35) follows.

Proof of Theorem 2.2.1: The singular case. For β > 0 we set ωβ = e−βvβ where vβ is the solution of (2.2.9). Then ωβ satisﬁes −divg

β 2 ω 2 + |∇ω|2

p−2 2

∇ω

p−2 = βλ(β) β 2 ω 2 + |∇ω|2 2 ω

ω=0

in S on ∂S.

(2.2.41) The conclusion will follow if we prove that there exists a unique β > 0 such that λ(β) = Λ(β) := β(p − 1) + p − N . The mapping β → (p − 1)β − λ(β) is continuous and increasing with lim (p − 1)β − λ(β) = −∞ and

β→0

lim (p − 1)β − λ(β) = ∞.

β→∞

Therefore there exists a unique βS > 0 such that (p − 1)β − λ(β) = N − p, which ends the proof. The regular case. We set β = −β with β > 0. Then the spherical pharmonic equation becomes −divg

β 2 ω 2 +|∇ω|2

p−2 2

p−2 ∇ω = β Λ (β ) β 2 ω 2 +|∇ω|2 2 ω ω =0

in S on ∂S (2.2.42)

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where Λ (β ) = β (p − 1) + N − p. If we set v = − β1 ln ω, then (2.2.9) becomes p−2 p−2 −divg 1 + |∇v|2 2 ∇v + β (p − 1) 1 + |∇v|2 2 |∇v|2 p−2 = −Λ (β ) 1 + |∇v|2 2 in S lim v(x) = ∞

in ∂S.

ρ(x)→0

(2.2.43) Thanks to Theorem 2.2.3, the remaining of the proof in the regular case goes as in the singular case, except that we conclude in proving, by the same limit argument, that there exists a unique β := βS > 0 such that (p − 1)β − λ(β ) = p − N . This ends the proof of Theorem 2.2.1. Remark. There is no known relation between βS and βS except in the case p = N where the N -Laplace equation is invariant by conformal transforx ) is N -harmonic in the mations (see Section 1.4). Thus if x → |x|−βS ω( |x| x x N ) cone CS = {x ∈ R \ {0} : |x| ∈ S} and vanishes on ∂CS , x → |x|βS ω( |x| endows the same properties, thus βS = −βS . 2.2.2

The 2-dimensional case

When N = 2, (2.2.2) becomes −

β 2 ω 2 + ω 2

p−2 2

ω

p−2 = β(β(p − 1) + p − 2) β 2 ω 2 + ωθ2 2 ω, (2.2.44)

on S 1 ≈ R/2π. If we set Y = ω /ω, then Y satisﬁes

β+1 β2 − 2 Y 2 + β2 Y + β(β − β0 )

Yθ = 1,

(2.2.45)

with β0 = 2−p p−1 . This equation is completely integrable by quadratures and the research of 2π-periodic solutions yields the following two results corresponding to singular and regular solutions. Theorem 2.2.6 Assume N = 2 and p > 1. Then for any k ∈ N∗ there exist βk > 0 and ωk : R → R, with least period 2π/k such that uk (x) ∼ uk (r, θ) = r−βk ωk (θ)

(2.2.46)

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is p-harmonic in R2 \ {0}; βk is the positive root of 2 1 2 2 β − ββ0 . (β + 1) = 1 + k

(2.2.47)

The couple (ωk , βk ) is unique up to an homothety over ωk . Theorem 2.2.7 Assume N = 2 and p > 1. Then for any k ∈ N∗ there exist βk ≥ 1 and ωk : R → R, with least period 2π/k such that

uk (x) ∼ uk (r, θ) = rβk ωk (θ)

(2.2.48)

is p-harmonic in R2 ; βk is the root larger or equal to 1 of (β − 1)2 =

1−

1 k

2

β 2 + ββ0 .

(2.2.49)

The couple (ωk , βk ) is unique up to an homothety over ωk . Remark. We notice that there always hold β1 = 1. Actually whatever is the dimension, the function x → xj is p-harmonic in RN and separable. Thus, if S is an hemisphere of the sphere S N −1 , βS = 1. The regular 2-dimensional equation plays a fundamental role in constructing N -dimensional separable p-harmonic functions in RN . We consider the following representation of spherical coordinates in RN := {x = (x1 , ..., xN )}. x1 x2 .. .

= r sin θN −1 sin θN −2 ... sin θ2 sin θ1 = r sin θN −1 sin θN −2 ... sin θ2 cos θ1 (2.2.50)

xN −1 = r sin θN −1 cos θN −2 xN = r cos θN −1 , where θ1 ∈ [0, 2π] and θj ∈ [0, π] for j = 2, ..., N − 1. If f is deﬁned in RN , we write f (x) ∼ f (r, θ1 , ..., θN −1 ). We can also denote a point σ on S N −1 as σ = (θ1 , ..., θN −2 , θN −1 ) = (σ , θN −1 ). In that case σ stands for a general point on S N −2 . Deﬁnition 2.1 A spherical p-harmonic eigenfunction ω on S N −1 is separable if there exist a system of coordinates (θ1 , ..., θN −2 , θN −1 = (σ , θN −2 ) with θ1 ∈ [0, 2π] and θj ∈ [0, π] for j = 2, ..., N −1, b > 0 and φ ∈ C 2 (S N −2 ) such that ω(σ) = (sin θN −1 )b φ(σ ).

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The next theorem shows how separable spherical p-harmonic eigenfunctions on S N −1 can be generated from spherical p-harmonic eigenfunctions on S N −2 . Proposition 2.2.8 Assume N ≥ 3, p > 1 and ω is a spherical pharmonic eigenfunction on S N −1 with exponent β > 0. If ω(σ) = (sin θN −1 )β φ(σ ) for some system of coordinates and φ is a spherical pharmonic eigenfunction on S N −2 with the same exponent β > 0. Proof. The gradient of a function ω is expressed by 1 ∇ N −2 ω, sin θN −1 S

∇S N −1 ω = −ωθN −1 e +

x by the rotation of center 0, angle where e is derived from |x| x deﬁned by 0, |x| and eN . If we set

Q = β 2 ω 2 + ωθ2N −1 +

π 2

in the plan

1 2 |∇S N −2 ω| , sin θN −1 2

equation (2.2.42) reads −

1 (sin θN −1 )N −2

(sin θN −1 )N −2 Q

p−2 2

ωθN −1

θN −1

p−2 p−2 1 2 ∇ N −2 ω Q div N −2 = β Λ (β , N )Q 2 ω, S S 2 sin θN −1 (2.2.51) where Λ (β , N ) = β (p − 1) + N − p. If ω is a solution under the form ω(σ , θN −1 ) = (sin θN −1 )β φ(σ ), then −

Q

p−2 2

= (sin θN −1 )(β

−1)(p−2)

β 2 φ2 + |∇S N −2 φ|

2

p−2 2

,

hence 1

(sin θN −1 )N −2 Q

p−2 2

ωθN −1

(sin θN −1 )N −2 θN −1 = N − 2 + (β − 1)(p − 1) − (N − 1 + (β − 1)(p − 1)) sin2 θN −1 ) p−2 2 2 φ, × β (sin θN −1 )(β −1)(p−1)−1 β 2 φ2 + |∇S N −2 φ|

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and

p−2 1 2 ∇ N −2 ω Q div N −2 S S sin2 θN −1 p−2 2 β 2 φ2 + |∇S N −2 φ|2 ∇S N −2 φ . = (sin θN −1 )(β −1)(p−1)−1 divS N −2

Finally φ satisﬁes p−2 2 2 2 2 β φ + |∇S N −2 φ| −divS N −2 ∇S N −2 φ p−2 2 2 2 2 φ, = β Λ (β , N − 1) β φ + |∇S N −2 φ| on S N −2 . The proof follows.

(2.2.52)

Remark. We can iterate this decomposition and construct separable spherical p-harmonic eigenfunctions under the form

w(θ1 , ...θN −1 ) = (sin θN −1 )β (sin θN −2 )β ...(sin θ2 )β ω(θ1 ),

(2.2.53)

where ω in a solution of (2.2.44) (with β = −β ). Therefore ω = ωk and β = βk for some k ∈ N∗ given in Theorem 2.2.7. This result is not surprising since

x21 + x22 x1 =⇒ sin θ1 =

, sin θN −1 sin θN −2 ... sin θ2 = 2 |x| x1 + x22 thus the corresponding separable p-harmonic function u in RN is the trivial extension to RN of the 2-dimensional separable p-harmonic function. Furthermore, in the case p = N we can construct separable N -harmonic functions regular or singular at 0 in a similar way by performing conformal transformation. N −1 , the exponent βS , although unknown exIf S is the hemisphere S+ plicitly, satisﬁes a remarkable inequality [Bidaut-V´eron, Garcia-Huidobro, V´eron (2015)].

Theorem 2.2.9

Assume N ≥ 3 and p > 1. Then there holds then β

>

N −1 p−1

(i)

if 1 < p < 2,

(ii)

% $ N −1 N −p < β N −1 < . if 2 < p < N, then max 1, S p−1 p−1 + (2.2.54)

S

N −1 +

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Proof. The eigenvalue identity. Let ω > 0 be the solution of (2.2.42) corresponing to β = βSN −1 . Because of uniqueness it depends only on the variable θN −1 . For simplicity put θ = θN −1 , then ω := ω(θ) satisﬁes p−2 1 sinN −2 θ β 2 ω 2 + ω 2 2 ω − N −2 sin θ p−2 (2.2.55) = βΛ(β) β 2 ω 2 + ω 2 2 ω in (0, π2 ) ω( π2 ) = 0 and ω (0) = 0. This equation can be written −ω − (N − 2) cot θ ω − (p − 2)

β 2 ω + ω = βΛ(β)ω. β 2 ω 2 + ω 2

Since π2 (ω +(N −2) cot θ ω ) cos θ sinN −2 θdθ = (N −1) − 0

π 2

(2.2.56)

ω cos θ sinN −2 θdθ

0

and

N −1 (β + 1) , βΛ(β) + 1 − N = (p − 1) β − p−1

we obtain (2 − p)

π 2

β 2 ω + ω 2 ω ω cos θ sinN −2 θdθ 2 ω 2 + ω 2 β 0 π2 N −1 (β + 1) ω cos θ sinN −2 θdθ. = (p − 1) β − p−1 0

(2.2.57)

Reduction. It is clear that ω decreases in a right neighborhood of 0 and that it cannot have a local minimum in (0, π2 ). We introduce elliptic coordinates associated to the arc of ellipse, Er = {(x, y) ∈ R2 : x > 0, y < 0, x2 + β −2 y 2 = r2 }, by setting ω = r cos φ and − ω = βr sin φ with φ = φ(θ) and r = r(θ). The functions r and φ are C 2 , hence rθ cos φ − rφθ sin φ = −βr sin φ and rθ = r(φθ − β) tan φ. Replacing into (2.2.56) we derive ﬁrst, rθ − (p − 1) + φθ cot φ + (N − 2) cot θ + Λ(β) cot φ = 0, (2.2.58) r

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and next, (p − 1)(φθ − β) tan φ + (φθ − Λ(β)) cot φ = (2 − N ) cot θ.

(2.2.59)

Estimate of φθ . The previous equation can be written (p − 1)(φθ − β) tan2 φ + φθ − Λ(β) = (2 − N )

cos θ sin φ . sin θ cos φ

(2.2.60)

Since lim

θ→0

sin φ cos φ = lim φθ = φθ (0), θ→0 cos θ sin θ

Λ(β) we derive φθ (0) = N −1 from (2.2.60). Because p < N , then Λ(β) < β(N −1) and thus φθ (0) < β. Similarly, we expand φ near θ = π2 and obtain φθ ( π2 ) = β. The key estimate is the following:

φθ (θ) ≤ β

∀θ ∈ [0, π2 ],

(2.2.61)

and we distinguish two cases: (i) If Λ(β) ≤ β, then (2 − N ) cot θ = (p − 1)(φθ − β) tan φ + (φθ − Λ(β)) cot φ ≥ ((p − 1) tan φ + cot φ) (φθ − β), and (2.2.61) follows. (ii) If Λ(β) > β then 0 < (p − 2)β − (N − p), and this is possible only if p > 2. We claim that β≤

N −2 . p−2

(2.2.62)

Equivalently β(Λ(β) − β) ≤ N − 2. Since lim cot θ tan φ = limπ

θ→ π 2

θ→ 2

cos θ sin θ 1 = lim = , cos φ θ→ π2 φθ sin φ β

there holds (p − 1) (φθ (θ) − β) tan2 φ = Λ(β) − φθ (θ) + (2 − N ) =

cos θ sin φ cos φ sin θ

π 1 (β(Λ(β) − β) + 2 − N ) + o(1) as θ → . β 2 (2.2.63)

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¯ = β we have At a point θ¯ where φθ (θ) ¯ (p − 1) tan φ(θ) ¯ + cot φ(θ) ¯ ¯ := (p − 1)φθθ (θ) A(θ) =

1+

N −2 Λ(β) − β

2

¯ cot2 θ¯ β(β − Λ(β)) + (N − 2)(1 + cot2 θ)

(N − 2)2 + 2 − N cot2 θ¯ Λ(β) − β N −2 N −2 ¯ − (β(N − 1)−Λ(β)) cot2 θ. = (2 − p)(β + 1) β − p−2 Λ(β) − β (2.2.64) If (2.2.62) does not hold, there exists > 0 such that φθ > β on [ π2 − , π2 ). ¯ = β and φθθ (θ) ¯ ≥ 0. Hence Therefore there exists θ¯ ∈ (0 π2 ) such that φθ (θ) ¯ 0 ≤ A(θ) < 0, a contradiction. Thus (2.2.62) holds.

= β(β − Λ(β)) + N − 2 −

−2 Next, if β < Np−2 , it follows from (2.2.63) that there exists > 0 such π that φθ < β on [ 2 − , π2 ). Therefore, if (2.2.61) does not hold, there exists 0 < θ1 < θ2 ≤ π2 − such that φθ (θ1 ) = φθ (θ2 ) = β and φθθ (θ2 ) ≤ 0 ≤ φθθ (θ1 ). Since N −2 N −2 − (β(N − 1) − Λ(β)) cot2 θi , A(θi ) = (2 − p)(β + 1) β − p−2 Λ(β) − β

we have A(θ2 ) ≤ 0 ≤ A(θ1 ) and A(θ2 ) − A(θ1 ) =

N −2 (β(N − 1) − Λ(β)) (cot2 θ1 − cot2 θ2 > 0, Λ(β) − β

since the function cot is decreasing. Contradiction, thus (2.2.61) holds. −2 Finally, if β = Np−2 and (2.2.61) does not hold, the maximum of φθ on is larger than β and achieved at some θ1 ∈ (0, π2 ). Then there exists [0, ¯ = β and φθθ (θ) ¯ ≥ 0. In this case, ¯ θ ∈ (0, π2 ) such that φθ (θ) π 2)

¯ =− 0 ≤ A(θ)

N −2 (β(N − 1) − Λ(β)) cot2 θ¯ < 0, Λ(β) − β

which is again a contradiction, and (2.2.61) holds in all the cases. End of the proof. Because r2 = β 2 ω 2 + ωθ2 and rθ = r(φθ − β) tan φ, we derive rrθ = (β 2 ω + ωθθ )ωθ = r2 (φθ − β) tan φ.

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By (2.2.61) β 2 ω + ωθθ ≥ 0 and since ωθ ≤ 0, β 2 ω + ωθθ 2 ω ω ≥ 0 =⇒ β 2 ω 2 + ωθ2 θ

0

π 2

β 2 ω + ωθθ 2 ω ω cos θ sinN −2 θdθ > 0. (2.2.65) β 2 ω 2 + ωθ2 θ

This implies the right-hand part of inequalities (2.2.54) (i)-(ii). Jointly with (2.2.4) (i) we obtain the result in (ii). 2.2.3

The case p → ∞

2.2.3.1

The spherical inﬁnite harmonic eigenvalue problem

The inﬁnite Laplacian Δ∞ admits the following form in spherical coordinates in RN , 1 1 1 2 2 2 2 2 − 2 ∇ ur + 2 |∇ u| , ∇ u. −∇ |∇u| .∇u(x) = − ur + 2 |∇ u| r r r r (2.2.66) Then separable solutions under the form u(x) = u(r, σ) = r−β ω(σ) may exist provided ω satisﬁes 1 2 2 ∇ |∇ ω| .∇ω + β(2β + 1) |∇ ω| ω + β 3 (β + 1)ω 3 = 0, 2

(2.2.67)

on S N −1 , an expression which can be deduced directly from the expression (2.2.2). If ω > 0 we set ω = e−βv and v satisﬁes 1 2 4 2 − ∇ |∇ v| .∇v + β |∇ v| + (2β + 1) |∇ v| + β + 1 = 0. 2

(2.2.68)

The next result [Bidaut-V´eron, Garcia-Huidobro, V´eron (2017)] shows that there exist separable inﬁnite harmonic functions in RN \ {0}. Theorem 2.2.10 Assume S S N −1 is a C 2 domain. Then for any γ > 0 there exist λ(γ) > 0 and a positive function v ∈ C 1 (S) viscosity solution 1 2 4 2 − ∇ |∇ v| .∇v + γ |∇ v| + (2γ + 1) |∇ v| + λ(γ) = 0 in S 2 (2.2.69) lim v(σ) = ∞. ρ(σ)→0

Proof. The proof is a delicate adaptation of Theorem 2.2.3 and we indicate only the two main estimates. For , θ > 0 we introduce the uniformly

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elliptic problem 1 2 4 2 −Δ v − ∇ |∇ v| .∇v + γ |∇ v| + (2γ + 1) |∇ v| + θv = 0 in S 2 lim v(σ) = ∞, ρ(σ)→0

(2.2.70) and set 1 2 4 2 P,γ v = −Δ v − ∇ |∇ v| .∇v + γ |∇ v| + (2γ + 1) |∇ v| + θv. 2 If a solution v of (2.2.70) depends only on the distance ρ to ∂S, in a neighborhood Nρ0 = {σ ∈ S : ρ(σ) < ρ0 } of ∂S, then it satisﬁes −v + (N − 2)Hv − v 2 v + γv 4 + (2γ + 1)v 2 + θv = 0 in Nρ0 lim v(ρ) = ∞, ρ→0

(2.2.71) where H is the mean curvature of the submanifold Sρ = {σ ∈ S : ρ(σ) = ρ}. Step 1: Construction of supersolutions and subsolutions. If a is a positive parameter, we introduce the singular ODE in (0, 1) y + y 2 y − γy 4 − ay 2 = 0 lim y (ρ) = −∞.

(2.2.72)

ρ→0

It can be completely integrated in y (which is negative). If we set y = −ζ −1 , there holds ( ( γ a 1 ζ + − arctan ζ = ρ, (2.2.73) a γ a a γ provided a > γ. Since 0 ≤ arctan x ≤ x for x ≥ 0, we infer ζ(ρ) ≥ γρ ⇐⇒ 0 ≥ y (ρ) ≥ −

1 . γρ

Using this estimate, we derive from (2.2.73) ( ( γ a √ √ ζ(ρ) ≤ tan(ρ aγ) ⇐⇒ y (ρ) ≤ − cot(ρ aγ). a γ From this estimate we obtain that for ρ0 ∈ (0, 1), √ sin ρ0 aγ 1 ρ0 1 ln ≤ y(ρ) − y(ρ0 ) ≤ ln √ γ sin ρ aγ γ ρ

∀ρ ∈ (0, ρ0 ].

(2.2.74)

(2.2.75)

(2.2.76)

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Assume (N − 2) |H| ≤ m in Nρ0 . Then for all τ ∈ (0, ρ0 ], any positive solution vτ of (2.2.71) in Nτ vanishing for ρ = τ satisﬁes P,γ (vτ ) = (N − 2)Hvτ + (2γ + 1 − a)vτ2 + θvτ ≥ |vτ | ((2γ + 1 − a) |vτ |) − m) ( a √ cot(τ aγ) − m . ≥ |vτ | (2γ + 1 − a) γ We choose a = a1 ∈ (1, 2γ + 1) and τ = τ1 ∈ (0, ρ0 ) such that ( a1 √ (2γ + 1 − a1 ) cot(τ aγ) ≥ m. γ Then vτ = vτ,a1 is a supersolution in Nτ . We choose now a = a2 > 2γ + 1, then P,γ (vτ ) ≤ |vτ | ((2γ + 1 − a) |vτ |) + m) + θvτ ( a √ cot(τ aγ) + θvτ . ≤ |vτ | m − (a − 2γ − 1) γ If we ﬁx a2 = 4γ + 2 − a1 , then ( a √ cot(τ aγ) ≤ −c < 0 m − (a − 2γ − 1) γ

∀τ ∈ (0, τ1 ],

which implies that vτ = vτ,a2 satisﬁes

vτ . P,γ (vτ ) ≤ vτ θ + vτ

Therefore there exists θ0 such that for any 0 < θ ≤ θ0 there holds P,γ (vτ ) ≤ −νvτ

in Nτ1 ,

(2.2.77)

for some ν > 0 depending on θ0 . Hence vτ,a1 and vτ,a2 are respectively supersolutions and subsolutions of P,γ (v) = 0 in Nτ1 and we extend them ∗ ∗ ∗ ) remain bounded by some to whole S by vτ,a1 and vτ,a2 in order P,γ (vτ,a j M1 > 0 in S\Nτ1 . Thus there exists M > 0 independent of and θ such that ∗ ∗ v = vτ,a + θ−1 M1 and v = vτ,a − θ−1 M1 are respectively supersolutions 1 2 and subsolutions of P,γ (v) = 0 in S. As in the proof of Theorem 2.2.3, ˙ + h) and for h > 0 small enough, we replace v and v by v h (σ) = v(ρ(σ) v h (σ) = v(ρ(σ) − h) which are respectively bounded subsolution in S and supersolutions in S \ Nh . By the maximum principle any C 2 solution v of problem (2.2.70) is larger than v h in S and dominated by v h in S \ Nh .

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Letting h to zero we infer that there exists M > 0 independent of , δ such that

(ii)

M ∗ ≤ v ≤ vτ,a 1 θ 1 + ln ρ ≤ M and γ

∗ − vτ,a 2

(i)

∗ vτ,a 2

+

M θ

in S

∗ vτ,a + 1 ln ρ ≤ M. 1 γ

(2.2.78)

Step 2: Gradient estimates. We claim that any C 3 solution of P,γ (v) = 0 in the geodesic ball BR (σ0 ) satisﬁes |∇ v(σ0 )| ≤

c , γR

(2.2.79)

where c = c(N ) > 0. We set z = |∇v|2 and deﬁne formally the linearized operator of Δ∞ at v by Bv (h) =

d 1 Δ∞ (v + th)t=0 = ∇ v, ∇ zg + ∇ (∇ v, ∇ hg ), ∇ vg , dt 2 (2.2.80)

and Lv (h) = Bv (h) + Δh.

(2.2.81)

If h = z we get Lv (z) =

1 2 |∇ z| + ∇ (∇ v, ∇ zg ), ∇ vg + Δz. 2

Since the equation satisﬁed by v can be written 1 ∇ v, ∇ zg = γz 2 + (2γ + 1)z + θv − Δv, 2 we obtain, by taking the gradient and multiplying by ∇ v, 1 ∇ v, ∇ (∇ v, ∇ zg )g = (2γz+2γ+1)∇ z, ∇ vg +θz−∇ (Δv), ∇ vg . 2 Since RiccS N −1 = (N − 2)g0 , Weinszenb¨ock’s formula reads 2 1 Δz = D2 v + ∇ (Δv), ∇ vg + (N − 2)z. 2

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110

2 This yields, after some computation and using (Δv)2 ≤ (N − 1) D2 v , Lv (z) =

≥

1 2 |∇ z| + 2 ((2γz + 2γ + 1)∇ z, ∇ vg + 2θz) 2 2 + 2 D2 v + 2(N − 2)z

(2.2.82)

1 2 |∇ z| + (Δw)2 + 4γ 2 z 3 − c0 , 2 N −1

for some c0 = c0 (N, γ) > 0. If ξ ∈ Cc2 (B R (σ0 )), Z = ξ 2 z satisﬁes Lv (Z) = ξ 2 Lv (z) + zLv (ξ 2 ) + 2∇ v, ∇ ξ 2 g ∇ v, ∇ zg + 2∇ z, ∇ ξ 2 g , (2.2.83) where 1 Lv (ξ 2 ) = ∇ z, ∇ ξ 2 g + ∇ v, ∇ (∇ v, ∇ ξ 2 g )g + Δξ 2 2 ≥ ∇ z, ∇ ξ 2 g − z D2 ξ 2 + Δξ 2 . Next we choose ξ such that 0 ≤ ξ ≤ 1 and R |∇ξ| + R2 D2 ξ ≤ c1 , then (z + 2)ξ (z + 2)2 ξ2 2 |∇ z| ≤ |∇ z| + c2 (z + 2) ∇ z, ∇ ξ 2 g ≤ c1 , R 8 R2 2 z Δξ 2 − z D2 ξ 2 ≤ c3 (z + 2) , R2

|∇ v, ∇ zg | ≤

√

z |∇ z| ,

√ ∇ v, ∇ ξ 2 g ≤ 2ξ |∇ v, ∇ ξg | ≤ 2c1 ξ z , R and 2 2 2∇ v, ∇ ξ 2 g ∇ v, ∇ zg ≤ 4c1 ξz |∇ z| ≤ ξ |∇ z|2 + c3 z . R 8 R2

At a point σ1 ∈ BR (σ0 ) where Z achieves its maximum, we have Lv (z)(ζ0 ) ≤ 0, which implies 1 2 ξ 2 2 c4 (z + 2)2 |∇ z| + (Δv)2 + 4γ 2 z 3 − c0 ≤ |∇ z| + . ξ2 2 N −1 4 R2 Imposing R ≤ 1 and < 12 , we multiply by ξ 4 and derive c5 (ξ 2 z)2 + 1 1 3 2 ξ 6 (Δv)2 2 2 3 ξ ∇z + + 4γ (ξ z) ≤ + c0 . 4 N −1 R2

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Because 4γ 2 (ξ 2 z)3 ≤ we derive, if we assume ξ(σ0 ) = 1, c7 z(σ0 ) ≤ Z(σ1 ) ≤ 2 R

c5 (ξ 2 z)2 c6 + 2, R2 R % $ c5 , with c7 = max c6 , γ

which is (2.2.79). The remaining of the proof of the theorem is based upon the existence of an ergodic constant and follows of the ideas developed in [Porretta, V´eron (2009)]. From this result it is possible to derive the existence of separable inﬁnite harmonic functions in the cone CS generated by S. The next result is the extension of Theorem 2.2.1 in the case p = ∞. Theorem 2.2.11 Let S S N −1 be a C 2 domain. Then there exist two real numbers βS > 0 and βS < 0 and at least two positive Lipschitz continuous functions deﬁned in S, ωS and ωS such that (r, σ) → u(r, σ) :=

r−βS ωS (σ) and (r, σ) → u (r, σ) : r−βS ωS (σ) are inﬁnite harmonic in CS and vanish on ∂CS \ {0}. The two functions ωS and ωS are viscosity solutions of (2.2.67) in S with zero boundary value on ∂S where β = βS and β = βS respectively. Furthermore βS and βS are unique. Remark. 1- Uniqueness of ωS and ωS , up to homothety is unknown.

2- Separable inﬁnite harmonic functions can be generated as in Proposition 2.2.8. 2.2.3.2

The 1-dim equation

If N = 2, (2.2.67) becomes −ωθθ ωθ2 = β 3 (β + 1)ω 3 + (2β + 1)ωθ2 ω,

(2.2.84)

in S 1 ∼ (0, 2π). This equation can be completely integrated by putting Y = ωθ [ ] ω , see Bhattacharya (2005) . The next result characterize the separable inﬁnite harmonic functions in R2 \ {0}. k2 k2 Theorem 2.2.12 For any k ∈ N∗ set βk = 1+2k and β˜k = 2k−1 . There π π ˜ k , positive on (0, k ), such that exist two k anti-periodic function ωk and ω

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x x x → |x|−βk ωk ( |x| ) and x → |x|βk ω ˜ k ( |x| ) are inﬁnite-harmonic respectively 2 2 in R \ {0} and R . The functions ωk and ω ˜ k are unique up to the multiplication by a constant.

Remark. 1- It is noticeable that the exponents βk or β˜k are rational numbers, contrary to the case p ∈ (1, ∞) \ {2} where they are quadratic real numbers. 2- It is proved in [Aronson (1984)] that the function 4

4

(x1 , x2 ) → x13 − x23 , is inﬁnite harmonic in R2 . It corresponds to k = 2, β˜k = 4 4 (cos θ1 ) 3 − (sin θ1 ) 3 , θ ∈ [0, 2π].

4 3

and ω ˜ 2 (θ1 ) =

Contrary to the case p < ∞, equation (2.2.84) is also valid for separable inﬁnite harmonic functions in a cone with vertex 0 generated by a spherical annulus or a spherical cap for functions ω satisfying some symmetry. If we use the representation of S N −1 (2.2.50) we deﬁne for 0 ≤ κ ≤ α ≤ π the spherical annulus Sκ,α centered at the north pole σN by Sκ,α = {σ ≈ (θ1 , ..., θN −1 ) : 0 ≤ θ1 < 2π, 0 ≤ θj < π if 2 ≤ j ≤ N − 2 and κ < θN −1 < α} , (2.2.85) and the spherical cap Sα Sα = {σ ≈ (θ1 , ..., θN −1 ) : 0 ≤ θ1 < 2π, 0 ≤ θj < π if 2 ≤ j ≤ N − 2 and 0 ≤ θN −1 < α} .

(2.2.86)

The following results are proved in [Bidaut-V´eron, Garcia-Huidobro, V´eron (2017)]. Proposition 2.2.13 Assume 0 ≤ κ ≤ α ≤ π and set ν = 12 (α − κ). Then there exist two positive separable inﬁnite hamonic functions in the cone CSκ,α under the form u(r, σ) = r−β ω(θN −1 ) vanishing on ∂CSκ,α \ {0} where ω is a solution of (2.2.84) in (κ, α) with θ = θN −1 satisfying ω(0) = ω(α) = 0,

(2.2.87)

with β = βSκ,α or β = βS κ,α , where βSκ,α =

π2 4ν(π + ν)

and βSκ,α = −

π2 . 4ν(π − ν)

(2.2.88)

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The exponents βSκ,α and βS κ,α are unique and the solutions ω of (2.2.84), (2.2.87) unique up to an homothety. Proposition 2.2.14 Assume 0 < α ≤ π. Then there exist two positive separable inﬁnite hamonic functions in the cone CSα under the form u(r, σ) = r−β ω(θN −1 ) vanishing on ∂CSα \ {0} where ω is a solution of (2.2.84) in (0, α) with θ = θN −1 satisfying ωθ (0) = 0 and ω(α) = 0,

(2.2.89)

with β = βSκ,α or β = βS κ,α if α < π, where βSκ,α =

π2 4α(π + α)

and βSκ,α = −

π2 . 4α(π − α)

(2.2.90)

The exponents βSα and βS α are unique and the solutions ω of (2.2.84), (2.2.89) unique up to an homothety. Remark. It is noticeable that in the case α = π there exists a separable inﬁnite hamonic function uD in RN \ D where D = {λeN : λ < 0} under the form uD (r, σ) = r− 8 ω(θN −1 ) 1

r > 0 , 0 ≤ θN −1 < π,

where ω is computable (not easy) since Y =

8ωθ 3ω

(2.2.91)

satisﬁes

8Y 3 = tan θ. 3Y 4 − 6Y 2 − 1 2.3 2.3.1

Boundary singularities of p-harmonic functions Boundary singular solutions

In all this section we assume that p > 1 and Ω is a C 2 domain with 0 ∈ ∂Ω and we consider equations with singular potential and shift of the type p−2

−Δp u + |∇u|

c(x), ∇u + d(x) |u|

p−2

u=0

in Ω.

(2.3.1)

We assume that c, d are measurable and locally bounded functions in Ω\{0}. In many applications to boundary singularity problems we assume that c and d satisfy p

|x| |c(x)| + |x| |d(x)| ≤ C0 for some C0 ≥ 0.

∀x ∈ Ω \ {0},

(2.3.2)

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The following Carleson estimate is an extension of Proposition 1.5.12. Proposition 2.3.1 Let R0 > 0, u ∈ C 1 (Ω \ {0}) be a positive solution of (2.3.2) which vanishes on ∂Ω \ {0} ∩ B2R0 and c, d satisfy (2.3.2). Then there exist positive constants c1 and R1 < R0 depending on p, C0 and Ω such that 1 u(x) u(x) u(y) ≤ ≤ c1 , c1 ρ(x) ρ(y) ρ(x) for all x, y ∈ Ω such that

1 2

(2.3.3)

|x| ≤ |y| ≤ 2 |x| ≤ R1 .

As a consequence we obtain the classical boundary Harnack inequality. Corollary 2.3.2 Let R0 > 0. If u, u ˜ ∈ C 1 (Ω \ {0}) are positive solutions of (2.3.2) vanishing on ∂Ω \ {0} ∩ B2R0 and c, d satisfy (2.3.2). Then there exist positive constants c2 = c2 (Ω, p, C0 ) and R1 = R1 (Ω, p, C0 ) < R0 such that for any a ∈ ∂Ω \ {0} ∩ BR1 there holds 1 u(x) u(x) u(y) ≤ ≤ c2 c2 u ˜(x) u ˜(y) u ˜(x)

∀x, y ∈ Ω ∩ B 2|a| (a).

(2.3.4)

3

As a variant of (2.3.4) it is easy to derive from (2.3.3) that for any a ∈ ∂Ω \ {0} there holds, u(x) u(y) 1 u(x) ≤ ≤ c3 c3 ρ(x) ρ(y) ρ(x)

∀x, y ∈ Ω ∩ B 2|a| (a).

(2.3.5)

3

If we combine this result with Harnack inequality we infer, Corollary 2.3.3 Let u, u ˜ ∈ C 1 (Ω \ {0}) be positive solutions of (2.3.2) which vanishes on ∂Ω\{0} and c, d satisfy (2.3.2). Then there exist positive constants c4 and r0 , depending on Ω and p such that for all r ∈ (0, r0 ] there holds % % $ $ u(x) u(x) : x ∈ Ω ∩ Γr ≤ c4 inf : x ∈ Ω ∩ Γr , (2.3.6) sup u ˜(x) u ˜(x) where Γr = B r \ B r2 . Proof. Since c, d satisfy (2.3.2) the classical Harnack inequality stipulates that for any b ∈ Ω, any θ > 1 and s > 0 such that B θs (b) ⊂ Ω and any nonnegative solution v of (2.3.2), there holds v(x) ≤ cv(y)

∀x, y ∈ Bs (b),

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115

where c = c(p, N, C0 , θ) > 0. Thus u(x) u(y) ≤ c2 u ˜(x) u ˜(y)

∀x, y ∈ Bs (b).

A simple geometric construction shows that there exist r0 > 0 and k0 ∈ N∗ such that for any r ∈ (0, r0 ] one can ﬁnd at most k0 + 1 points bj ∈ Ω such r that |bj | = 3r 4 , b0 , bk0 ∈ ∂Ω, |bj − bj+1 | ≤ 4 for 1 ≤ j < k0 , |b0 − b1 | = r 5r |bk0 −1 − bk0 | ≥ 2 , ρ(bj ) ≥ 12 and Γr ⊂ B r2 (b0 )

B r2 (0)

b 0 −1

B r3 (bj ).

j=1

We set rj = r3 if j = 1, ..., b0 − 1, r0 = rb0 = r2 . If x, y ∈ Γr , there exist indices jx , jy in {0, 1, ..., b0} such that x ∈ Brjx (bjx ) and y ∈ Brjy (bjy ). Therefore u(y) u(x) ≤ c2k0 . u ˜(x) u ˜(y)

(2.3.7)

Deﬁnition 2.3.4 Let Ω be bounded. We say that d ∈ L∞ loc (Ω \ {0}) satisﬁes the comparison hypothesis if for any > 0 and any two solutions u, u ˜ ∈ C 1 (Ω \ {0}) of (2.3.1) there holds ˜ u≥u ˜ in (∂Ω ∩ Bc ) ∪ (Ω ∩ ∂B ) =⇒ u ≥ u

in Ω ∩ Bc .

(2.3.8)

This assumption is satisﬁed if d ≥ 0. If d is signed and satisﬁed (2.3.2) p the comparison hypothesis holds provided |x| d− (x) is small enough. The main tool for estimating positive singular solutions is the following Theorem 2.3.5 Assume that Ω is bounded and d satisﬁes the comparison hypothesis. If u, u ˜ ∈ C 1 (Ω \ {0}) are nonzero nonnegative functions satisfying (2.3.1) with c = 0 and vanishing on ∂Ω \ {0}, there exists κ > 0 such that u ˜ ≤ κu. Proof. Since a nonnegative solution is either positive or identically zero by Harnack inequality, u and u ˜ are positive in Ω. Let us assume that no such κ ˜(xn ) ≥ nu(xn ). exists. Then for any n ∈ N∗ there exists xn ∈ Ω such that u Up to a subsequence xn → ξ ∈ Ω. Clearly ξ ∈ Ω is impossible because u and u ˜ are continuous and positive in Ω. We denote by ξn the projection of

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116

xn on the boundary. We ﬁrst assume that ξ ∈ ∂Ω \ {0} thus ξn = 0 for n large enough. Then ˜(ξn ) u(xn ) − u(ξn ) u ˜(xn ) − u ≥n . ρ(xn ) ρ(xn ) Since u ˜(xn ) − u ∂u ˜(ξ) u(xn ) − u(ξn ) ∂u(ξ) ˜(ξn ) →− > 0 and →− > 0, ρ(xn ) ∂n ρ(xn ) ∂n by Hopf lemma, we derive a contradiction. At the end we assume that ξ = 0 and we set rn = |xn |. By Corollary 2.3.3 there holds % $ u ˜(x) u˜(xn ) ≤ c4 min : x ∈ Ω , |x| = rn . n≤ u(xn ) u(x) By the local comparison theorem, it yields u ˜ ≥ nu in Ω ∩ Brcn which leads to a contradiction. Proposition 2.3.6 Under the assumption of Theorem 2.3.5 any nonnegative unbounded solution u ∈ C 1 (Ω \ {0}) of (2.3.1) tends to inﬁnity at 0 in every closed cone C0 with vertex 0 such that C0 ∩ Br0 ⊂ Ω ∪ {0}. Proof. Actually we prove a stronger result, namely |x|u(x) = ∞. x→0 ρ(x) lim

(2.3.9)

Assume it is not true, then there exists a sequence {xn } converging to 0 and a constant m > 0 such that rn u(xn ) ≤ ρ(xn )m. By Corollary 2.3.3 % $ |x|u(x) : |x| = rn ≤ c4 m, sup ρ(x) where rn = |xn |. Thus u(x) ≤ cm for all x ∈ Ω such that |x| = rn . By the maximum principle u(x) ≤ cm for all x ∈ Ω such that rn ≤ |x| ≤ r1 . Letting n to inﬁnity yields a contradiction. Such a function u is called a boundary singular solution. In the next theorem we prove that whenever it is nonempty the set of positive boundary singular solutions is a half straight line. Theorem 2.3.7 Assume that the assumptions of Theorem 2.3.5 are fulﬁlled and there exists a positive boundary singular solution u. Assume also

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either 1 < p ≤ 2 and d ≤ 0 or p ≥ 2 and lim inf x→0

ρ(x) |∇u(x)| > 0. u(x)

(2.3.10)

Then any other positive boundary singular solution is proportional to u. Proof. We assume that v is a nonzero nonnegative singular solution. Then there exists k > 0 such that v ≤ ku := uk . For > 0 we deﬁne κ by κ = inf{c > 0 : v(x) ≤ cu(x) : ∀x ∈ Ω, |x| = }. Thus v ≤ κ u on |x| = . Since d satisﬁes the comparison hypothesis, v ≤ κ u in Ω ∩ Bc , the mapping → κ is nonincreasing with limit κ as → 0 and v ≤ κu in Ω. We proceed by contradiction in assuming v = κu in Ω. Step 1: We claim that v < κu in Ω. Because of (2.3.10) there exists δ > 0 such that ∇u never vanishes in Ω ∩ Bδ . We ﬁrst prove that v(x) < κu(x) := uκ (x) in Ω ∩ Bδ . If it is not true the contact set Ξ of the graphs of v and uκ over Ω ∩ Bδ is not empty. The function w = uκ − v satisﬁes Lw + Dw = 0,

(2.3.11) up−1 −v p−1

. There exists a where L is a linear elliptic operator and D = κ w neighborhood of any ξ ∈ Ξ where L is not degenerate and D is bounded. By the strong maximum principle w is identically zero in this neighborhood. Therefore the set Ξ is open. This implies Ξ = Ω ∩ Bδ and in particular v = κu in Ω ∩ ∂Bδ . Since uniqueness follows from the comparison hypothesis, equality holds in whole Ω, a contradiction. Up to replacing δ by a smaller value, we have in particular v(x) < κu(x) for all x in Ω ∩ ∂Bδ . Let ξ ∈ ∂v κ ∂Ω∩Bδ . Then ∂u ∂n (ξ) and ∂n (ξ) are both negative. If p ≥ 2, D is bounded. Thus Hopf boundary lemma applies to w wich satisﬁes ∂w ∂n (ξ) < 0. If 1 < p < 2 and d is nonpositive, w is L-superharmonic, thus we still get ∂uκ ∂w ∂v ∂n (ξ) < 0. Because v(x) < κu(x) on Ω ∩ ∂Bδ and ∂n < ∂n on ∂Ω ∩ ∂Bδ there exists κ < κ such that v(x) ≤ κ u(x) on Ω ∩ ∂Bδ . By the comparison principle it implies v(x) ≤ κ u(x) in Ω ∩ Bδc . Since we can replace δ by a ∂v κ smaller δ ∈ (0, δ) we conclude that the inequalities v < κu and ∂u ∂n < ∂n hold in Ω and ∂Ω \ {0} respectively. Step 2: End of the proof. By the deﬁnition of κ, there exists a sequence

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{xn } converging to 0 such that $ % v(x v(xn = max : |x| = |xn | := rn → κ u(xn ) u(x)

as n → ∞.

Set an = max{u(x) : |x| = rn } and n = −rn n0 where n0 is the normal outward unit vector to ∂Ω at 0. For n large enough ρ(n ) ≤ rn ≤ 54 ρ(n ). By Proposition 2.3.1 1 u(y) u(n ) an ≤ ≤ c1 ρ(y) ρ(n ) ρ(n )

∀y ∈ Ω s.t.

rn ≤ |y| ≤ 2rn , 2

which implies 5ca n max u(y) : y ∈ Ω ∩ (B 2rn \ B r2n ) ≤ . 2

(2.3.12)

−1 We deﬁne un (x) = a−1 n u(rn x) and vn (x) = an v(rn x) and set dn (x) = p rn −1 rn d(rn x) for x ∈ Ω := rn Ω. Then un and vn are positive solutions of

−Δp w + dn wp = 0

in Ωn ,

(2.3.13)

and they vanish on Ωn \ {0}. By (2.3.12) both un and vn are uniformly ˜ n := Ωn ∩ (B 2 \ B 1 ). By the regularity results of Section 1-3, bounded in Ω 2 1 there exists a subsequence {nj } such that unj and vnj converge in the Cloc topology in Ωn ∩ (B 2 \ B 21 to some functions U and V and dnj converges in the weak-star topology of L∞ (Ωn ∩ (B 2 \ B 1 ) to some d˜ and U and V loc

2

are solutions of ˜ p=0 −Δp W + dW

in H ∩ B2 \ B 12 .

Noticing that |∇un (x)| =

rn |∇u(rn x)| ρ(rn x) |∇u(rn x)| rn u(rn x) , , = an u(rn x) an ρ(rn x)

and since by Corollary 2.3.3 rn u(rn x) rn u(xn ) |xn | 1 ≥ = ≥ , an ρ(rn x) an c4 ρ(xn ) c4 ρ(xn ) c4 it follows from (2.3.10) that |∇un (x)| ≥

γ c4

∀x ∈ H ∩ B2 \ B 12 .

(2.3.14)

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Set ξn =

xn rn

119

and $ min

v(x : |x| = rn u(x)

% = mn ≤ κ.

We can assume that ξn → ξ ∈ H ∩ S N −1 and mn → m ≤ κ. Furthermore V (ξ) = κU (ξ) and, if ξ ∈ ∂H, ∂U ∂V (ξ) = κ (ξ) < 0. ∂n ∂n The non-degeneracy inequality (2.3.14) implies that the strong maximum principle or Hopf boundary lemma are valid. Thus V and κU coincide in a neighborhood of ξ. By connectedness the coincidence set of V and κU is the whole set H ∩ B2 \ B 12 which yields m = κ. Therefore for any > 0 there exists n ∈ N such that n ≥ n implies (κ − )u(x) ≤ v(x) ≤ κu(x)

∀x ∈ Ω s.t. |x| = rn .

The comparison hypothesis implies (κ−)u(x) ≤ v(x) in Ω∩Brnc and ﬁnally, letting rn → 0 and → 0, v = κu. Remark. The existence of positive p-harmonic functions with a boundary singularity at 0 can be seen as a particular case of Theorem 3.3.11 in Chapter 3, with a geometric constraint on the domain Ω when 1 < p < N , since we have to assume that Ω is contain in a half-space and that ∂Ω ∩ Bδ = ∂H ∩ Bδ for some δ > 0. We end this section in proving an a priori estimate valid for positive solutions of (2.3.1) with a boundary singularity. Theorem 2.3.8 Assume p > 1, Ω is bounded and c, d satisfy (2.3.2). Then there exist θ > 0, r0 > 0 and c = c(Ω, p) such that if A ∈ Ω veriﬁes ρ(A) ≥ r0 , any positive solution u ∈ C 1 (Ω \ {0}) of (2.3.1) satisﬁes u(x) ≤ cu(A)ρ(x)|x|−θ−1

∀x ∈ Ω.

(2.3.15)

Proof. We recall that for β > 0, Ωβ = {x ∈ Ω : ρ(x) ≤ β}, Ωβ = Ω \ Ωβ . There exists r0 > 0 such that every x ∈ Ωβ admits a unique projection on ∂Ω denoted by σ(x), thus ρ(x) = |x − σ(x)|. If a ∈ ∂Ω and t ≤ r0 we set Nt (a) = a − tna where na is the normal outward unit vector at a. Then ρ(Nt (a)) = t and σ(Nt (a)) = a. We put A = −r0 n0 , A1 = −2−1 r0 n0 ,..., Ak = −2−k r0 n0 .

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120

Step 1: Estimate in Br0 . By Proposition 2.3.1 there holds u(Ak ) u(x) ≤ c1 ρ(x) ρ(Ak )

∀x ∈ B 2−k r0 \ B2−k−1 r0 ,

(2.3.16)

and in particular u(Ak+1 ) u(Ak ) u(A) ≤ c1 ≤ ck1 ρ(Ak+1 ) ρ(Ak ) r0

∀k = 1, ... .

(2.3.17)

Therefore u(Ak ) u(A) u(x) ≤ c1 ≤ ck1 ρ(x) ρ(Ak ) r0

∀x ∈ B 2−k r0 \ B2−k−1 r0 .

(2.3.18)

If x ∈ B r0 , there exists k ∈ N such that 2−k−1 r0 < |x| ≤ 2−k r0 . Thus r0 r0 < k ≤ log2 . log2 2|x| |x| Combined with (2.3.18) it yields u(x) ≤ ρ(x)

r0 |x|

c ln ln 2

u(A) r0

∀x ∈ B r0 ∩ Ω.

(2.3.19)

Step 2: Estimate in Ωr1 . We assume that 0 < r1 ≤ r30 . Since Ω is bounded, there exists an integer k1 such that any two points x and y in Ωr1 can be joined by a connected chain of at most k0 closed balls B r21 (bj ) of radius r21 and center bj . Since B 2r1 (bj ) ⊂ Ω, we have by Harnack inequality 3

u(x) ≤ ck0 u(y). This implies in particular u(x) ≤ ck1 u(A).

(2.3.20)

Step 3: Estimate in Ωr1 \ Br0 . If x ∈ Ωr1 \ Br0 , there exists a ∈ ∂Ω such that |x − a| ≤ r1 . Since r1 ≤ r30 , then |a| > 2r30 . Hence, by (2.3.5) and (2.3.20), c3 ck1 u(a − r1 na ) u(x) ≤ u(A). ≤ c3 ρ(x) r0 r0 Combining (2.3.19), (2.3.20) and (2.3.21) we obtain (2.3.15).

(2.3.21)

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2.3.2

121

Rigidity results for singular p-harmonic functions in the half-space N

Let 1 < p ≤ N and u ∈ C 1 (R+ \ {0}) be a positive solution of −Δp u = 0 u=0

in RN + on ∂RN + \ {0}.

(2.3.22)

By Theorem 2.2.1 there exists a positive solution of (2.3.22) under the form

VRN (r, σ) = r +

−β

S

N −1 +

ωS N −1 (σ) +

N −1 ∀(r, σ) ∈ R+ × S+ .

(2.3.23)

The next results are extensions of Theorem 2.3.7 when the domain is a half-space. We prove that under some growth assumptions, all the positive . p-harmonic functions vanishing ∂RN + \ {0} are multiple of VRN + N

Theorem 2.3.9 Assume p > 1 and u ∈ C 1 (R+ \ {0}) is a nonnegative N p-harmonic function in RN + which vanishes on ∂R+ \ {0} and such that (i) lim sup |x|

βS N −1 +

x→0

u(x) < ∞ and (ii) lim |x|−1 u(x) = 0. |x|→∞

(2.3.24)

. Then there exists k ≥ 0 such that u = kVRN + Proof. Step 1: We claim that u(x) → 0 when |x| → ∞. We recall that βS N −1 = −1 and x → W (x) := xN is the regular nonnegative separable +

N p-harmonic function in RN N −1 . + which vanishes on ∂R+ . We put β+ := βS+ By (2.3.4)

u(x) ≤ c2

u(|x| eN ) u(|x| eN ) W (x) = c2 xN W (|x| eN ) |x|

if |x| ≥ 1.

This implies that for any > 0 there exists R > 0 such that for all R ≥ R , u(x) ≤ sup u(y) + xN |y|=1

∀1 ≤ |x| ≤ R.

Letting R → ∞ and → 0 yields u(x) ≤ sup u(y) = C |y|=1

∀|x| ≥ 1.

The proof of Lemma 1.5.7 admits several generalizations and Lemma 3.3.2 and Lemma 3.4.15 in next section provide very useful and general ones.

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122

However it is easy to see, using the usual scaling techniques and regularity results of Section 1-3, that there holds for some α ∈ (0, 1), (i)

|∇u(x)| ≤ c(N, p)C|x|−1

(ii)

|∇u(x) − ∇u(x )| ≤ c(N, p, α)C|x|−1−α |x − x | , α

(2.3.25)

for all x, x such that 2 ≤ |x| ≤ |x |. If {rn } is an increasing sequence −1 , tending to inﬁnity, we set un (x) = u(rn x). Then 0 ≤ un ≤ C in RN + ∩ Br n and from (2.3.25), (i)

|∇un (x)| ≤ c(N, p)C|x|−1

(ii)

|∇un (x) − ∇un (x )| ≤ c(N, p, α)C|x|−1−α |x − x |α ,

(2.3.26)

if 2rn−1 ≤ |x| ≤ |x | ≤ 2 |x|. The sequence {un } is relatively compact in 1 topology of RN the Cloc + \ {0} and bounded by C. Up to a subsequence it converges in this topology to a bounded nonnegative p-harmonic function N Z in RN + which vanishes on ∂R+ \ {0}. Since Z is bounded its possible singularity at 0 is removable (this can be seen by performing a reﬂexion through ∂RN + ). By Theorem 1.5.13, Z(x) = axN for some a ≥ 0. The boundedness of Z implies a = 0. Therefore u(x) → 0 when |x| → ∞. Step 2: End of the proof. We apply again boundary Harnack inequality to in B1+ \ {0}. Because of (2.3.24)-(i), we see that there exists u and VRN + C > 0 such that (x), u(x) ≤ C VRN +

(2.3.27)

+ + in BR for > 0 and for all x ∈ B1+ \ {0}. Comparing u and (C + )VRN + R ≥ R large enough, we derive from the previous estimate, Step 1 and the + . Letting → 0 implies that comparison principle that u ≤ (C + )VRN + . (2.3.27) holds in RN + Finally, we use a variant of Lemma 1.5.7 valid if μp is replaced by |.|−β , and obtain the following estimates with c = c(N, p, α) > 0:

(i)

|∇u(x)| ≤ c|x|−β−1

(ii)

|∇u(x) − ∇u(x )| ≤ c|x|−β−1−α |x − x | , α

(2.3.28)

for all x, x ∈ RN + such that |x| ≤ |x |. Let k ≥ 0 be the supremum of the c ≤ C such that (2.3.27) holds with c instead of C . We can assume that are k > 0 otherwise u = 0. If the graphs of the two functions u and kVRN + N

tangent at some point x0 ∈ R+ \ {0}, it follows from the strong maximum

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123

N principle if x0 ∈ RN . + or Hopf lemma if x0 ∈ ∂R+ \ {0} that u = kVRN + never vanishes. If the two graphs are not tangent in Notice that ∇VRN + N

R+ \ {0}, they are asymptotically tangent either at 0 or at inﬁnity. In the ﬁrst case there exist two sequences {kn } decreasing to k and {xn } converging to 0 such that u(xn ) = kn . VRN (xn ) + We set rn = |xn | and un (x) = rnβ u(rn x). Then un is positive and pN harmonic in RN + , vanishes on ∂R+ \ {0} and satisﬁes the same estimates (2.3.28) as u. Up to a subsequence N 1 Cloc (R+

xn rn

N −1

→ ξ ∈ S+

and un → U in

the \ {0}) topology, where U is a positive p-harmonic function in which vanishes on ∂RN . Furthermore RN + + \ {0} and such that U ≤ kVRN + N −1

are tangent at ξ ∈ S + . As above it implies the graphs of U and kVRN + . Then, for > 0, there exists n ∈ N such that, for n ≥ n , U = kVRN + (x) ≤ u(x) ≤ (k + )VRN (x) (k − )VRN + +

∀x s.t. |x| = rn .

By the maximum principle the above inequality holds for all x such that vanish |x| ≥ rn ; notice that here we use also the fact that both u and VRN + ≤ u ≤ (k + )V N at inﬁnity. Letting n → ∞ yields (k − )VRN R+ which + implies the claim in this case. are tangent We are left with the case where the graphs of u and kVRN + only at inﬁnity. It means that if we deﬁne k by u(x) V (x) |x|≤ RN +

k = sup

∀ > 0,

then k < k and it is a increasing function of . Put k = lim sup x→0

u(x) u(rn ξn ) = lim < k, n→∞ VRN (rn ξn ) VRN (x) + + N −1

where |ξn | = 1, and the sequence {rn } decreases to 0 while ξn → ξ ∈ S + . We set un (x) = rnβ u(rn x). As above there exists a positive p-harmonic N function U in RN + which vanishes on ∂R+ \ {0} such that un → U and N −1

and ξn = xrnn → ξ ∈ S + . As in the ﬁrst case it implies that U = k VRN + for every > 0 there exists n ∈ N such that for n ≥ n , (x) ≤ u(x) ≤ (k + )VRN (x) (k − )VRN + +

∀x s.t. |x| = rn .

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Therefore this inequality holds for |x| ≥ rn and thus in RN + . This implies contradicting the tangency assumption. that u = k VRN + If p = N , βS N −1 = 1, βS N −1 = −1 and the result holds without the + − upper estimate at x = 0. N

Theorem 2.3.10 Assume p = N and u ∈ C 1 (R+ \ {0}) is a nonnegative N N -harmonic function in RN + which vanishes on ∂R+ \ {0} and satisﬁes lim |x|−1 u(x) = 0.

(2.3.29)

|x|→∞

. Then there exists k ≥ 0 such that u = kVRN + (x) = |x| Proof. We recall that VRN +

−2

xN . As in the proof of Theorem 2.3.9−2

Step 1, u(x) → 0 when |x| → ∞. Set u˜(y) = u(x) with y = |x| x. Then N ˜(y) → 0 u ˜ is N -harmonic in RN + and vanishes on ∂R+ \ {0}. Furthermore u when y → 0. By Theorem 1.5.13 u ˜(y) = kyN for some k ≥ 0. This implies the claim. Remark. In the case p = 2 the Kelvin transform u ˜(y) = |x|2−N u(x) with −2 y = |x| x leaves the Laplace equation invariant. Therefore the above proof can be adapted to this case. This yields the following: any nonnegative N harmonic function u in RN + which vanishes on ∂R+ \ {0} and satisﬁes −N (2.3.29) is a multiple of the Poisson kernel x → |x| xN . 2.4

Notes and open problems

2.4.1. The uniqueness of the ﬁrst eigenfunction of the p-Laplacian is due to Lindqvist [Lindqvist (1990-1992)]. Other proofs either involving some regularity conditions on the boundary or topological condition on Ω have been obtained by Anane [Anane (1987)], Barles [Barles (1988)] or Guedda and V´eron [Guedda, V´eron (1988)]. The uniqueness proof presented here is an adaptation of Belloni and Kawohl’s argument [Belloni, Kawhol (2002)]. 2.4.2. There exist geometric lower bounds on the ﬁrst positive eigenvalue λ1 (p, M ) of a compact complete manifold. It is proved in [Matei (2000)] that if the Ricci curvature of M satisﬁes Ricg ≥ (1 − N ) there exists a constant c = c(k, p) > 0 such that λ1 (p, M ) ≤ cN,p (hM + hpM )

(2.4.1)

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125

where hM is deﬁned in (2.1.61). If the sectional curvature of M is nonnegative, then λ1 (p, M ) ≤

22p+2 (N + p)N +p . (N − 1)N −1 pp (diam(M ))p

(2.4.2)

2.4.3. The existence of the regular couple (βS , ωS ) in Theorem 2.2.3 when S is a smooth subdomain of S N −1 is due to Tolksdorf [Tolksdorﬀ (1983)]. His proof of existence is speciﬁc to the case the equation on a domain ofS N −1 . An other proof, more ﬂexible, is obtained by Porretta and V´eron [Porretta, V´eron (2009)]. N −1

c 2.4.4. If S ⊂ S N −1 is a domain such that cR eron 1,p (S ) > 0, [Gkikas, V´ (2017)] prove the existence of a singular couple (β S , ωS ) (and a regular couple (β¯S , ωS ) as well) in approximating S by an increasing sequence of smooth subdomain {Sk }. They prove that a subsequence of normalized solutions {ωk } of the corresponding spherical p-harmonic eigenvalue problem in Sk with β = κk converges weakly in W01,p (S) and that {∇ω k } converges in Lploc (S) to some solution ω ∈ W01,p (S) of (2.2.6), with β = limk→∞ βk .

If S coincides with the interior of its closure it is possible to approximate S from outside by a decreasing sequence of smooth subdomains {S k }. A subsequence of the corresponding spherical p-harmonic eigenvalue problem 1,p (S) to some solution ω ∈ in S k with β = β k converges weakly in Wloc 1,p W (S) of (2.2.2) in S with β =: β = limk→∞ β k . The fact that ω ∈ W01,p (S) is a consequence of Netrusov’s theorem [Adams, Hedberg (1999), Th 10.1.1]. The exponents β and β are respectively the maximal and the minimal possible exponents corresponding to singular separable solutions in CS . If S is Lipschitz then β = β and there exists c > 0 such that cω ≤ ω ≤ c−1 ω. The fact that ω = λω is an open problem. Deep results in connexion with the characterization of the Martin boundary for p-harmonic functions in a domain can be found in [Lewis, Nystr¨ om (2010)] and [Lewis (2012)]. 2.4.5. It is proved by Ancona [Ancona (2012)] that the quotient of two positive harmonic functions in a cone CS generated by any spherical domain S, and vanishing on ∂CS is constant. As a consequence, any such harmonic function u is separable under the form u(r, σ) = λrbS ωS (σ),

(2.4.3)

where λ1,S is the ﬁrst eigenvalue of −Δ in W01,2 (S), bS > 0 depends explicitely of λ1,S and λ is an arbitrary positive real number. When p = 2 it

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is conjectured that any positive p-harmonic function u in C S vanishing on ∂CS is separable and under the form

u(r, σ) = λr−βS ωS (σ).

(2.4.4)

If u is singular at 0, u it is conjectured that u(r, σ) = λr−βS ωS ) at least in the case where the ωS are unique up to homothety. 2.4.6. The eigenvalue problem for the inﬁnite Laplacian is solved by [Juutinen, Lindqvist, Manfredi (1999)] and then extended by [Juutinen (2007)]. Local gradient estimates of inﬁnite harmonic functions are obtained by [Evans, Smart (2011)] where it is proved the diﬀerentiability of the viscosity ﬁrst eigenfunction; uniqueness is an open problem. The existence of separable inﬁnite harmonic function in a cone is proved by [Bidaut-V´eron, Garcia-Huidobro, V´eron (2017)]. They prove existence of such solutions in any cone, regular or not. 2.4.7. When p = 1, existence of separable 1-harmonic functions in RN \ {0} under the form u(r, σ) = r−β ω(σ) leads to the spherical 1-harmonic eigenvalue problem on S N −1 that satisﬁes ω ∇ ω ω −div = (1 − N )β 2 2 β 2 ω 2 + |∇ ω| β 2 ω 2 + |∇ ω|

(2.4.5)

on S N −1 . A necessary condition for existence of a solution in S ⊂ S N −1 vanishing on ∂S is β < 0. The construction of solutions seems an open problem. 2.4.8. Given a d- dimensional compact Riemannian manifold without boundary (M d , g), it is an open problem to prove the existence of couples (β, ω) where β ∈ R and ω ∈ C 1 (M ) which satisﬁes p−2 p−2 2 2 2 2 β 2 ω 2 + |∇ω| ∇ω = βΛ(β) β 2 ω 2 + |∇ω| ω (2.4.6) −divg in M . If β = 0 any constant satisﬁes (2.4.6). If β = 0 any solution satisﬁes

β ω + |∇ω| 2

2

2

p−2 2

ωdvg = 0.

(2.4.7)

M

If there exist an involutive isometry τ of M and two relatively open smooth domains M and M = τ (M ) such that ∂M = ∂M = {σ ∈ M : τ (σ = σ)} and M = M ∪ M ∪ ∂M , it is possible to construct such a couple in

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127

considering (β, ω ) where ω is a positive solution of (2.4.6) in M vanishing on ∂M and then by extending ω to whole M is putting # if σ ∈ M ω (σ) ω(σ) = −ω ◦ τ (σ) if σ ∈ M . 2.4.9. The study of p-Laplace equations perturbed by a potential of the following type −Δp u + d(x)|up−1 |u = 0

(2.4.8)

x is a natural problem. When the potential is written d(x) = D( |x| )|x|−p it is possible to look for singular separable solutions under the form u(r, σ) = r−β ω(σ) and ω is a solution of p−2 p−2 div β 2 ω 2 + |∇ ω|2 2 ∇ ω + βΛp (β) β 2 ω 2 + |∇ ω|2 2 ω (2.4.9) − D(σ)|ω p−1 |ω = 0.

The study of this spherical equation in a domain Ω which boundary contains 0 could be a way to analyze the boundary behaviour of solutions of (2.4.8) vanishing on ∂Ω \ {0}. 2.4.10.

A similar problem is the research of separable solutions of − Δp u = 0 |∇u|

p−2

∂u p−2 + d(x) |u| u=0 ∂n

in RN + in ∂RN + \ {0},

(2.4.10)

x )|x|1−p . If u(r, σ) = r−β ω(σ), then assuming that d(x) = D( |x|

2 2 p−2 β ω + |∇ ω|2 2 ∇ ω p−2 N −1 + βΛp (β) β 2 ω 2 + |∇ ω|2 2 ω = 0 in S+ 2 2 p−2 N −1 β ω + |∇ ω|2 2 ∇ ω, eN + D(σ)|ω p−1 |ω = 0 in ∂S+ .

div

(2.4.11)

2.4.11. The construction of p-harmonic functions in a bounded C 2 domain with a boundary singularity when N > p > 1 and Ω is not ﬂat near this singularity is still an open problem. We conjecture existence and uniqueness of such singular solutions (up to homothety). It could be also interesting to ﬁnd a way diﬀerent from super and sub solutions to construct such

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solutions. In this direction, it is worth noting that VRN (r, σ) := r−β ω(σ) + (where β = βS N −1 and ω = ωS N −1 ) deﬁned in RN + satisﬁes + + p−2 PV , ∇ζdx = T (ζ), (2.4.12) ∇VRN ∇VRN + + H

N

where P V stands for principal value, for all ζ ∈ C01 (R+ ) such that ζ(x) = ζ(r, σ) = rΛ(β) ψ(σ)

N −1

where ψ ∈ C01 (S +

),

(we recall that Λ(β) = β(p − 1) + p − N ) and where T is the functional deﬁned by p−2 ∇VRN , eN ζdS T (ζ) = lim ∇VRN + + →0 ∂B + (2.4.13) 2 2 p−2 β ω + |∇ ω|2 2 (βωσ.eN − ∇ ω, eN g ) ψdσ. = N −1 S+

In [Lewis (2012), Th 3.9] it is shown that in a Reifenberg-ﬂat domain the set of positive boundary singular solutions is a half-line. The proof is based on a sharp form of boundary Harnack inequality for p-harmonic functions. 2.4.12. In [Bhattacharya (2005)] it is proved that all the nonnegative inﬁnity harmonic functions in a half-space vanishing on the boundary except 1 one point a are comparable to the singular one x → |x − a|− 3 . His proof is based upon Harnack inequality and boundary Harnack inequality. Other results concerning blow-up estimates of inﬁnity harmonic functions with a boundary singularity are obtained in [Bhattacharya (2004)].

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Chapter 3

Quasilinear equations with absorption

In this section we present several aspects of the isolated singularities problem for solutions of −Δp u + g(u, |∇u|) = 0,

(3.0.1)

either in a punctured domain Ω := Ω \ {one point} (internal singularity), or lying at the boundary ∂Ω \ {one point} of a domain Ω (boundary singularity). The function g is either |u|q−1 u or |∇u|q and the interest lies in the case where the reaction term g(u, ∇u) dominates the diﬀusion term Δp u by its growth at inﬁnity, which means q > p − 1 > 0. In the cases we consider here the reaction is called an absorption term. The common thread underlying the description is the existence of explicit separable solutions of (3.0.1), although in many cases the proof of their existence is far from simple. The second step relies on existence of a priori estimates of the possible blow-up near the singularity. 3.1

Singular solutions with power absorption in RN

Let q > p − 1 > 0. Existence of separable solutions of −Δp u + |u|q−1 u = 0,

(3.1.1)

in RN \ {0} under the form (in spherical coordinates) u(x) = u(r, σ) = p and ω satisﬁes r−β ω(σ) yields β = βp,q = q+1−p 2 2 p−2 −div βp,q ω + |∇ ω|2 2 ∇ ω + |ω|q−1 ω p−2 2 2 − βp,q Λ(βp,q ) βp,q ω + |∇ ω|2 2 ω = 0 in S N −1 , (3.1.2) 129

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130

where, with the notations of Chapter 2, pq −N . q+1−p (3.1.3) A necessary condition for the existence of a solution to (3.1.2) is Λ(βp,q ) > 0 which is equivalent to p βp,q Λ(βp,q ) = βp,q ((p − 1)βp,q + p − N ) = q+1−p

(i)

0 < p − 1 < q < q1 :=

(ii)

0 p − 1 > 0 and u ∈ L∞ loc (BR0 ) ∩ Wloc (BR0 ) which satisﬁes, for some constants a > 0, b ≥ 0,

−Δp u+ + auq+ ≤ b

in D (BR0 ).

(3.1.6)

1q b + , a

(3.1.7)

Then u(x) ≤

c aR0p

1 q+1−p

where c = c(N, p, q) > 0. Proof. Let R < R0 and

p

v(x) = λ(Rp − |x|p )− q+1−p + μ.

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Since v(x) = v(r) is radial with respect to 0, a straightforward calculation yields −Δp v + av q ≥ b

in BR ,

provided aμq = b and aλq+1−p = c(N, p, q)Rp . By the comparison principle v ≥ u in BR and in particular 1 1q b c(N, p, q) q+1−p + . v(0) = aRp a

The result follows by letting R → R0 .

Theorem 3.1.3 Assume Ω is a domain containing 0, g : R → R is a continuous function satisfying lim inf r−q1 g(r) > 0, r→∞

lim sup |r|−q1 g(r) < 0, r→−∞

(3.1.8)

1,p and u ∈ L∞ loc (Ω \ {0}) ∩ Wloc (Ω \ {0}) satisﬁes

−Δp u + g(u) = 0,

(3.1.9)

in Ω \ {0}. Then u can be extended as a solution of (3.1.9) in Ω. The proof is based upon the following result. Lemma 3.1.4 Let Ω be a domain containing 0, 1 < p < N and q ≥ q1 . 1,p If u ∈ L∞ loc (Ω \ {0}) ∩ Wloc (Ω \ {0}) satisﬁes −Δp u + auq ≤ b

a.e. in {x ∈ Ω : u(x) ≥ 0},

(3.1.10)

for some constants a > 0, b ≥ 0. Then u+ ∈ L∞ loc (Ω) Proof. Set R0 = dist (0, ∂Ω) and we can assume R0 > 1. It follows from Lemma 3.1.2 that for any x ∈ BR there holds 1 q+1−p 1q b c + . (3.1.11) u(x) ≤ p a|x| a Let ζ ∈ C0∞ (Ω), ζ ≥ 0, θ ∈ C 1 (R) ∩ L∞ (R) be a nondecreasing function, vanishing on (−∞, 0] and increasing on [0, ∞) such that 0 ≤ θ ≤ sign+ , 1 and μ ≥ ab q , then p−2 |∇u| ∇u.∇(ζ p θ(u − μ)) + (auq − b)ζ p θ(u − μ) dx ≤ 0. Ω

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132

Therefore p ζ p |∇(u − μ)| θ (u − μ)dx Ω

θ(u − μ)ζ p−1 |∇(u − μ)|

+p

p−2

∇(u − μ)∇ζdx ≤ 0.

Ω

Using H¨ older’s inequality and letting θ to converge to sign+ yields,

p

p

ζ p |∇(u − μ)+ | dx ≤ p Ω

Ω

(u − μ)p+ |∇ζ| dx.

(3.1.12)

Next, let {ηn } ⊂ C ∞ (Ω) be a sequence of molliﬁers, i.e. nonnegative functions such that 0 ≤ ηn ≤ 1, # ηn =

0

if |x| ≤

1

if

1 n

1 2n

or |x| ≥

≤ |x| ≤

R0 2 ,

# and |∇ηn | ≤ cn. Let θ be as above and μ ≥ max We set Δ= {x :

1 2n

≤ |x| ≤

1 n}

2R0 3

b 1q a

) ,

R0 2

max ≤|x|≤

2R0 3

u(x) .

and

p−2

Xn = −

θ(u − μ) |∇u|

∇u.∇ηn dx.

Ω N

older’s inequality, Then |Xn | ≤ c1 θ L∞ n1− p ∇(u − μ)+ p−1 Lp (Δn ) by H¨ and therefore 0 ≤ θ(u − μ)(auq − b)ηn dx ≤ Xn . Ω

Using (3.1.12) with ζ = ηn and the a priori estimate (3.1.7), we obtain |Xn | ≤ c1 pn

p2 −N p

(u − μ)+ p−1 Lp (Δn ) ≤ cn

p2 −N p

p2 −p

n q+1−p −

N (p−1) p

.

(3.1.13)

But p2 − p N (p − 1) pq p2 − N + − = −N p q+1−p p q+1−p =

pq pq1 − ≤ 0, q + 1 − p q1 + 1 − p

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hence Xn remains bounded independently of n. Letting θ converge to sign+ we derive sign+ (u − μ)(auq − b)dx ≤ lim inf |Xn | < ∞. (3.1.14) 0≤ n→∞

B R0 2

Therefore u+ ∈ Lq (B R0 ). Notice that if q > q1 , Xn → 0 when n → ∞, 2 which implies u ≤ μ, however the proof of the upper bound on u+ diﬀers according the position of q with respect to p. In the case q ≥ p, then p−1

p−1

(u − μ)+ Lp (Δn ) ≤ (u − μ)+ Lq (Δn ) |Δn | ≤ c (u − μ)+ p−1 Lq (Δn ) n

(q−p)(p−1) pq

N (p−q)(p−1) pq

.

This implies Xn → 0 and u ≤ μ. In the case q < p, then q1

1− q

(u − μ)+ Lp (Δn ) ≤ (u − μ)+ Lpq1 (Δn ) (u − μ)+ L∞p(Δn ) ≤ c (u − μ)+ Lq (Δn ) nN (p−1)( q − p ) 1

p−1

1

N (p−1)( q − q ) 1 , ≤ c (u − μ)+ p−1 Lq (Δn ) n 1

by (3.1.13). Then Xn → 0 and again u ≤ μ.

1

Proof of Theorem 3.1.3. If g satisﬁes (3.1.8), then (3.1.6) holds for some constants a > 0 and b ≥ 0. Thus u+ ∈ L∞ loc (Ω) by Lemma 3.1.4. Similarly 1,p (Ω). It is straightforward that u ∈ Wloc (Ω) by using the same u− ∈ L∞ loc test functions as above and the result follows by Theorem 1.3.9. Estimate (3.1.7) is a particular case of a more general result proved in [Vazquez (1981)]. Proposition 3.1.5 Assume g : R → R+ is a continuous nondecreasing function such that g(0) > 0 which satisﬁes lim g(s) = 0 and s→−∞

a

∞

ds

0, and γ + > 0 for example. Case 1: p = N . Set

ur (x) =

u(rx) μp (r)

∀x ∈ B 1r \ {0}.

(3.1.29)

Then N −p

N −1

−Δp ur + rN g(r− p−1 ur , r− p−1 |∇ur |) = 0

in B 1r \ {0},

(3.1.30)

and let ξr such that |ξr | = 1 and γ(r)μp (rξr ) = u(rξr ). By assumption N −1 |ur (x)| ≤ cμp (x) and |∇ur (x)| ≤ c|x|− p−1 , therefore it follows from the weak singularity assumption that N −p

N −1

rN g(r− p−1 ur , r− p−1 |∇ur |) → 0 as r → 0, uniformly on any compact set K ∈ RN \ {0}, provided r is smaller than N −p N −1 rK := (sup{|z|, z ∈ K})−1 . Because ur and rN g(r− p−1 ur , r− p−1 |∇ur |) remain eventually locally bounded in RN \ {0}, Theorem 1.3.9 applies and we conclude that ∇ur remains locally bounded in C α (RN \ {0}). Therefore the set of functions {ur } is relatively compact in the C 1,α local topology of RN \ {0} for all 0 < α < α. If {rn } is a sequence converging to 0 and 1,α U ∈ C 1,α is such that {urn } converges to U in Cloc (RN \ {0}), then U is N p-harmonic in R \ {0} and it satisﬁes |U (x)| ≤ cμp (x)

in RN \ {0}.

(3.1.31)

It follows from Corollary 1.5.9 that U = λμp for some λ ∈ [−c, c]. If we assume that the sequence {rn } is the one for which the lim sup is achieved, then lim sup x→0

u(x) u(rn ξ) = lim sup = γ+ = γ, r →0 μp (x) n |ξ|=1 μp (rn )

(3.1.32)

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137

and the convergence u(rn ξ) , rn →0 μp (rn )

(3.1.33)

γ+ = lim

holds uniformly on {ξ : |ξ| = 1}. Since g satisﬁes (3.1.22), it follows by the maximum principle that u is positive in the annulus {x : rp ≤ |x| ≤ rn } for any p > n; hence there exists 0 < R0 < 1 such that u > 0 in BR0 \ {0}. Since γμp is a supersolution of equation (3.0.1), for any > 0, there exists r > 0 such that r → 0 when → 0 and ∀x ∈ B 1 \ {Br }.

u(x) ≤ (γ + )μp (x) + max u(y) |y|=1

Letting → 0 yields in B 1 \ {0}.

u ≤ γμp + max u(y) |y|=1

Next, there exists a sequence {rn } converging to 0 such that lim inf x→0

u(x) u(rn ξ) = lim = γ− ≥ 0. inf μp (x) rn →0 |ξ|=1 μp (rn )

(3.1.34)

By the compactness of {ur } in C 1,α locally in RN \ {0}, we can assume that the sequence {rn } is such that the convergence γ− = lim

rn →0

u(rn ξ) , μp (rn )

holds uniformly on {ξ : |ξ| = 1}. Therefore, since u is p-subharmonic and nonnegative in BR0 \ {0} we derive from the maximum principle that for any > 0, u is bounded from above in BR0 \Brn by the p-harmonic function v = (γ− + )μp + max{u(y) : |y| = R0 }. Letting rn → 0 and → 0 yields u(x) ≤ γ− μp + max{u(y) : |y| = R0 }

∀x ∈ BR0 \ {0}.

(3.1.35)

Since γ− ≤ γ+ it follows from (3.1.33) and (3.1.35) that γ+ = γ− = γ and 1 topology, (3.1.26) (3.1.25) holds. Since the convergence holds in the Cloc follows. Next, assume (3.1.24)-(i) holds and let ζ ∈ Cc∞ (Ω), then p−2 p−2 |∇u| ∇u, ∇ζ + g(u, |∇u|)ζ dx = − ζ |∇u| ∇u, ndS. Ω\B 1

n

∂B 1

n

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Because of (3.1.25) and (3.1.26), g(u, |∇u|) is integrable in B1 and ζ |∇u|p−2 ∇u, ndS = |γ|p−2 γζ(0), lim n→∞

∂B 1

n

which yields (3.1.27). Case 1: p = N . We put ur (x) =

u(rx) μN (r)

∀x ∈ B(2r)−1 \ {0}.

(3.1.36)

Then |ur (x)| ≤ c if 0 < r|x| ≤ 12 , and −ΔN ur + rN g (μN (r)ur , μN (r)|x||∇ur |) = 0

in B(2r)−1 \ {0}. (3.1.37)

Furthermore, since we can assume r ≤ 12 , |μN (ξ)| . |ur (ξ)| ≤ c 1 + |μN (r)| Notice the right-hand side of the above expression remains locally bounded in RN \ {0} and it converges to c when r → 0. Next |∇ur (ξ)| ≤

c . μN (r)|ξ|

Because of assumption (3.1.24), the term rN g (μN (r)ur , μN (r)|x||∇ur |) remains locally bounded in RN \{0}. Therefore, proceeding as in the previous case, we conclude that the set of functions U := {ur }0 0 there

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139

exists a radial function u ∈ C 1 (B R \ {0}) satisfying in D (BR )

−Δp u + g(u, |∇u|) = |γ|p−2 γδ0 u=0

(3.1.38)

on ∂BR .

Furthermore u is nonnegative and it satisﬁes lim

|x|→0

u(x) = cN,p γ. μp (x)

(3.1.39)

If we assume moreover that (t, s) → g(s, t) is nondecreasing, then u is the unique solution satisfying (3.1.39); we denote it by uγ . It is a nondecreasing function of γ and R. Proof. Since u(x) = u(|x|) = u(r), we look for positive solutions of N −1 −|u |p−2 (p − 1)u + in (0, R) u + g(u, |u |) = 0 r u(r) (3.1.40) = cN,p γ lim r→0 μp (r) u(R) = 0. Step 1: Existence when p = N . Set t = − ln r, τ := − ln R; then τ < t < ∞, and if u(r) = v(t), (3.1.40) is equivalent to −(|v |N −2 v ) + e−N t g(v, et |v |) = 0 lim

r→0

in (τ, ∞)

v(t) = γN γ t v(τ ) = 0.

(3.1.41)

The function w1 (t) = γN γ(t − τ ) is a supersolution for (3.1.41). For θ > 0, let vθ be the maximal solution of (|vθ |N −2 vθ ) = e−N t g(vθ , et |vθ |)

in (τ, Tmax )

vθ (τ ) = 0 and vθ (τ ) = θ. The function vθ is increasing and larger than θ(t − τ ). By the maximum principle the curves of w1 and vθ cannot intersect, thus vθ ≤ w1 . The Cauchy-Lipschitz theorem and the fact that θ(t − τ ) ≤ vθ (t) ≤ w1 (t) imply that the maximal solution is deﬁned on [τ, ∞) and there exists limt→∞ vθ (t) = (θ) ≤ γN γ. More precisely ∞ ((θ))p−1 = θp−1 + e−N t g(vθ (t), et |vθ (t)|)dt. (3.1.42) τ

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Because θ → vθ (t) is continuous and increasing for any t ≥ τ , we infer from (3.1.24)-(ii) that the integral term in (3.1.42) is a continuous function of θ. Since (cN γ) ≥ γN γ, it follows from the mean value theorem that there exists θ∗ ≤ cN γ such that (θ∗ ) = γN γ. p−N

p−N

Step 2: Existence when 1 < p < N . Set t = r p−1 , τ := R p−1 ; thus τ < t < ∞, and if v(t) = u(r), (3.1.40) is equivalent to p p(N −1) −1 −p −p − N t− N −p g v, Np−1 t p−1 |v | = 0 in (τ, ∞) −(|v |p−2 v ) + Np−1 v(t) = γN,p γ r→0 t lim

v(τ ) = 0. (3.1.43) The remaining of the proof, based upon convexity arguments and the fact that linear functions are supersolutions, is the same as in Step 1. The function vθ is the solution of the initial value problem associated to equation (3.1.43) with vθ (τ ) = 0 and vθ (τ ) = θ, and (θ) is the limit of vθ (t) when t → ∞. In particular p ∞ p(N −1) −1 N −p −p − N p−1 p−1 =θ + p−1 τ − N −p g vθ , Np−1 t p−1 |vθ | dτ, ((θ)) τ

(3.1.44) and we infer that problem (3.1.43) admits a solution. Step 3: Identity (3.1.38) holds. In both cases p = N and 1 < p < N , relation (3.1.39) holds by construction. As in the proof of Theorem 3.1.7, the validity of (3.1.39)-(3.1.44) implies that the convergence of u(rx) μp (r) holds locally in the C 1 topology. Thus (3.1.26) holds. This implies (3.1.38). Step 4: Uniqueness. The two mappings θ → vθ and θ → vθ are nondecreasing. Since (s, t) → g(s, t) is nondecreasing in s and t, the correspondence θ → (θ) deﬁned by (3.1.42) is increasing. It is continuous, hence there exists a unique θ∗ ∈ R+ such that (θ∗ ) = γN,p γ. Notice that θ∗ < γN γ. Remark. Uniqueness of solutions of (3.1.38) in a bounded domain with prescribed boundary value (non necessarily radially symmetric) can be obtained if one can prove that (3.1.39) holds. This can be obtained either with a sign assumption on u or with the assumption that |u| ≤ cμp , using Theorem 3.1.7. In that case the method is to use the maximum principle in order to prove that two solutions u and u∗ of the same problem satisfy u ≤ (1 + )u∗ and u∗ ≤ (1 + )u for any > 0.

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Proposition 3.1.9 Under the assumptions of Proposition 3.1.8, we as−p such that g satisﬁes sume moreover that there exists α > Np−1 g(s, t) = λ(α+1)(p−1)+1 g(λ−α s, λ−α−1 t)

∀s > 0, t > 0 λ > 0, (3.1.45)

that s → s1−p g(s, αs) is increasing on R+ and the following limits hold: lim s1−p g(s, αs) = 0,

(i)

s→0

(ii)

(3.1.46)

lim s1−p g(s, αs) = ∞.

s→∞

We denote by uγ,R the solution of (3.1.38). Then (γ, R) → uγ,R is increasing and uγ,R converges to u∞,R as γ → ∞. Furthermore u∞,R converges to u∞,∞ : u∞ when R → ∞, u∞ (x) = κ|x|−α where κ is the only positive zero of the mapping c → c1−p g(c, αc) − αp−1 ((p − 1)α + p − N ),

(3.1.47)

and there holds 0 ≤ u∞ − u∞,R ≤ κR−α

∀R > 0.

Proof. For λ > 0, set Tλ [u](x) = λα u(λx). Then Tλ [uγ,R ] = u

(3.1.48) λ

p−N +α p−1 γ, R λ

.

Since (γ, R) → uγ,R is increasing, with limit u∞,R at inﬁnity when γ → ∞, we obtain Tλ [u∞,R ] = u∞, R λ

∀λ ∈ R+ .

When R → ∞, uγ,R → uγ,∞ := uγ and u∞,R → u∞,∞ := u∞ . Thus Tλ [u∞ ] = u∞

∀λ ∈ R+ .

(3.1.49)

This proves that u∞ is self-similar and in particular, u∞ (r) = u∞ (1)r−α := κr−α

∀r ∈ R+ .

(3.1.50)

Next, κr−α is a particular radial solution of (3.0.1), therefore, using (3.1.45), κp−1 r−(α+1)(p−1)+1 αp−1 ((p − 1)α + p − N ) = g(κr−α , ακr−α−1 ) = r−(α+1)(p−1)+1 g(κ, ακ). This implies that κ satisﬁes the identity αp−1 κp−1 ((p − 1)α + p − N ) = g(κ, ακ).

(3.1.51)

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142

−p Because of (3.1.46) and α > Np−1 , κ is uniquely deﬁned. Notice also that the functions uγ,R are smaller than κr−α . This implies that the limits of uγ,R at inﬁnity exist and for any γ > 0 and R > R > 0

0 < uγ,R (r) < uγ,R (r) < uγ,R (r) + uγ,R (R)

∀r ∈ (0, R].

This implies (3.1.48).

The next result, typical of quasilinear equation with strong absorption, shows the dichotomy between the two types of possible behaviour for a singular solution of (3.1.1) when g(u, |∇u|) = |u|q−1 u. This phenomenon will be encountered in many other situations. Theorem 3.1.10 Let Ω and p be as in Theorem 3.1.7 and p−1 < q < q1 . If u ∈ C 1 (Ω \ {0}) is a positive solution of (3.1.1) in Ω \ {0}, then the following alternative occurs: (i) either lim |x|βp,q u(x) = ωc ,

x→0

where βp,q =

p q+1−p

(3.1.52)

1

p−1 and ωc = (βp,q Λ(βp,q )) q+1−p ,

(ii) or there exists γ ≥ 0 such that lim

x→0

u(x) = γ, μp (x)

(3.1.53)

and there holds −Δp u + uq = γ p−1 δ0

in D (Ω).

(3.1.54)

Proof. Without any loss of generality we can assume B 1 ⊂ Ω. Since g(u, |∇u|) = |u|q−1 u and p − 1 < q < q1 , the assumptions (3.1.23), (3.1.22), p = βp,q hold. Furthermore κ = ωc (see (3.1.24) and 3.1.37) with α = q+1−p (3.1.2), (3.1.3)). If u ≤ cμp , assertion (ii) follows from Theorem 3.1.7, thus we assume that lim sup x→0

u(x) = ∞. μp (x)

(3.1.55)

We write (3.0.1) under the form −Δp u + D(x)up−1 = 0

with D(x) = (u(x))q+1−p .

(3.1.56)

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Since u satisﬁes (3.1.7), there holds D(x) ≤ c|x|−p in B1 \ {0}. By Harnack inequality, (see e.g. Theorem 1.3.6) there holds max

|x−y|≤ |x| 2

u(y) ≤ c

min

|x−y|≤ |x| 2

∀x ∈ B 21 \ {0},

u(y)

for some c = c(N, p) > 0. This implies ∀x, y ∈ B 12 \ {0} s.t. |x| = |y|.

u(x) ≤ c13 u(y)

(3.1.57)

Therefore (3.1.55) implies lim inf x→0

u(x) = ∞, μp (x)

(3.1.58)

and there exist sequences {rn } converging to 0 and {γn } converging to ∞ such that inf u(x) = γn μp (rn ).

(3.1.59)

|x|=rn

By the comparison principle and with the notations of Proposition 3.1.9, we obtain u(x) ≥ uγn ,1 (x)

∀ rn ≤ |x| ≤ 1.

(3.1.60)

If we ﬁx |x| and let n → ∞, we ﬁnally infer u(x) ≥ u∞,1 (x)

∀x ∈ B 1 \ {0},

(3.1.61)

which implies, using (3.1.48) and (3.1.7), ωc |x|−βp,q − ωc ≤ u(x) ≤ c|x|−βp,q

∀x ∈ B 1 \ {0}.

(3.1.62)

There are several ways for obtaining the sharp upper bound for u, but all are based upon the uniqueness of positive separable solutions of (3.0.1) in RN \ {0}. For > 0 we denote by U the solution of −Δp u + uq = 0 lim u(x) = ∞.

c

in B

(3.1.63)

|x|→

The existence of this solution follows from (3.1.7); it is radially symmetric and constructed as the increasing limit, when n and R tend to ∞, of the

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sequence {U,n,R } of the minimal solutions of −Δp u + uq = 0 u(x) = n u(x) = 0

c

in BR B on ∂B on ∂BR .

(3.1.64)

Although it is easy to prove that the solutions of (3.1.63) is unique, this property will not be used. Because of the minimality, the function p Tr [U ](x) := U,r (x) = r q+1−p U (rx) satisﬁes (3.1.63) with replaced by r , hence it is equal to U r (x). Then ωc |x|−βp,q − ωc ≤ u(x) ≤ U + max{u(z)}

∀x ∈ B 1 \ B .

|z|=1

(3.1.65)

Put ur (x) = rβp,q u(rx). Then ur satisﬁes the same equation (3.0.1) as u in Ωr := 1r Ω, and (3.1.65) is transformed into ωc |x|−βp,q − ωc rβp,q ≤ ur (x) ≤ U r (x) + rβp,q max{u(z)} ∀x ∈ B r1 \ B r . |z|=1

(3.1.66) If we let → 0, U converges to U0 and there holds ωc |x|−βp,q − ωc rβp,q ≤ ur (x) ≤ U0 (x) + rβp,q max{u(z)} |z|=1

∀x ∈ B r1 \ {0}.

(3.1.67) Moreover, since Tr [U ] = U r , we derive that Tr [U0 ] = U0 , by letting → 0. This means that U0 is a positive self-similar solution of (3.0.1) in RN \ {0}. Hence it is separable, and since it is bounded from below by ωc |x|−βp,q , it follows from Lemma 3.1.1 that it is equal to ωc |x|−βp,q . Relation (3.1.52) follows now from (3.1.67). Notice that the gradient of ur satisﬁes |∇ur (x)| ≤ c|x|−βp,q −1

∀x ∈ B 1r \ {0}

|∇ur (x) − ∇ur (x)| ≤ c|x − x |α |x|−βp,q −1−α ∀ 0 < |x| ≤ |x | ≤ 1r . (3.1.68) Combined with Ascoli’s theorem, these estimates imply that the limit of rβp,q u(r, .), when r → 0, occurs in the C 1 (S N −1 ) topology. In the case p = 2, q1 = q1,2 = NN−2 ; the expression of Theorem 3.1.10 gives a sharper result since not only positive solutions can be classiﬁed, but N +1 also signed solutions, provided q ≥ q˜1,2 := q˜1 = N −1 . This exponent is called the isotropy threshold. In this case βp,q = β2,q =

2 q−1

and

1

ωc = (β2,q (β2,q + 2 − N )) q−1 .

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The exponent q˜1 corresponds to the threshold of isotropic behaviour in the problem. The separable solutions are written under the form u(x) = u(r, σ) = r−β2,q ω(σ) and ω satisﬁes −Δ ω − β2,q (β2,q + 2 − N )ω + |ω|q−1 ω = 0

in S N −1 .

(3.1.69)

Lemma 3.1.11 Assume q˜1 ≤ q < q1 , then the only solutions of (3.1.69) are the three constants ωc , −ωc and 0. Proof. The importance of q˜1 comes from the fact that if q ≥ q˜1 , then β2,q (β2,q + 2 − N ) ≤ N − 1 = λ1 (2, S N −1 ). For any function v = v(r, σ), we set 1 v(r, σ)dS(σ), v¯(r) = N −1 |S | S N −1 and η = ω − ω ¯ . Then −Δ η − β2,q (β2,q + 2 − N )η + |ω|q−1 ω − |ω|q−1 ω = 0. ¯ = η = 0, Since ω − ω −

S N −1

ηΔ ηdvg0

≥ (N − 1)

S N −1

Moreover S N −1

η 2 dvg0 .

(|ω|q−1 ω − |ω|q−1 ω)ηdvg0 =

S N −1

(|ω|q−1 ω − |¯ ω |q−1 ω ¯ )ηdvg0

(|¯ ω |q−1 ω ¯ − |ω|q−1 ω)ηdvg0

+ S N −1

=

S N −1

(|ω|q−1 ω − |¯ ω |q−1 ω ¯ )ηdvg0

≥2

1−q S N −1

|η|

q+1

dvg0 . (3.1.70)

Therefore

(N − 1 − β2,q (β2,q + 2 − N ))

S N −1

2

η dvg0 + 2

1−q S N −1

|η|

This yields η = 0, thus ω is constant and the result follows.

q+1

dvg0 ≤ 0.

Theorem 3.1.12 Let Ω be as in Theorem 3.1.7 and q˜1 ≤ q < q1 . If u ∈ C 1 (Ω \ {0}) is a solution of −Δu + |u|q−1 u = 0,

(3.1.71)

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in Ω \ {0}, then there exists = limx→0 |x|βp,q u(x), and ∈ {ωc , −ωc , 0}. If = 0 there exists γ ∈ R such that lim

x→0

u(x) = γ, μ2 (x)

(3.1.72)

and there holds −Δu + |u|q−1 u = γδ0

in D (Ω).

(3.1.73)

Proof. Step 1: We claim that ∈ {ωc , −ωc , 0}. The method of proof is completely diﬀerent from the one of Theorem 3.1.10 and is based upon the reduction of the problem to an autonomous equation in a cylinder where the techniques associated to dynamical systems analysis can be used. We still assume B 1 ⊂ Ω and we put u(r, σ) = r−β2,q v(t, σ), with ln r = t ∈ (−∞, 0]. Then v satisﬁes 2q + 2 q−1 vt + Δ v + β2,q (β2,q + 2 − N )v − |v| vtt + N − v = 0, (3.1.74) q−1 in (−∞, 0]×S N −1 . Since |u(x)| ≤ c|x|−β2,q , v is bounded in (−∞, 0]×S N −1 . By standard a priori estimates in Lp and H¨older spaces, all the derivatives of v up to the order 3 are bounded in C δ ((−∞, 1]×S N −1 ) for some δ ∈ (0, 1) (precisely, if 1 < q < 2, δ = E[q − 1], the integer part of q − 1; if q ≥ 2, then δ = 1). We set 2 1 2 |v|q+1 dvg0 . |∇ v| − β2,q (β2,q + 2 − N )v 2 − vt2 + E(v) = 2 S N −1 q+1 The next identity is the energy estimate: 2q + 2 dE(v) = N− vt2 dvg0 . dt q−1 S N −1

(3.1.75)

Therefore −1 2q + 2 N− vt2 dvg0 dt = E(v)(−1) − E(v)(τ ), q−1 S N −1 τ for τ < −1. Thanks to the previous regularity estimates, E(v)(t) is uni= 0, it follows formly bounded on (−∞, −1]. Since N − 2q+2 q−1

−1 τ

S N −1

vt2 dvg0 dt < ∞.

(3.1.76)

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Diﬀerentiating (3.1.74) with respect to t yields 2q + 2 vtt + Δ vt + β2,q (β2,q + 2 − N )vt − q |v|q−1 vt = 0. vttt + N − q−1 Then −1 −1 2q + 2 2 N− vtt dvg0 dt − q vt2 |v|q−1 dvg0 dt q−1 S N −1 S N −1 τ τ = E(vt )(−1) − E(vt )(τ ). Using (3.1.76) and the boundedness of v, we derive −1 2 vtt dvg0 dt < ∞. τ

(3.1.77)

S N −1

Finally, since vt and vtt are uniformly continuous we have that lim (|vt (t, .)| + |vtt (t, .)|) = 0,

t→∞

uniformly in S N −1 .

(3.1.78)

We deﬁne the negative trajectory of v in C 2 (S N −1 ) by T− [v] = {v(t, .)}, t≤0

and its α-limit set by A[T− [v]] =

*

{v(t, .)}.

τ p − 1. Then there exists a constant c = c(N, p, q) > 0 such that any nonnegative u ∈ C 1 (Ω) solution of (3.4.1) in Ω satisﬁes max{u(x) : x ∈ B R (a)} ≤ c min{u(x) : x ∈ B R (a)}. 2

2

(3.4.56)

Proof. We assume that a = 0 and write (3.4.1) as −Δp u + C(x)|∇u(x)|p−1 = 0,

(3.4.57)

with C(x) = |∇u(x)|q+1−p . By (3.4.41), |C(x)| ≤ cR−1 with c = c(N, p, q) > 0. Next we set uR (y) = u(Ry). Then −ΔuR + RC(Ry)|∇uR (x)|p−1 = 0

in B1 .

Since RC(Ry) is bounded in B1 independently of R and y, we derive from Theorem 1.3.6 max{u(x) : x ∈ B 12 } ≤ c min{u(x) : x ∈ B 12 }. Then (3.4.56) follows.

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As a consequence of Proposition 3.4.13 we have Corollary 3.4.14 Assume Ω is as in Theorem 3.4.12, q > p − 1. Then there exist a constant c = c(N, p, q) > 0 such that for any R > 0 such that R < ρ(0) and for any nonnegative solution u ∈ C 1 (Ω \ {0}) solution of (3.4.1) in Ω \ {0}, there holds max{u(z) :

R 8

≤ |z| ≤

R 8}

≤ c min{u(z) :

R 8

≤ |z| ≤

R 8.

(3.4.58)

Remark. Since the estimate of the gradient does not depend on the sign of the solution of (3.4.1), the Harnack inequality holds also for a negative solution u in Ω \ {0}. It reads max{−u(z) :

R 8

≤ |z| ≤

R 8}

≤ c min{−u(z) :

R 8

≤ |z| ≤

R 8.

(3.4.59)

The next result is a version of Lemma 3.3.2 adapted to the study of isolated singularities is a domain. Its proof is much easier since we need not perform a reﬂexion and we can use directly scaling and the regularity results of Theorem 1.3.9. Again, the result holds for any signed solution. Lemma 3.4.15 Assume R > 0, 1 < p ≤ N and p − 1 < q ≤ p. Let u ∈ C 1 (B R \ {0}) be a solution of (3.4.1) in BR \ {0} which satisﬁes ∀x ∈ BR \ {0},

|u(x)| ≤ φ(|x|)

(3.4.60)

where φ : R∗+ → R+ is continuous, nonincreasing and veriﬁes φ(rs) ≤ γφ(r)φ(s)

∀r, s ∈ (0, r0 ],

(3.4.61)

for some γ, r0 > 0. Then there exist constants c1 > 0 and α ∈ (0, 1) such that φ(|x|) |x|

∀x ∈ B R

(i)

|∇u(x)| ≤ c1

(ii)

|x − x |α φ(|x|) |∇u(x) − ∇u(x )| ≤ c1 |x|1+α

2

∀ x, x ∈ B R , |x| ≤ |x |. 2

(3.4.62) Proof of Theorem 3.4.12. By scaling we can assume that B 1 ⊂ Ω. Step 1: We claim that if u satisﬁes lim inf |x|→0

u(x) < ∞, μp (x)

(3.4.63)

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then statements (i) of Theorem 3.4.12 hold. Because of (3.4.63) there exist k ≥ 0 and a sequence {rn } converging to 0 such that inf

lim

n→∞ |x|=rn

u(x) = k, μp (rn )

(3.4.64)

(remember that μp is a radial function). Using (3.4.58), we derive lim sup

n→∞ |x|=r

n

u(x) = k˜ ≤ ck. μp (rn )

(3.4.65)

For > 0 and M = max{u(x) : |x| = 1}, the p-harmonic function v = (k˜ + )μp + M dominates the sub-p-harmonic function u on |x| = rn for n large enough and on |x| = 1. By the maximum principle u ≤ v in B 1 \ Brn . Letting n → ∞ and → 0 implies, up to changing the multiplicative constant ck, 0 ≤ u(x) ≤ mμp

in B1 \ {0}.

(3.4.66)

By Lemma 3.4.15, mμp (|x|) |x|

(i)

|∇u(x)| ≤ c1

(ii)

|∇u(x) − ∇u(x )| ≤ c1

∀x ∈ B 12 m|x − x |α μp (|x|) |x|1+α

∀ x, x ∈ B 12 , |x| ≤ |x |. (3.4.67)

Next we deﬁne y → ur (y) by ur (y) =

u(ry) μp (r)

∀y ∈ B 1r \ {0}.

It satisﬁes in the above domain, q

−Δp ur + (μp (r))q+1−p rp−q |∇ur | = 0.

(3.4.68)

Furthermore the following estimates hold from (3.4.66)-(3.4.67), (i) (ii) (iii)

mμp (r|y|) μp (r) mμp (r|y|) |∇ur (y)| ≤ c1 |y|μp (r)

0 ≤ ur (y) ≤

∀y ∈ B 1r \ {0} ∀y ∈ B 2r1 \ {0}

mμp (r|y|)|y − y |α |∇ur (y) − ∇ur (y )| ≤ c1 |y|1+α μp (r) 1 , 0 < |y| ≤ |y |. ∀ y, y ∈ B 2r

(3.4.69)

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When y remains in a compact subset K of RN ∗ , there exists rK > 0 such that μp (r|y|) remains bounded for 0 < r ≤ r . By estimates (3.4.69) (ii)-(iii) the K μp (r) set of functions {ur }0 0 there exists nj such that if rnj ≤ rnj we have u(x) ≤ (k + )μp (x) for |x| = rnj . By comparison, u(x) ≤ (k + )μp (x) + M

∀x ∈ B1 \ Brnj .

Letting rnj → 0 we obtain that the above inequality holds in B1 \ {0}. Letting → 0 and using (3.4.65) yields k = k˜ and again u ≤ μ ˜p + M in B1 \ {0}. Therefore u(x) = k. x→0 μp (x) lim

(3.4.70)

The C 1 convergence implies # lim |x|

N −1 p−1

x → 0 x → e |x|

∇u(x) =

k p−N p−1 e

if 1 < p < N

−ke

if p = N,

(3.4.71)

and as a consequence |∇u|q ∈ L1loc (Ω). When p = N , there holds u(rnj ξ) =θ rnj →0 μp (rnj ) lim

uniformly for |ξ| = 1.

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Hence k = θ and we conclude as in the previous case. Finally, for any δ > 0 and ζ ∈ Cc∞ (Ω), we have ∂u ζdS. |∇u|p−2 ∇u, ∇ζ + |∇u|q dx = |∇u|p−2 ∂n Ω\Bδ |x|=δ Using (3.4.71), we infer that (3.4.54) holds. Step 2: We claim that if u satisﬁes lim inf |x|→0

u(x) = ∞, μp (x)

(3.4.72)

then statement (ii) of Theorem 3.4.12 holds. Put m = min u(x). Since for |x|=1

any γ > 0 the function uγ,m (see deﬁnition in Corollary 3.4.5) is a solution of (3.4.1) in B1∗ which has no critical points, is smaller than u on ∂B1 and at x = 0 because is satisﬁes (3.4.24)-(i), it is smaller than u. Letting γ → ∞ yields u(x) ≥ Us (|x|) + m = μN,p,q |x|−γp,q + m

∀x ∈ B1∗ .

(3.4.73)

The upper bound is obtained by a scaling technique. We adapt the argument used in the proof of Theorem 3.4.11, and for > 0 we denote by w ˜ the solution of w − r

(q+1−p)(p−1) N −1

q

|w| p−1 = 0

in (, ∞)

w() = −∞.

(3.4.74)

Then w ˜ = −

q+1−p (N −1)(qc −q)

p−1 (N −1)(qc −q) − q+1−p (N −1)(qc −q) p−1 p−1 r − .

(3.4.75)

Therefore, the function u ˜ deﬁned by ∞ 1 − q+1−p (N −1)(qc −q) (N −1)(qc −q) N −1 q+1−p p−1 p−1 r − s− p−1 ds, u ˜ (r) = M + (N −1)(qc −q) r

has no critical point and dominates u for |x| = 1 and |x| = ; hence it dominates u in B1 \ B . Letting → 0 leads to ∞ 1 − q+1−p (N −1)(qc −q) N −1 q+1−p p−1 u(x) ≤ u ˜0 (|x|) = M + r s− p−1 ds (N −1)(qc −q) ≤ Us (x) + M. This yields (3.4.55).

|x|

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3.4.4.3

Global singular solutions

The following rigidity result holds: Theorem 3.4.16 Assume 0 < p − 1 < q < p ≤ N and u ∈ C 1 (RN \ {0}) is a nonnegative solution of (3.4.1) in RN \ {0}. I- If q ≥ qc , then u is a constant. II- If p − 1 < q < qc , then (i) either u is a constant, (ii) or there exist γ > 0 and c ≥ 0 such that u(x) = uγ,c (|x|) ∞ 1 − q+1−p (N −1)(qc −q) N −1 − q+1−p q+1−p p−1 =c+ + |˜ γ | p−1 s− p−1 ds. (N −1)(qc −q) s |x|

(3.4.76) (ii) or there exists c ≥ 0 such that u(x) = Us (x) + c = μN,p,q |x|−γp,q + c.

(3.4.77)

Proof. If q ≥ qc the result follows from Theorem 3.4.11 and Corollary 3.4.10. If p − 1 < q < qc , we have the following estimate by Theorem 3.4.7 |∇u(x)| ≤ CN,p,q |x|− q+1−p 1

∀x ∈ RN \ {0}.

Since for any a ∈ S N −1 , d 1 d u(ta) ≤ CN,p,q t− q+1−p =⇒ u(ta) ∈ L1 (1, ∞), dt dt there exists ca = lim u(ta). Next, if |x| = t = 0, t→∞

1 x |u(x) − u(ta)| = u(t |x| ) − u(ta) ≤ πCN,p,q t− q+1−p +1 → 0 as t → ∞. Then c

x |x|

= ca , which implies that there exists c = lim u(x). By The|x|→∞

orem 3.4.12 either (3.4.53) or (3.4.55) holds. In the ﬁrst case γ must be positive other while u would be regular and therefore constant by Corollary 3.4.10. Then, for any ∈ (0, γ), the function uγ+,c+ (see (3.4.76)) dominates u at x = 0 and |x| → ∞. It has no critical values so it dominates u in RN \ {0}. Then u ≤ uγ,c . In the same way u ≥ uγ,c. In the second case

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we notice that (1 + )Us is a supersolution while (1 − )Us is a subsolution. Hence (1 − )Us + c − ≤ u ≤ (1 + )Us + c + .

This yields u = Us + c. 3.4.4.4

Isolated singularities of negative solutions

If u is a negative solution of (3.4.1) we set u˜ = −u and deal with positive solutions of q

˜ = |∇˜ u| , −Δp u

(3.4.78)

in B1 \ {0}. Theorem 3.4.17 Assume Ω ⊂ RN is a domain containing 0, 1 < p ≤ N ˜ ∈ C 1 (Ω \ {0}) is a nonnegative solution of (3.4.78) and p − 1 < q < qc . If u in Ω \ {0}, there exists γ ≥ 0 such that lim

x→0

u ˜(x) = γ, μp (x)

(3.4.79)

and the following equation q

−Δp u ˜ − |∇˜ u| = cN,p γδ0 , p−1 −p . holds in D (Ω) with cN,p = N ωN Np−1

(3.4.80)

Proof. We can assume that B 1 ⊂ Ω. Step 1: We claim that (3.4.80) holds. If 1 < p < N , it follows from q ˜ ∈ L1doc (Ω); hence |∇˜ u| ∈ L1loc (Ω) and there Theorem 1.5.14 that Δp u exists γ ≥ 0 such that (3.4.80) holds. If p = N , then u ∈ Lr (B1 ) for all r < ∞ and ∇u ∈ Ls (B1 ) for all s < N . Since qc = N , it implies that ˜ = g ∈ Lm (B1 ) for all m ∈ [1, Nq ). We assume ﬁrst that −Δp u ˜(x) = m < ∞. lim inf u x→0

Then there exists a sequence {rn } converging to 0 such that lim

inf u ˜(x) = m.

n→∞ |x|=rn

By (3.4.59) it follows ˜(x) = M ≤ cm < ∞, lim sup u

n→∞ |x|=r

n

(3.4.81)

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218

where c = c(N, q) > 0. Let > 0, there exists n ∈ N such that for n ≥ n there holds u ˜(x) ≤ M := M + if |x| = rn . Then N q |∇(˜ u − M )+ | dx ≤ |∇˜ u| (˜ u − M )+ dx B1 \Brn

B1 \Brn

|∇˜ u|

+ ≤

p−1

(˜ u − M )+ dS

∂B1

B1 \Brn

Nq N |∇(˜ u − M )+ | dx

N N −q

B1 \Brn }

|∇˜ u|

+

(˜ u − M )+

p−1

(˜ u − M )+ dS.

p−1

(˜ u − M )+ dS

1− Nq dx

∂B1

Letting n → ∞ and → 0 yields N N N −q |∇(˜ u − M )+ | dx ≤ (˜ u − M )+ dx B1

B1

|∇˜ u|

+

(3.4.82)

∂B1

< ∞. If we assume now that lim sup u ˜(x) ≥ cM,

(3.4.83)

x→0

there exists a sequence {rn } converging to 0 such that lim

inf u ˜(x) ≥ M.

n→∞ |x|=rn

u) = M − inf{˜ u, M }, then PM (˜ u) = −χu≤M and Set PM (˜ ˜ |∇˜ u|N dx + |∇˜ u|q PM (˜ u)dx (B1 \Br )∩{˜ u≤M}

(B1 \Br )∩{˜ u≤M}

n

n

|∇˜ u|

=−

N −2

∂B1

Hence

N

B1 ∩{˜ u≤M}

|∇˜ u| dx ≤ M

|∇˜ u|

N −1

∂u ˜ PM (˜ u)dS. ∂n

dS.

(3.4.84)

∂B1

Therefore, if (3.4.81) and (3.4.83) hold, ∇˜ u ∈ Lp (B1 ). If (3.4.83) does not hold, it implies in particular that u ˜ remains uniformly bounded in B1 \ {0}.

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c Let η ∈ C ∞ 1(RN ) such that 0 ≤ η ≤ 1, η = 0 in B , η = 1 in B2 and |∇η | ≤ 3χB2 \B , then N |η ∇˜ u| dx − N u ˜ |η ∇˜ u|N −1 |∇η | dx B1

B1

q

|η | |∇˜ u| u ˜dx +

≤

N

B1

|∇˜ u|

N −2

∂B1

∂u ˜ u ˜dS. ∂n

Hence

1− N1

N

B1

|η ∇˜ u| dx ≤ cN ˜ u L∞

N

B2 \B

|η ∇˜ u| dx

B1

|∇˜ u|

+

N −2

∂B1

This implies that ∇˜ u ∈ LN (B1 ) and N q |η ∇˜ u| dx ≤ |∇˜ u| u ˜dx + B1

B1

q

|η |N |∇˜ u| u ˜dx

+

|∇˜ u|N −2

∂B1

∂u ˜ u˜dS. ∂n

∂u ˜ u ˜dS. ∂n

(3.4.85)

Next we denote by T the distribution associated to the locally integrable function −Δp u ˜ − |∇˜ u|q . Its support is the point 0. Let ζ ∈ C0∞ (Ω) and η as above. Taking η ζ as test function, we get N −2 N −2 T (η ζ) = |∇˜ u| ∇˜ u, ∇ζη dx + |∇˜ u| ∇˜ u, ∇η ζdx Ω

Ω

q

η ζ |∇˜ u| dx.

− Ω

Since N −2 |∇˜ u| ∇˜ u, ∇η ζdx ≤ c(N ) ζ L∞

1− N1 N

B2 \B

Ω

|∇˜ u| dx

,

which tends to 0 when → 0, then 1− N1 1 N N |T (ζ)| ≤ |G| |∇˜ u| dx ∇ζ L∞ + |∇˜ u|q dx ζ L∞ , G

G

where G denotes the support of ζ. Therefore T is a distribution of order 0 or 1 and it can be written as T = aδ0 −

N j=1

aj

∂δ0 . ∂xj

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∂ζ (0) = −aj , and for > 0, set ∂xj

Take ζ ∈ C0∞ (B1 ) such that ζ(0) = a and ζ (x) = ζ( x ). Then |∇˜ u|

T (ζ ) =

N −2

q

∇˜ u, ∇ζ dx −

Ω

ζ |∇˜ u| dx = a2 + Ω

But N −2 |∇˜ u| ∇˜ u, ∇ζ dx ≤ c(N ) Ω

N a2j j=1

.

N1 |∇˜ u| dx ∇ζ L∞ → 0 as → 0. N

B

Similarly q

ζ |∇˜ u| dx → 0 as → 0. Ω

Therefore a = aj = 0 for all j = 1, ..., N . This implies that (3.4.80) holds with γ = 0. Finally we are left with ˜(x) = ∞, lim inf u

(3.4.86)

x→0

in which case (3.4.80) holds by Theorem 1.5.14. ˜ Step 2: We claim that (3.4.79) holds. With C(x) = − |∇˜ u(x)|q+1−p , equation (3.4.78) becomes p−1 −Δp u ˜ + C˜ |∇˜ u(x)| =0 N (p−1)

in Ω \ {0}.

(3.4.87)

N (p−1)

q+1−p ,∞ q+1−p − When 1 < p < N , C˜ ∈ Lloc (Ω) ⊂ Lloc (Ω) for any > 0. Since (p−1) + q < qc , Nq+1−p > N , thus C˜ ∈ LN (Ω) for > 0 small enough. As a loc consequence the positive singularities are isotropic by Theorem 1.3.7 which means that either u ˜ is regular near 0, in which case the coeﬃcient γ of the Dirac mass in (3.4.78) is zero, or there exists c > 0 such that N −p

˜(x) ≤ c ≤ |x| p−1 u N −

1 c

∀x ∈ B 1 \ {0}.

(3.4.88)

q+1−N ,∞ + If p = N , C˜ ∈ Lloc (Ω) ⊂ LN loc (Ω) for , small enough since N −1 < q < N . Then 1 1 1 ≤ u˜(x) ≤ ln ∀x ∈ B 12 \ {0}. (3.4.89) c ln |x| c |x|

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The proof of (3.4.77) presents some similarity with the one of Theorem 3.4.12, with a major modiﬁcation at the end since the equation has no comparison principle. Let γ ∈ [c, c−1 ] such that there exists a sequence {xn } tending to 0 verifying γ = lim

n→∞

u ˜(xn ) . μp (xn )

(3.4.90)

We denote rn = |xn |. If 1 < p < N we set u ˜(rx) . μp (r)

u ˜r (x) = Then u ˜r satisﬁes

˜r = (μp (r))q+1−p rp−q |∇˜ ur | −Δp u

q

in B 1r \ {0},

and the estimates (3.4.69) are valid provided ur is replaced by u ˜r . Up to 1 urnk } converges in the Cloc a subsequence {rnk } of {rn } tending to 0, {˜ N topology of R \ {0} to some nonnegative p-harmonic function v which satisﬁes the same upper and lower bounds as u ˜, but in whole RN \ {0}. Then it is a positive multiple of μp . Since (3.4.90) holds, v = γμp and u ˜rnk → γμp . Because the convergence is locally in C 1 , we have N −1

lim rnp−1 ∇u(rnk ξ) = − k

rn →0

N −p γξ p−1

uniformly for |ξ| = 1.

Using (3.4.78) p−2 q |∇u| ∇u, ∇ζ − |∇u| dx = Ω\Brn

|x|=rnk

k

p−2

|∇u|

(3.4.91)

∂u ζdS, ∂n

where ζ ∈ C0∞ (Ω). By (3.4.91) the right-hand side of the above identity converges to N ωN

N −p p−1

p−1 γζ(0) = cN,p γζ(0).

By (3.4.80), this limit is cN,p γζ(0). We obtain γ = γ; ﬁnally (3.4.79) holds since the limit is uniquely deﬁned. The proof when p = N is based upon similar ideas with the help of the scaling used in the proof of Theorem 3.4.12.

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3.4.5 3.4.5.1

Geometric estimates Gradient estimates on a Riemannian manifold

The technique used in proving the gradient estimates extends to a geometric setting. We assume that (M, g) is N -dimensional Riemannian manifold, T M its tangent bundle, ∇g is the covariant gradient, ., .g the scalar product expressed in the metric g = (gij ), Riccg the Ricci tensor and Secg the sectional curvature, and Δg the Laplacian. We consider the following equation on M q

−Δp u + |∇g u| = 0,

(3.4.92)

which expands in −Δg u −

D2 u∇g u, ∇g u 2

|∇g u|

+ |∇g u|

q+2−p

= 0.

(3.4.93)

2

In order to obtained bound from above on z = |∇g u| we recall Weitzenb¨ock formula (combined with Schwarz inequality),

2 1 Δg |∇u|2 = D2 u + ∇u, ∇g Δg ug + Riccg (∇u, ∇u) 2 1 (Δg u)2 + ∇u, ∇g Δg ug + Riccg (∇g u, ∇g u). ≥ N

(3.4.94)

We write (3.4.93) under the form Δg u = −

q+2−p 1 ∇g z, ∇g u + z 2 = 0. 2 z

(3.4.95)

With the help of H¨older’s inequality (3.4.35) is replaced by Δg z + (p − 2)

2a2 q+2−p D2 z∇g u, ∇g ug 1 ∇g z, ∇g u2g ≥ z − z N N a2 z2 −

∇g z, ∇g u2g p − 2 |∇g z| + (p − 2) 2 z z2 2

+ (q + 2 − p)z q+2−p ∇g z, ∇g ug + Riccg (∇g u, ∇g u), (3.4.96) where a > 0. The operator A is deﬁned by (3.4.36) (in the geometric setting) and ellipticity condition (3.4.37) holds. A natural assumption on the curvature of the manifold is to assume Riccg (ξ, ξ) ≥ (1 − N )B 2 |ξ|2

∀ξ ∈ T M,

(3.4.97)

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for some B ≥ 0. Then we deﬁne L by L(z) = A(z) + c1 z q+2−p − c2

|∇g z|2 − (N − 1)B 2 z, z

(3.4.98)

where c1 , c2 > 0 are chosen such that L(z) ≤ 0, which is possible by (3.4.96), (3.4.97). Lemma 3.4.18 Assume q > p − 1 > 0 and let a ∈ M , R > 0 and B ≥ 0 ˜ 2 for some such that (3.4.97) holds in BR (a). Assume also Secg ≥ −B ˜ B ≥ 0 if p > 2, or rM (a) ≥ R if 1 < p < 2 where rM (a) is the convexity radius of a. We set r(x) = dg (x, a). Then there exists c = c(N, p, q) > 0 such that the function − 2 w(x) = λ R2 − r2 (x) q+1−p + μ, (3.4.99) satisﬁes L(w) ≥ 0

in BR (a),

(3.4.100)

provided 1 2 λ ≥ cR q+1−p B 2 R2 + Bp R + 1 q+1−p ,

(3.4.101)

˜ and where Bp = B + (p − 2)+ B, 1 μ ≥ (N − 1) B 2 q+1−p .

(3.4.102)

Proof. The proof is a combination of some geometric inequalities and standard computations. The following classical one can be found in [Ratto, Rigoli, V´eron (1994), Lemma 1] Δg r ≤ (N − 1) coth(Br) ≤ Then Δg w ≤

8N q+1−p

×

R2 − r 2

N −1 (1 + Br). r

(3.4.103)

− 2(q+2−p) q+1−p

q+3−p 2 2 2 r + R − r (1 + Br) . q+1−p

(3.4.104)

If r(x) ≤ rM (a) the ball Br(x) is convex and D2 r ≥ 0 (see e.g. [Sakai (1997), ˜ 2 , there holds [Sakai (1997), Lemma 2.9] IV-5]). Furthermore, if Secg ≥ −B ˜ ˜ coth(Br) ˜ ≤ B (1 + Br). ˜ D2 r ≤ B r

(3.4.105)

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Therefore 0≤

D2 w∇g w, ∇g w

|∇g w|2 − 2(q+2−p) 2 q+1−p 8 R2 − r 2 r (q + 3 − p) 2 ˜ . + R − r2 (1 + Br ≤ q+1−p q+1−p (3.4.106) It follows A(w) = −Δg w − (p − 2)

D2 w∇g w, ∇g w 2

|∇g w| 2(q+2−p) 2 − q+1−p R2 + (R2 − r2 )Bp r , ≥ −kλ R − r2

(3.4.107)

where k = k(n, p, q) > 0. Since wq+2−p ≥ λq+2−p (R2 − r2 )−

2(q+2−p) q+1−p

+ μq+2−p ,

there holds − 2(q+2−p) q+1−p L(w) ≥ λ R2 − r2 −k(R2 + (R2 − r2 )Bp r + c1 λq+1−p − 2 q+2−p − c3 λr2 (R2 − r2 )−2 q+3−p − (N − 1)B 2 λ R2 − r2 q+1−p + c2 μq+2−p − (N − 1)B 2 μ, (3.4.108) where c3 = c3 (N, p, q) > 0. We ﬁx c2 μq+1−p ≥ (N − 1)B 2 ,

(3.4.109)

and then λ in order to have c1 λq+1−p ≥ k(R2 + (R2 − r2 )Bp r + c3 r2 2

∀r ∈ [0, R).

It suﬃces to choose λq+1−p ≥ c4 R2 (1 + RBp ),

(3.4.110)

for some c4 = c4 (N, p, q) > 0. At end we require also − 2(q+2−p) − 2 c1 λq+1−p 2 q+1−p R − r2 ≥ (N − 1)B 2 R2 − r2 q+1−p , 2 and we impose, for some c5 = c5 (N, p, q) > 0, λq+1−p ≥ c5 B 2 R4 .

(3.4.111)

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Combining (3.4.110) and (3.4.111) we ﬁx λq+1−p = c6 R2 (1 + B 2 + RBp )

(3.4.112)

and we get (3.4.100) with (3.4.100) and (3.4.100).

This result allows to have an upper bound on the norm of the gradient of a solution of (3.4.92). Proposition 3.4.19 Let q > p − 1 > 0 and Ω be an open subset of (M, g) in which there holds Riccg ≥ (1 − N )B 2 for some B ≥ 0. Assume also ˜ 2 in Ω if p > 2, or rM (x) ≥ dist g (x, ∂Ω) for any x ∈ Ω if Secg ≥ −B 1 < p < 2. Then any C 1 solution u of (3.4.92) in Ω satisﬁes |∇g u(x)| ≤ c

Bp 1 + + B2 2 ρ∂Ω (x) ρ∂Ω (x)

1 2(q+1−p)

,

(3.4.113)

where ρ∂Ω (x) = dist g (x, ∂Ω) and c = c(N, p, q) > 0. Proof. Let a ∈ Ω and R < distg (a, ∂Ω). If w is as in Lemma 3.4.18, then A(z − w) + c1 (z q+2−p − wq+2−p ) − (N − 1)B 2 (z − w) |∇g w|2 |∇g z|2 − ≤ 0, − c2 z w

(3.4.114)

in any connected component G of the set of x ∈ Ω such that z(x) > w(x). If c1 (q + 2 − p)(w(a))q+2−p > (N − 1)B 2 , it follows by the mean value theorem and the fact that w is minimal at a, that c1 (z q+2−p − wq+2−p ) − (N − 1)B 2 (z − w) > 0

in G.

(3.4.115)

1

Since w(a) ≥ μ ≥ ((N − 1)B 2 ) q+1−p and q + 2 − p > 1, this condition is fulﬁlled by choosing the right μ as in (3.4.109). We conclude as in the proof of Theorem 3.4.71 that G = ∅. Therefore z ≤ w in BR (a). In particular, z(a) ≤ w(a) = λR− q+1−p + μ, 4

(3.4.116)

with μ and λ given respectively by (3.4.101) and (3.4.102) with equality, which yields (3.4.113). Remark. Since Riccg (x)(ξ, ξ) = (N − 1) V Secg (x)(V ) where V denotes the set of two planes in Tx M which contain the unit vector ξ, there holds ˜ 2 =⇒ Riccg (1 − N )B ˜ 2. Secg ≥ B

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However, in the previous estimate, the long range estimate on ∇g u depends only on the Ricci curvature. 3.4.5.2

Growth of solutions and Liouville type results

Corollary 3.4.20 Assume (M, g) is a complete noncompact Ndimensional Riemannian manifold such that Riccg ≥ (1 − N )B 2 and let q > p − 1 > 0. Assume also rM (x) = ∞ if 1 < p < 2, or that the sectional curvature Secg satisﬁes for some a ∈ M |Secg (x)| = 0, dist g (x, a) distg (x,a)→∞ lim

(3.4.117)

if p > 2. Then any solution u of (3.4.92) satisﬁes 1

|∇g u(x)| ≤ cN,p,q B q+1−p

∀x ∈ M.

(3.4.118)

In particular, u is constant if Riccg ≥ 0, while in the general case u has at most a linear growth with respect to dist g (x, a).

Proof. The proof follows from (3.4.113).

Deﬁnition 3.4.21 A N-dimensional Riemannian manifold (M, g) is asymptotically ﬂat if there exist a compact set K ⊂ M and two positive constants τ and A such that M \ K is diﬀeomorphic to RN \ B for some ball B, and in this chart, (i)

gij (x) → δij

as ρB (x) → ∞

(ii)

ρτB (x) |∇g gij (x)| + ρτB+1 (x) |∇k gij (x)| ≤ A,

(3.4.119)

where ρB (x) = dist g (x, B), for all x ∈ M \ K and all i, j, k, = 1, ..., N . This implies (see Chapter 1) that the Riemann tensor satisﬁes −1 |Riemg (x)| ≤ c(N )Aρ−τ (x) B

∀x ∈ M \ K,

(3.4.120)

and all the curvatures have the same decay at inﬁnity. In many applications τ > 1, therefore (3.4.119) (i) is a consequence of (3.4.119) (ii). A classical example of asymptotically ﬂat metric on RN \{a} for a ∈ RN , N ≥ 3, is the Schwarzchild metric deﬁned by gij (x) =

1+

m (N − 2)|x − a|

N4−2 δij ,

(3.4.121)

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where m > 0 is an important geometric invariant called the ADM mass (see e.g. [Arnowitt, Deser, Misner (1961)]). Corollary 3.4.22 Assume (M, g) is an asymptotically ﬂat N-dimensional Riemannian manifold with exponent τ and 0 < p−1 < q < min{p+ τ −1 2 , p}. Then any solution u ∈ C 1 (M ) of (3.4.92) in M is asymptotically constant. Proof. We ﬁx a point a ∈ K and denote SR (a) = {x ∈ M : distg (x, a) = R}. Since M is asymptotically Euclidean, there exists R0 such that K ⊂ B R0 (a) := B and any point x ∈ M \ B is connected to some ax ∈ SR0 (a) := S by a unique geodesic curve issued of ax . This geodesic curve depends on ax and its initial speed X at t = 0. In Chapter 1 it is denoted by t → expax (tX) with x = expax (dist g (x, ax )X) = expax (ρB (x)X) since ρB (x) = dist g (x, ax ). Furthermore, for any x ∈ M \ B, rM (x) ≥ ρB (x). Let x such that ρB (x) > 2R0 and R ≤ 12 ρB (x), we have from (3.4.120) Riccg (y) ≥ (1 − N )b2 R−1−τ Secg (y) ≥ −˜b2 R−1−τ

(i) (ii)

∀y ∈ BR (x) ∀y ∈ BR (x),

(3.4.122)

for some b, ˜b > 0. Therefore we can apply estimates (3.4.113) in BR (x) and obtain 1 2(q+1−p) − 3+τ 2 (x) + ρ−(1+τ ) (x) (x) + ρ . |∇g u(x)| ≤ c ρ−2 B B B

(3.4.123)

Therefore $ % 1+τ 1 − 2(q+1−p) − q+1−p (x), ρB (x) . |∇g u(x)| ≤ c max ρB

(3.4.124)

Let x such that ρB (x) > 2R0 . If y is a point on the curve {expax (tX)}t>0 such that ρB (x) ≥ ρB (y) > 2R0 , there holds

ρB (x)

u(x) = u(y) +

ρB (y)

= u(y) +

ρB (x)

ρB (y)

d u ◦ expax (tX)dt dt ∇g u(expax (tX),

d expax (tX) g dt. dt

Using (3.4.123) and integrating yields $ % 1− 1+τ 1− 1 dist g (x, y) ≤ c max ρB q+1−p (y), ρB 2(q+1−p) (y) .

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If q < min{p + τ, p + 1}, dist g (x, y) → 0 when ρB (y) → ∞ on γ˜x , and there exists m(x) = lim u ◦ expax (tX).

(3.4.125)

t→∞

For R > 2R0 , the sphere SR (a) is endowed with the induced metric gR . If (x, y) ∈ SR (a), there exists a geodesic γx,y,R on SR (a) which connect them. We denote by dSR (a) (x, y) the geodesic distance between x and y induced on SR (a). Then γx,y,R (0) = x and γx,y,R (dSR (a) (x, y)) = y and clearly dSR (a) (x, y) ≤ cR,

(3.4.126)

for some c depending on g. Next

dSR (a) (x,y)

u(x) − u(y) = 0

=

0

dSR (a) (x,y)

d u ◦ γx,y,R (t)dt dt ∇g u(expax (tX),

dγx,y,R (t) g dt. dt

Then 1+τ 1 dSR (a) (x, y) ≤ max R1− q+1−p , R1− 2(q+1−p) , which tends to 0 when R → ∞, hence m(x) = m(y).

The previous results applies to positive p-harmonic functions in considering the transformation u ), v = exp(− p−1

(3.4.127)

which is a one to one correspondence between C 1 positive p-harmonic functions v on M and C 1 solutions of −Δp u + |∇g u|p = 0

on M.

(3.4.128)

Proposition 3.4.23 Let the assumptions of Corollary 3.4.20 be satisﬁed. Then any positive p-harmonic function v satisﬁes v(y)e−cN,p B distg (x,y) ≤ v(x) ≤ v(y)ecN,p B distg (x,y) In particular, if Riccg ≥ 0, v is constant.

∀x, y ∈ M. (3.4.129)

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Proof. The second assertion follows from (3.4.118) with B = 0. For the ﬁrst assertion, we consider a geodesic curve γ from x to y and deﬁne u = (1 − p) ln v. Then u(x) − u(y) =

0

Since

distg (x,y) dγ(t) u ◦ γ(t)dt = g dt. ∇g u(γ(t)), dt dt 0

distg (x,y) d

∇g u(γ(t)), dγ(t) g ≤ |∇g u(γ(t))| ≤ cN,p B, dt

we derive (3.4.129).

Remark. In the particular case of p-harmonic functions (1 < p ≤ N ) on an asymptotically ﬂat manifold Corollary 3.4.22 does not apply. The next result shows that two types of behaviors might occur. Proposition 3.4.24 Assume (M, g) is asymptotically ﬂat with τ ≥ 1. If v is a positive p-harmonic function on M with 1 < p ≤ N , then v(x) admits a limit, ﬁnite or inﬁnite, when x tends to inﬁnity on M . Proof. We apply (3.4.123) in BR (x) for |x| ≥ 2R ≥ 2R0 and obtain that u = (1 − p) ln v satisﬁes |∇g u(x)| ≤ c(dist g (x, a))−1 ,

(3.4.130)

where a ∈ BR0 is ﬁxed. It implies that v satisﬁes Harnack inequality at inﬁnity. Therefore there exists c > 0 such that v(x) ≤ cv(y) ∀x, y s.t. R0

0 there exists a sequence {xk,n } ⊂ M such that dist g (xk,n , a) ↑ ∞ and v(xk,n ) ≥ k. For |xk,n | ≥ 2R0 , there holds v(x) ≥ c−1 k

∀x s.t. dist g (xk,n , a) = dist g (x, a).

(3.4.133)

In particular, if n > 1, (3.4.132) is valid if dist g (x, a) = dist g (xk,n , a) and if dist g (x, a) = dist g (xk,n , a). Since c−1 k is p-harmonic, we derive by the

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maximum principle v(x) ≥ c−1 k if dist g (xk,1 , a) ≤ distg (x, a) ≤ dist g (xk,n , a). Letting n → ∞ and then k → ∞ we obtain lim v(x) = ∞. distg (x,a)→∞ Next we assume lim sup v(x) = m < ∞. distg (x,a)→∞

(3.4.134)

Then for any > 0 exists r such that m − ≤ sup{v(x) : dist g (x, a) ≥ r} < m +

∀r ≥ r ,

(3.4.135)

and we can impose r ≥ 2R0 . Then there exists a sequence {xn, } such that dist g (xn, , a) = rn, ≥ 2n r , (i)

v(xn, ) =

sup

v(x)

distg (x,a)=rn,

(ii)

m − ≤ v(xn, ) < m +

∀n ∈ N∗ .

(3.4.136)

The function x → m + 2 − v is p-harmonic and positive in {x ∈ M : dist g (x, a) ≥ r1, }. By Harnack inequality (3.4.131) there holds m + − v(y) ≤ c (m + − v(xn, )) ≤ 2c

(3.4.137)

for all y such that distg (y, a) = rn, with n ≥ 2. This implies v(y) ≥ m − (2c + 1) ∀y s.t. dist g (y, a) = rn, .

(3.4.138)

From the maximum principle in Θ,n := {x ∈ M : r1, ≤ distg (y, a) ≤ rn, } we derive that v(y) ≥ m − (2c + 1) in Θ,n . Letting n → ∞ yields v(y) ≥ m − (2c + 1) ∀y s.t. dist g (y, a) ≥ r2, .

(3.4.139)

Thus lim inf v(x) ≥ m − (2c + 1). distg (x,a)→∞

(3.4.140)

Since is arbitrary, combining (3.4.134) and (3.4.140) we infer the claim.

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231

Boundary singularities of quasilinear Hamilton-Jacobi type equations

Let Ω ⊂ RN be a domain with a C 2 boundary containing 0. This section is concerned with the properties of the functions u ∈ C(Ω \ {0}) ∩ C 1 (Ω) which satisfy −Δp u + |∇u|q = 0 u=0

in Ω on ∂Ω \ {0},

(3.5.1)

in the range of exponents 0 < p − 1 < q < p. 3.5.1

Separable solutions

The Ariane’s shread for this study is the understanding of the same problem when Ω = RN + . Separable solutions have the form u(x) = u(r, σ) = N −1 . Then ω is a solution of r−γp,q ω(σ) with r > 0, σ ∈ S+ 2 2 p−2 q 2 2 ω + |∇ ω|2 2 ∇ ω + γp,q ω + |∇ ω|2 2 −div γp,q p−2 2 2 N −1 (3.5.2) − γp,q Λ(γp,q ) γp,q ω + |∇ ω|2 2 ω = 0 in S+ N −1 on ∂S+ .

ω=0

The critical exponent q˜c is the value of q which veriﬁes the identity βS N −1 = γp,q = +

βS N −1 p−q + ⇐⇒ q := q˜c = p − . q+1−p βS N −1 + 1

(3.5.3)

+

Theorem 3.5.1

Assume p > 1, then the following holds.

(i) If p > q ≥ q˜c there exists no nontrivial nonnegative solution of (3.5.2). (ii) If p − 1 < q < q˜c there exists only one positive solution ω := ωh of (3.5.2). Proof. We recall some ideas and statements introduced in the proof of Theorem 3.2.1. 2 2 p−2 q 2 2 η + |∇ η|2 2 ∇ η + γp,q η + |∇ η|2 2 T (η) = −div γp,q (3.5.4) p−2 2 2 2 2 − γp,q Λ(γp,q ) γp,q η + |∇ η| η.

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Any solution is C 1 up to the boundary. By the strong maximum principle N −1 N −1 N −1 2 ω > 0 in S+ and ∂ω . Thus γp,q ω 2 + |∇ ω|2 > 0 in S + . ∂n < 0 on ∂S+ Non-existence. We set θ = γp,q /βS N −1 ∈ (0, 1] and assume that there exists +

a nontrivial positive solution ω. Put β = βS N −1 and η = φθS N −1 = φθ , then +

+

∇ φ

p−2 β 2 φ2 + |∇ φ|2 2 p−2 − θp (p − 1)(θ − 1)φ(θ−1)(p−1)−1 β 2 φ2 + |∇ φ|2 2 |∇ φ|2 p−2 − γp,q Λ(γp,q )θp−2 φ(θ−1)(p−1) β 2 φ2 + |∇ φ|2 2 φ q + θq φ(θ−1)q β 2 φ2 + |∇ φ|2 2 . (3.5.5) Using the equation satisﬁed by φS N −1 we have T (η) = −θp−1 φ(θ−1)(p−1) div

+

p−2 T (η) = θp−1 φ(θ−1)(p−1) β (Λ(β) − Λ(γp,q )) β 2 φ2 + |∇ φ|2 2 φ p−2 − θp (p − 1)(θ − 1)φ(θ−1)(p−1)−1 β 2 φ2 + |∇ φ|2 2 |∇ φ|2 q + θq φ(θ−1)q β 2 φ2 + |∇ φ|2 2 ,

(3.5.6)

since γp,q Λ(γp,q ) = βΛ(γp,q )θ. One has Λ(γp,q ) ≤ Λ(β)θp−1 because γp,q ≤ β, and therefore T (η) > 0. The function φ being deﬁned up to an homothety, we can assume that it is the smallest one larger than ω. N −1 or only on Therefore, the graphs of φ and ω are tangent, either in S+ N −1 ∂S+ . Using the strict maximum principle or Hopf boundary lemma in the same way as in the proof of Theorem 3.2.1, we reach a contradiction. Uniqueness. The proof based either on the strong maximum principle or Hopf boundary lemma is similar to the one in Theorem 3.2.1. Existence. We write (3.5.2) under the form A[ω] = B(ω, ∇ω) ω=0

N −1 in S+ N −1 on ∂S+ ,

(3.5.7)

where A[ω] = −div a(ω, ∇ω), a(r, ξ) = (γp,q r2 + |ξ|2 )

p−2 2

ξ,

and B(r, ξ) = γp,q Λ(γp,q )(r2 + |ξ|2 )

p−2 2

q

r − (γp,q )(r2 + |ξ|2 ) 2 .

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The operator A is a Leray-Lions operator as in Theorem 3.2.1 and B is a N −1 × T S N −1 which satisﬁes Caratheodory function in S+ p

q

|B(r, ξ)| ≤ (γp,q + 1)Λ(γp,q )(γp,q r2 + |ξ|2 ) 2 + (γp,q r2 + |ξ|2 ) 2 ≤ C(r)(1 + |ξ|p )

(3.5.8)

for some increasing function C. Clearly the constant function μN,p,q = 1 −1 γp,q Λ(γp,q ) q+1−p is a positive solution of (3.5.2). The subsolution is looked for under the form η = φθS N −1 := φθ where θ = γp,q /βS N −1 := γp,q /β > 1 +

+

and φS N −1 is a positive solution of (3.2.4). Using the expression of T (η) + in (3.5.6) we see that it is negative provided φ is replaced by φ, for > 0 small enough, chosen also such that φ < μN,p,q . By Theorem 1.4.7 there exist a solution ω of (3.5.7) such that φ ≤ ω ≤ μN,p,q . 3.5.2 3.5.2.1

Boundary isolated singularities Removable singularities

Theorem 3.5.2 Assume q˜c ≤ q < p ≤ N , g : R+ → R+ is a continuous function which satisﬁes inf

s>0

g(s) > 0, sq˜c

(3.5.9)

and Ω ⊂ RN is a bounded C 2 domain such that 0 ∈ ∂Ω and Ω ∩ Bδ ⊂ Bδ+ for some δ > 0. If u ∈ C(Ω \ {0}) ∩ C 1 (Ω) is a nonnegative solution of −Δp u + g(|∇u|) = 0 u=0

in Ω on ∂Ω \ {0},

(3.5.10)

it is identically zero. Proof. We recall that we have denoted H + = RN + . We proceed by contradiction and assume that such a function u is nontrivial. First we can assume that u satisﬁes −Δp u + |∇u|q˜c ≤ 0 u=0

in Ω on ∂Ω \ {0},

(3.5.11)

and that Ω ⊂ RN + . We denote by M the supremum of u(x) for |x| = δ. By ˜ = (u − M )+ the maximum principle u ≤ M in Ω \ Bδ+ . The function u N

is deﬁned by extension to R+ \ {0} by 0 outside Bδ+ is a subsolution of N

problem (3.5.10) in R+ \ {0}. For > 0 and n ∈ N, large enough so that

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Ω ⊂ Bn , there exists k > 0 such that u ˜(x) ≤ kVn (x) for x ∈ ∂B ∩H where Vn is the p-harmonic function deﬁned in Bn+ := H + ∩ Bn which vanishes on ∂Bn+ \ {0} and satisﬁes lim |x|β v(r, σ) = φS N −1 (σ);

r→0

+

(3.5.12)

it has been constructed in Proposition 3.3.5. We put H+ = H + ∩ Bc and consider the following problem −Δp v + |∇v|

q˜c

in H+ ∩ Bn on ∂(H+ ∩ Bn ).

=0 v = kVn (x)

(3.5.13)

The function kVn is a supersolution of this problem while u˜ is a subsolution and one has u ˜ ≤ kVn . By Theorem 1.4.8 there exists a solution v := v,n,k of (3.5.13) and there holds u ˜ ≤ v,n,k ≤ kVn ≤ kVRN . By the construction + of the solution, the mapping k, n, ) → vk,n, is nondecreasing in k and n and nonincreasing in . By Corollary 3.4.9 the following estimate holds |vk,n, (x)| ≤ cN,p,q (|x| − )−γp,q

∀x ∈ H ∩ Bn .

(3.5.14)

The scaling transformation v → Jλ [v] deﬁned by Jλ [v](x) = λγp,q˜c v(λx), for λ > 0, transform the problem (3.5.13) with parameters (k, n, ) into the similar problem with parameters (λγp,q˜c k, nλ , λ ), thus Jλ [vk,n, ] = vλγp,q˜c k, nλ , λ .

(3.5.15)

N

1 (R+ \ {0}) to some function v∞,0 Since vk,n, converges eventually in Cloc by the previous a priori estimate and Lemma 3.3.2, there holds u ˜ ≤ v∞,0 and

Jλ [v∞,0 ] = v∞,0 .

(3.5.16)

This implies that v∞,0 is self similar under the form v∞,0 (x) = v∞,0 (r, σ) = r−γp,q ω(σ) where ω is a nonnegative solution of (3.5.2) with q = q˜c . Then ω = 0 and u ≤ M in Ω. Since u vanishes on ∂Ω \ {0} it can be extended by continuity to the zero function. When p = N the removability holds without the geometric constraint N −q and q˜c = N − 12 . on Ω. In that case γN,q = q+1−N Theorem 3.5.3 Let Ω ⊂ RN be a bounded C 2 domain such that 0 ∈ ∂Ω and g : R+ → R+ is a continuous function which satisﬁes inf

s>0

g(s) sN − 2 1

> 0.

(3.5.17)

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If u ∈ C(Ω \ {0}) ∩ C 1 (Ω) is a nonnegative solution of −ΔN u + g(|∇u|) = 0 u=0

in Ω on ∂Ω \ {0},

(3.5.18)

it is identically zero. Proof. Without loss of generality it can be supposed that u satisﬁes −ΔN u + |∇u|

N − 12

≤0

in Ω.

(3.5.19)

As in the proof of Proposition 3.3.6, we assume that B 12 (a ) is exterior to Ω and that ∂B 21 (a ) is tangent to ∂Ω at 0. We put a = nδ where n is

the normal outward unit vector to ∂Ω at 0, a ˜ = 2a , and denote by J1a˜ the inversion of center a ˜ and power 1. The ball B 12 (a ) is transformed by

J1a˜ into the half space H + passing through 0, with boundary T0 ∂Ω and containing −n. Then U = u ◦ J1a˜ satisﬁes ˜ | −ΔN U + |x − a

−1

|∇U |N − 2 ≤ 0 U =0 1

in Ω on ∂Ω \ {0},

(3.5.20)

−1

where Ω = J1a˜ (Ω). Moreover |x − a ˜ | ≤ ν for all x ∈ Ω for some ν > 0 N− 1 thus the inequality in (3.5.17) can be replaced by −ΔN U + ν |∇U | 2 ≤ 0. Because Ω ⊂ H − = −H + , it follows from Theorem 3.5.2 that U = 0 and thus u = 0. 3.5.2.2

Construction of singular solutions

In the subcritical case p − 1 < q < q˜c we can classify all the nonnegative solutions of (3.4.1) with an isolated singularity on the boundary. The geometric assumptions on the domain are diﬀerent according 1 < p < N or p = N . A ﬁrst tool is the construction of solutions with weak singularities on the boundary. Theorem 3.5.4 Let 1 < p ≤ N , p−1 < q < q˜c and Ω ⊂ RN be a bounded C 2 domain such that 0 ∈ ∂Ω. We suppose also that Ω is included into the half-space H + and that there exists δ > 0 such that Ω ∩ Bδ = Ω ∩ Bδ . Then for any k ≥ 0 there exists a unique positive function w := wk ∈ C(Ω \ {0}) ∩ C 1 (Ω) satisfying (3.5.1) such that lim rβ w(r, σ) = kφS N −1 (σ)

r→0

N −1 ) topology. in the C 1 (S+

+

N −1 ∀σ ∈ S+ ,

(3.5.21)

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The proof of this result necessitates some intermediate constructions. Lemma 3.5.5 Let the assumptions on p, q and Ω of Theorem 3.5.4 be satisﬁed. If for some k > 0 there exists wk satisfying Theorem 3.5.4, then for any k > 0 such a function wk exists. Proof. We notice that for c > 1 (resp. 0 < c < 1) cwk is a supersolution (resp. a subsolution) of (3.4.1). Assume that wk exists. Because ckVΩ is a supersolution (independently of the value of c > 0), when 0 < c < 1 it follows by Theorem 1.4.6 that there exists a solution wck of (3.5.18)(3.5.19) between cwk and ckVΩ . When c > 1 we set τ = (γp,q − β)−1 and w∗ = Jcτ [wk ] where Jcτ [wk ](x) = cτ γp,q wk (cτ x)

τ

∀x ∈ Ωc :=

1 cτ

Ω.

1

Then w∗ is a solution of problem (3.5.18) in Ω cτ instead of Ω. In particular it satisﬁes the equation in B +δ ⊂ Bδ+ and there holds cτ

w∗ (x) = k. x→0 VRN + lim

If m = sup{w∗ (x) : |x| = c−τ δ}, the function (w∗ − m)+ is a subsolution of problem (3.5.18) in Ω. Consequently, there exists a solution wck of problem (3.5.1) in Ω which satisﬁes (w∗ − m)+ ≤ wck ≤ ckVΩ and wck ∈ C 1 (Ω \ {0}) by regularity. Hence |x|1+β |∇wck (x)| + |x|1+β+α

|∇wck (x) − ∇wck (y)| ≤ M, α |x − y|

for some α ∈ (0, 1), M > 0 and all x = y, |x| ≤ |y|. This implies that the N −1 set of functions {rβ wck }0 0 is small enough and the function g is Lipschitz continuous. The computation, which uses many asymptotic expansions, is very delicate and technical. It is performed with all details in [Bidaut-V´eron, GarciaHuidobro, V´eron (2015)] and we skip it here. (x) = δ0−β . The function Proof of Theorem 3.5.4. Set M = sup VRN + |x|=δ0

W = (W − M )+ initially deﬁned in Bδ+0 and extended by 0 in Ω \ Bδ+0 is a subsolution of problem (3.5.1). The function VΩ (see Proposition 3.3.5) is a supersolution and it is larger than W . By Theorem 1.4.6 there exists a solution w of the same equation satisfying W ≤ w ≤ VΩ . Thus w satisﬁes (3.5.21) with k = 1. Existence for all k > 0 follows from Lemma 3.5.5. Uniqueness follows from (3.5.21) and the comparison principle. In the case p = N no geometric assumption is needed for the corresponding existence result. Theorem 3.5.7 Let N − 1 < q < N − 12 and Ω ⊂ RN be a bounded C 2 domain such that 0 ∈ ∂Ω. Then for any k ≥ 0 there exists a unique positive function w := wk ∈ C(Ω \ {0}) ∩ C 1 (Ω) satisfying q

−ΔN w + |∇w| = 0 w=0

in Ω on ∂Ω \ {0},

(3.5.24)

such that lim rw(r, σ) = kσ

r→0

N −1 ∀σ ∈ S+ ,

(3.5.25)

N −1 ) topology. in the C 1 (S+

Proof. Lemma 3.5.5 is immediately extendable to the case p = N . The main step is the construction of a subsolution W which satisﬁes lim

r→0

W (r, σ) =1 VΩ (r, σ)

N −1 ∀σ ∈ S+

(3.5.26)

N −1 . We use the notations of Proposition 3.3.6. If locally uniformly on S+ − we denote by W a subsolution of the equation in B δ0 \ {0}, vanishing on

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Bδ0 ∩ ∂RN − \ {0} and satisfying lim rβ W (r, σ) = φS N −1 (σ) −

r→0

N −1

uniformly in S −

,

(3.5.27)

˜ = −n = 2a and power and we assume δ0 < 1. The inversion I1a˜ of center a − 1 (a) and B 1 transforms RN into the ball B \ {0} into a subdomain, that δ0 − 2 is called a lunula and is the intersection of B 21 (a) and a ball Brb (b) for δ2 ˜ = W ◦ I a˜ satisﬁes b = 0 2 n and rb = δ0 2 . Then W 1−δ0

1

1−δ0

q 2(q−N ) ˜ = 0, ˜ + |x − a ˜| −ΔN W ∇W ˜ satisﬁes in Γ. Since |x − a ˜| ≤ 1 in Γ, W q ˜ + ∇W ˜ ≤ 0. −ΔN W

(3.5.28)

˜ ˜ (x) : x ∈ ∂Γ ∩ RN If M = sup{W + }, the function (W − M )+ is a subsolution of problem (3.5.24); it is smaller than the supersolution VΩ and (3.5.27) holds. We conclude as in Theorem 3.5.4. The type of singularities exhibited by the solutions uk obtained in Theorem 3.5.4 and Theorem 3.5.7 are not the only ones since there exists also solution with strong singularities which are the limit of the previous ones. Theorem 3.5.8 Assume p, q and Ω satisfy the assumption of Theorem 3.5.4. Then there exists w∞ = lim wk , and w∞ is the unique positive k→∞

solution of (3.5.1) which satisﬁes lim rγp,q w(r, σ) = ωh (σ)

r→0

N −1 uniformly on S+ ,

(3.5.29)

where ωh is the unique positive solution of (3.5.2) obtained in Theorem 3.5.1. Proof. Uniqueness follows from (3.5.29) and the comparison theorem. Furthermore the monotonicity of k → wk follows from the same reference and be the function (r, σ) → WRN (r, σ) = r−γp,q ωh (σ). This (3.5.21). Let WRN + + N function dominates wk since Ω ⊂ R+ . Therefore there exists w∞ = lim wk k→∞

and w∞ ≤ WRN . We recall that Jλ [w] is deﬁned by Jλ [w](x) = λγp,q w(λx) + is self-similar under Jλ . In order to specify the dofor λ > 0. Then WRN + main, we denote by wk,B + the solution of (3.5.2) in Bδ+ instead of Ω which δ

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satisﬁes (3.5.21), thus δ → wk,B + is increasing and wk,B + ≤ wk which δ δ implies w∞,δ ≤ w∞ . Moreover, for any λ, δ > 0, Jλ [wk,B + ] = wλγp,q k,B +

λ−1 δ

δ

.

(3.5.30)

By letting successively k → ∞ and δ → ∞, this implies Jλ [w∞,B + ] = w∞,B + δ

λ−1 δ

=⇒ Jλ [w∞,RN ] = w∞,RN . + +

(3.5.31)

Thus w∞,RN is a nontrivial nonnegative separable solution of (3.5.1) in RN + + −γp,q N N vanishing on ∂RN \ {0}, therefore w (r, σ) = r ω (σ) = W (r, σ). h ∞,R+ R+ + In (3.5.30) we let λ → 0, take |x| = 1 and write x = σ, then lim λγp,q w∞,B + (λ, σ) = lim w∞,B +

λ→0

δ

λ→0

λ−1 δ

(1, σ) = w∞,RN (1, σ) = ωh (σ). +

(3.5.32) N −1 ) as in the proof of TheoFurthermore this convergence holds in C 1 (S+ rem 3.3.13. In the case p = N we have a similar result with no geometric assumption on Ω. Theorem 3.5.9 Let p = N , N − 1 < q < N − 12 and Ω ⊂ RN be a bounded C 2 domain such that 0 ∈ ∂Ω. Then there exists w∞ = lim wk k→∞

and w∞ is the unique positive solution of (3.5.28) which satisﬁes lim rγN,q w(r, σ) = ωh (σ)

r→0

N −1 eventually uniformly on S+ ,

(3.5.33)

where ωh is the unique positive solution of (3.5.2) obtained in Theorem 3.5.1. Proof. Let B := B 21 (a) and B := B 12 (a ) the inner and outer tangent balls to ∂Ω at 0 (see Proposition 3.3.6). For distinction we denote by wk,Ω the solution of (3.5.1) which satisﬁes (3.5.29). Then wk,B ≤ wk,Ω ≤ wk,B c , in the domains corresponding to each couple of solutions. Since Jλ [wk,B ] = wλγN,q k,B λ , Jλ [wk,Ω ] = wλγN,q k,Ωλ and Jλ [wk,B c ] = wλγN,q k,B c λ we derive Jλ [wk,B ] = wλγN,q k,B λ ≤ Jλ [wk,Ω ] ≤ Jλ [wk,B c ] = wλγN,q k,B c λ . Furthermore Jλ [wk,B ] ≤ Jλ [wk ,B ] if λ ≤ λ and λγN,q k ≤ λγN,q k and similarly Jλ [wk,B c ] ≤ Jλ [wk ,B c ] if λ ≤ λ and λγN,q k ≤ λγN,q k . If we

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denote by w∞,Ω , w∞,B and w∞,B c the respective limits of wk,Ω , wk,B and wk,B c when k → ∞ (which exist my monotonicity), there holds w∞,B ≤ w∞,Ω ≤ w∞,B c .

(3.5.34)

Jλ [w∞,B c ] ≤ Jλ [w∞,Ω ] ≤ Jλ [w∞,B ],

(3.5.35)

Then

and (i)

Jλ [w∞,B ] ≤ Jλ [w∞,B ]

if λ ≤ λ

(ii)

Jλ [wk,B c ] ≤ Jλ [wk ,B c ]

if λ ≤ λ .

(3.5.36)

By replacing λ by λν we can rewrite (3.5.35) as Jν [Jλ [w∞,B c ]] ≤ Jν [Jλ [w∞,Ω ]] ≤ Jν [Jλ [w∞,B ]] .

(3.5.37)

Because of the monotonicity with respect to λ, the following two limits exist when λ → 0, ˜ B c = lim Jλ [w∞,B c ]. ˜ B = lim Jλ [w∞,B ] and W W λ→0

λ→0

(3.5.38)

Applying Lemma 3.3.2 to w∞,B and w∞,B c with φ(|x|) = c|x|−γN,q we infer the following series of estimates (i) |∇Jλ [w∞,B ](x)| ≤ c2 |x|−γN,q −1 (ii) |∇Jλ [w∞,B ](x) − ∇Jλ [uB ∞ ](y)| ≤ c2

∀x ∈ B λ |x − y|α ∀x, y ∈ B λ |x|γN,q +1+α |x| ≤ |y|

(iii) Jλ [w∞,B ](x) ≤ c2 |x|−γN,q −1 (dist (x, ∂B λ ))α

∀x ∈ B , (3.5.39)

and (i) |∇Jλ [u∞,B c ](x)| ≤ c2 |x|−γN,q −1 c

(ii) |∇Jλ [u∞,B c ](x) − ∇Jλ [uB ∞ ](y)| ≤ c2

∀x ∈ B c λ |x − y|α ∀x, y ∈ B c λ |x|γN,q +1+α |x| ≤ |y|

(iii) Jλ [u∞,B c ](x) ≤ c2 |x|−γN,q −1 (dist (x, ∂B c ))α

∀x ∈ B c . (3.5.40)

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Therefore the sets of functions {Jλ [u∞,B ]}λ>0 and {Jλ [u∞,B c ]}λ>0 are 1 topology. By uniqueness the limits in (3.5.38) equicontinuous in the Cloc ˜ B c are positive solutions of ˜ B and W hold in this topology. Hence W −ΔN w + |∇w|q = 0 w=0

in RN + on RN + \ {0}.

(3.5.41)

Since Jν [Jλ [w∞,B c ]] = Jλν [w∞,B - c ] .and Jν [Jλ [w∞,B ]] - = .Jλν [w∞,B ], it ˜ B c = W ˜ B c and Jν W ˜B = W ˜ B for all follows by letting λ → 0 that Jν W ˜ B c and W ˜ B are nonnegative nontrivial separable solutions ν > 0. Hence W . of (3.5.41), hence they are equal to WRN + Applying the same estimates to Jλ [u∞,Ω ] we infer (i) |∇Jλ [u∞,Ω ](x)| ≤ c2 |x|−γN,q −1 (ii) |∇Jλ [u∞,Ω ](x) − ∇Jλ [u∞,Ω ](y)| ≤ c2

∀x ∈ B λ |x − y|α ∀x, y ∈ B λ |x|γN,q +1+α |x| ≤ |y|

(iii) Jλ [u∞,Ω ](x) ≤ c2 |x|−γN,q −1 (dist (x, ∂B λ ))α

∀x ∈ B . (3.5.42) By equicontinuity there exist a subsequence {λk } and a function W such 1 topology of RN that Jλk [u∞,Ω ] converges eventually to U in the Cloc + . The function U is a positive solution of (3.5.41) which vanishes on ∂RN + \ {0}. , hence (3.5.33) holds. Using (3.5.35) we obtain that W = WRN + 3.5.2.3

Global solutions

This connexion between solutions with weak and strong boundary singularities combined with boundary Harnack inequality allows us to give a complete classiﬁcation of solutions of (3.4.1) with a boundary isolated singularity. Theorem 3.5.10 Assume 1 < p < N , p − 1 < q < q˜c and Ω ⊂ RN is a C 2 domain such that 0 ∈ ∂Ω and Ω ∩ Bδ = Bδ+ for some δ > 0. If w ∈ C(Ω \ {0}) ∩ C 1 (Ω) is a nonnegative solution of (3.5.1). Then the following alternative holds, (i) either there exists k ≥ 0 such that lim r

r→0

βS N −1 +

u(r, .) = kφS N −1 , +

N −1 ), furthermore u = 0 if k = 0 and Ω ⊂ RN in C 1 (S+ +,

(3.5.43)

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(ii) or lim rγp,q u(r, .) = ωh ,

(3.5.44)

r→0 N −1 ). in C 1 (S+

Proof. We recall that wk,B + denotes the solution of (3.5.1) with Ω = Bδ+ δ which satisﬁes (3.5.29). Step 1. Firstly we assume that lim inf x→0

w(x) = k < ∞. VRN (x) +

(3.5.45)

By Theorem 3.5.4 (3.5.43) is equivalent to lim inf x→0

w(x) = k. w1,B + (x)

(3.5.46)

δ

This liminf is achieved following a sequence {xn } converging to 0. We write the equation under the form −Δp w + |∇w|

p−2

c(x), ∇w = 0

(3.5.47)

with c(x) = |∇w|q−p ∇w. By (3.4.32), |c(x)| ≤ |x|−1 . Then we can apply boundary Harnack inequality separately to w and w1,B + (x) under the form given by Proposition 2.3.1 δ and derive w(xn ) ρ(xn ) w(x) w(xn ) = ≥c w1,B + (xn ) ρ(xn ) w1,B + (xn ) w1,B + (x) δ

δ

∀x ∈ Ω s.t. |x| = |xn | .

δ

(3.5.48) Therefore there exists k ≥ k such that w(x) ≤ wk ,B + (x) + m ≤ k VRN +m + δ

∀x ∈ Ω s.t. |x| = |xn | ,

where m is the supremum of w on |x| = δ. By the comparison principle this c for any n. therefore it holds in whole Bδ+ . inequality holds in Bδ+ ∩ B|x n| Applying again Theorem 3.5.4 and the comparison principle we obtain w ≤ wk ,B + + m ≤ k VRN + m ≤ M |x|−β + δ

in Bδ+ ,

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243

where we set β := βS N −1 . We also write φ := φS N −1 and deﬁne +

+

wr (y) = rβ w(ry)

∀y ∈ B + δ . r

Then wr satisﬁes −Δp wr + rp−q−β(q+1−p) |∇wr |q

in B + δ . r

(3.5.49)

Applying Lemma 3.3.2 with |w(x)| ≤ M |x|−β , we derive the estimates (i) |∇wr (x)| ≤ c2 |x|−β−1

∀x ∈ B +δ

|x − y|α (ii) |∇wr (x) − ∇wr (y)| ≤ c2 β+1+α |x|

∀x, y ∈ B +δ , |x| ≤ |y|.

2r

(3.5.50)

2r

+

1 δ \ {0}, topology of B 3r Consequently {wr } is relatively compact in the Cloc N

and in particular there exist a function W continuous in R+ \{0}, C 1 in RN + and vanishing on ∂RN + \{0} and a subsequence {rnk } of {rn } := {|xn |} such N

1 (R+ \ {0}) topology. Since that wrnk converges eventually to W in the Cloc p − q − β(q + 1 − p) > 0, W is a positive p-harmonic function in RN + which −β N vanishes on ∂R+ \ {0} and is dominated by M |x| . By Theorem 2.3.9, W , which is kVRN by (3.5.45) and the deﬁnition of {rn }. is a multiple of VRN + + and thus Uniqueness implies that wrn → kVRN +

lim sup w(rn , ξ) = k.

rn →0 |ξ|=r

(3.5.51)

n

By the comparison principle for any > 0 there exists n ∈ N such that (x) + m for all n ≥ n and all x ∈ Bδ+ \ Brn . Letting w(x) ≤ (k + )VRN + n → ∞ and → 0, we derive (x) + m w(x) ≤ kVRN +

∀x ∈ Bδ+ .

(3.5.52)

Jointly with (3.5.45) we infer (3.5.43). Step 2. Next we assume lim inf x→0

w(x) w(x) = ∞ = lim . x→0 VRN (x) VRN (x) + +

(3.5.53)

By the comparison principle w ≥ wk,B + in Bδ+ , for any k > 0. Therefore δ

w(x) ≥ w∞,B + (x) δ

∀x ∈ Bδ+ .

(3.5.54)

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244

Next we put wr (x) = Jr [w](x) = rγp,q w(rx). Then wr satisﬁes (3.4.1) in and it vanishes on ∂RN B+ + ∩ B δ+ \ {0}. It satisﬁes also the regularity δ+ r

r

estimates (i) |∇wr (x)| ≤ c2 |x|−γp,q −1

∀x ∈ B +δ

|x − y|α (ii) |∇wr (x) − ∇wr (y)| ≤ c2 γp,q +1+α |x|

∀x, y ∈ B +δ , |x| ≤ |y|.

2r

(3.5.55)

2r

Therefore there exists a subsequence {rn } converging to 0 such that {wrn } N 1 converges in the Cloc topology of R+ \ {0} to a function W which is a N nonnegative solution of (3.4.1) in RN + and vanishes on ∂R+ \ {0}. In order to obtain an estimate from above of w besides the a priori estimate (3.4.40) which is not precise enough, we see that for any ∈ (0, δ), w is dominated in Bδ+ \ B+ by the solution U,B + + m, where U,B + is the limit δ δ when → ∞ of the solutions u := U,B + , of δ

q

−Δp u + |∇u| = 0 u=0 u=

in Bδ+ \ B+ N on (∂Bδ ∩ RN + ) ∪ (∂R+ ∩ (Bδ \ B )) N on ∂B ∩ R+ .

(3.5.56)

Existence of U,B + , follows from the fact that the constant function is δ a solution and the monotonicity with respect to from the comparison principle. Using (3.4.42) we have U ,B + , (x) ≤ cN,p,q (|x| − )−γp,q , δ

therefore U,B + , ↑ U,B + when → ∞ and U,B + is the minimal solution δ δ δ of −Δp u + |∇u|q = 0 u=0 u=∞

in Bδ+ \ B+ N in (∂Bδ ∩ RN + ) ∪ (∂R+ ∩ (Bδ \ B )) on ∂B ∩ RN +.

(3.5.57)

When → 0, U,B + ↓ UB + which is the minimal solution of δ

δ

q

−Δp u + |∇u| = 0 u=0 lim u(x) = ∞.

in Bδ+ N in (∂Bδ ∩ RN + ) ∪ (∂R+ ∩ (Bδ )

(3.5.58)

x→0

The minimality implies that for any r > 0, Jr [U,B + ] = U ,B + , then δ

δ r

r

Jr [UB + ] = UB + when → 0. Furthermore, if δ < δ, we have UB + ≤ UB + . δ

δ r

δ

δ

which is a positive This implies that UB + increases and converges to URN + δ

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N solution of (3.4.1) in RN + , vanishing on ∂R+ \ {0} and tending to inﬁnity nontangentially at 0. Moreover

] = U RN Jr [UB + ] = UB + =⇒ Jr [URN + + δ

δ r

∀r > 0.

is a nontrivial nonnegative self-similar solution of (3.4.1) in RN Then URN +, + \ {0}. Thus vanishing on ∂RN + (r, σ) = r−γp,q ωh (σ). U RN +

(3.5.59)

Since w ≤ UB + + m we derive δ

γ m wr ≤ Jr [UB + + m] = UB + + rp,q δ

δ r

in B + δ . r

(3.5.60)

In particular, if we take |x| = r and x = rξ and use the fact that w∞,B + is δ smaller than w and satisﬁes (3.5.29), we infer ωh (ξ) ≤ lim rγp,q w(r, ξ) ≤ lim UB + = ωh (ξ), r→0

r→0

(3.5.61)

δ r

which is (3.5.44).

In the case p = N a similar result holds and its proof follows the same ideas combined with the tools of Theorem 3.5.9. Theorem 3.5.11 Let p = N , N − 1 < q < N − 12 and Ω ⊂ RN be a bounded C 2 domain such that 0 ∈ ∂Ω. If w ∈ C(Ω \ {0}) ∩ C 1 (Ω) is a nonnegative solution of (3.5.1). Then the following alternative holds, N −1 ) with (i) either there exists k ≥ 0 such that (3.5.43) holds in C 1 (S+ βS N −1 = 1; furthermore u = 0 if k = 0, + (ii) or (3.5.44) holds with γN,q .

3.5.2.4

Entire singular solutions

As a consequence of Theorem 3.4.11, Theorem 3.5.10 and Theorem 3.5.11 we have the following rigidity results concerning entire solutions in RN +. Our ﬁrst result concerns signed solutions. Proposition 3.5.12 N C(R+

Assume 0 < p − 1 < q < p and u ∈ C 1 (RN +) ∩

N \ {0}) is a solution of (3.4.1) in RN + which vanishes on ∂R+ \ {0}.

(i) If u satisﬁes lim |x|

x→0

β

S

N −1 +

u(x) = 0,

(3.5.62)

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246

then u is bounded from above in RN +. (ii) If the singularity at 0 is removable, then u ≡ 0. N −1 }. If Proof. Set RN + = {x = (x , xN ) : xN > 0} = {(r, σ) : r > 0, σ ∈ S+ the singularity at 0 is removable then the function u˜ which extends u by reﬂexion through ∂RN + is a solution of (3.4.1). By the gradient estimate (3.4.32) it satisﬁes

|∇˜ u(x)| ≤ cN,p,q |x − y|− q+1−p 1

∀x, y ∈ RN .

(3.5.63)

Letting |y| → ∞ yields ∇˜ u(x) = 0, thus u ˜ is constant and equal to 0. Next we assume that the singularity at 0 is not removable; hence |∇u(r, σ)| ≤ cN,p,q r− q+1−p 1

∀r > 0.

(3.5.64)

Therefore, for all 0 < s < r, (i)

|u(r, σ) − u(s, σ)| ≤ cN,p,q (s−γp,q − r−γp,q )

N −1 ∀σ ∈ S+

(ii)

|u(r, σ) − u(r, σ )| ≤ πcN,p,q r−γp,q

N −1 ∀σ, σ ∈ S+ . (3.5.65)

Consequently there exists m = lim u(x) and we have |x|→∞

|u(x)| ≤ m + cN,p,q |x|−γp,q

∀x ∈ RN +.

(3.5.66)

If we assume that (3.5.62) holds, then for any > 0 there exists δ > 0 such that for all δ ∈ (0, δ ], |u(x)| ≤ |x|

−β

S

N −1 +

∀x ∈ Bδ+ .

Using Lemma 3.3.2 and the fact that u vanishes on ∂RN + \ {0} we derive |u(x)| ≤ c

xN |x|

β

S

N −1 +1 +

≤ c VRN (x) +

∀x ∈ B + δ .

(3.5.67)

2

Since c VRN + m + is a supersolution of (3.4.1) in RN + larger than u for + |x| = and |x| = R large enough, we derive by the comparison principle + m + . Letting → 0, δ → 0 and R → ∞ yields u ≤ m. that u ≤ c VRN + N

Corollary 3.5.13 Assume 0 < p − 1 and u ∈ C 1 (RN + ) ∩ C(R+ \ {0}) is N a nonnegative solution of (3.4.1) in R+ which vanishes on ∂RN + \ {0}. I- If p > q ≥ q˜c , then u ≡ 0.

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II- If p − 1 < q < q˜c and u(x) → 0 when x → ∞, then (i) either u ≡ 0 , (ii) either there exists k > 0 such that u = wk,RN + . (iii) or u = WRN + Proof. The ﬁrst case follows from Theorem 3.5.2 and Proposition 3.5.12 (i). In the second case either (3.5.43) or (3.5.44) holds. If (3.5.44) holds then for any > 0 u is dominated near x = 0, at inﬁnity and on the boundary by + (we recall that WRN (r, σ) = r−γp,q ωh (σ)). the supersolution (1 + )WRN + + Then + =⇒ u ≤ WRN . u ≤ (1 + )WRN + + both at Similarly u = (1 + )u + is a supersolution which dominates WRN + N in R+ and ﬁnally u = WRN . zero and at inﬁnity. Therefore u (r, σ) ≥ WRN + + In the case where (3.5.43) holds, then for any δ > 0 we construct the function wk,B + (see the deﬁnition in the proof of Theorem 3.5.8). Then δ by letting δ → ∞. On the other hand, u ≥ wk,B + which implies u ≥ wk,RN + δ since u(x) → 0 when |x| → ∞, there holds u(x) ≤ wk,B + (x) + max{u(x) : |x| = δ}. δ

, which ends the proof. This implies u ≤ wk,RN + 3.6

Notes and open problems

3.6.1. It is not known if Theorem 3.1.13 is valid in any dimension N , i.e. if the connected component H of H in (3.1.90) is a singleton ω. Since 4 be replaced by N − 2q+2 (3.1.99) is valid in any dimension provided − q−1 q−1 , it is not known if one can connect two elements of H with diﬀerent energy E, except in some trivial cases. It is not known if the energy E is quantized on H (one value on each connected component of H, all diﬀerent). 3.6.2. A necessary and suﬃcient conditions for the existence of large solutions to −Δu + u2 = 0

in Ω

(3.6.1)

is obtained by [Dhersin, Le Gall (1997)] using tools from probability theory. Later on [Labutin (2003)] extends to the equation −Δu + uq = 0

in Ω

(3.6.2)

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with any q > 1. A local version of Labutin’s conditions is given in [Marcus, V´eron (2009)]. Set Fm (x) = {y ∈ Ωc : 2−(m+1) ≤ |x − y| ≤ 2−m } for x ∈ RN and m ∈ Z, and deﬁne WΩc by WΩc =

∞

2n

2 q−1 C2,q (2n Fn (x)) ,

(3.6.3)

−∞

where C2,q is the Sobolev capacity. There is equivalence between (i) uΩ (x) → ∞ as x → y ∈ ∂Ω, (ii) WΩc (y) → ∞ as x → y ∈ ∂Ω. Furthermore any large solution u satisﬁes c−1 WΩc (x) ≤ u(x) ≤ cWΩc (x).

(3.6.4)

for some constant c = c(Ω, q) > 1. This inequality implies uniqueness. 3.6.3. The local continuous graph property (see Deﬁnition 3.1.28) can be viewed as a “local star-shapedness” property with respect to points at inﬁnity. Since uniqueness of large solutions of (3.1.17) holds when a domain is star-shaped, provided g satisﬁes (3.1.115), we conjecture that this uniqueness holds if ∂Ω has this local graph property. 3.6.4. A natural problem is to consider boundary diﬀusion operators associated to the p-Laplacian. Assuming that Ω is a smooth domain and 0 ∈ ∂Ω an interesting problem the study of properties of solutions of − Δp u = 0 |∇u|

p−2

∂u q−1 u=0 + |u| ∂n

in Ω in ∂Ω \ {0}.

(3.6.5)

The existence of separable solutions when Ω = RN + and q > p − 1 leads to N −1 with nonlinear diﬀusion on interesting new quasilinear problems on S+ N −1 , see Note 2.6.10 in Section 2. ∂S+ 3.6.5. The study of signed solutions of (3.1.2) on S N −1 , or (3.2.3) on N −1 appears to be completely open. It is not clear whether there exists S+ sign changing solutions whenever the exponent q is below the corresponding critical exponent, even if we are unable to construct such solutions. 3.6.6. When the domain Ω is piecewise smooth, it is natural to study boundary singularities of solutions concentrated at the irregular parts of the boundary. It is likely that in the case of conical boundary points a large part of the previous constructions of solutions and characterization of behaviour can be performed. See [Borsuk, Kondratiev (2007)].

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Chapter 4

Quasilinear equations with measure data

4.1

Equations with measure data: the framework

Let Ω ⊂ RN be a domain and A ∈ C(Ω × RN ) → RN a Caratheodory vector ﬁeld which satisﬁes for some p > 1 and any x ∈ Ω and ξ, ξ ∈ RN , (i)

A(x, ξ).ξ ≥ |ξ|p

(ii)

|A(x, ξ)| ≤ λ|ξ|p−1

(4.1.1)

A(x, ξ) − A(x, ξ ).ξ − ξ > 0

(iii)

if ξ = ξ .

We denote by M(Ω) (resp. Mb (Ω) the space of Radon measures on Ω (resp bounded Radon measure) and by M+ (Ω) (resp. M+ b (Ω)) its positive cone. Any nonnegative Radon measure is extended as a Borel measure with the same notation. If μ ∈ M(Ω), we consider the following problem −div A(x, ∇u) = μ in Ω.

(4.1.2)

If one wants to deﬁne a “local” solution of (4.1.2), the most natural way is to use the weak formulation, that is to look for functions satisfying A(x, ∇u), ∇ζdx = ζdμ ∀ζ ∈ C01 (Ω). (4.1.3) Ω

Ω

p−1

The validity of this expression needs |∇u| ∈ L1loc (Ω), but the diﬃculty comes from the fact that is 1 < p < 2, ∇u, in the sense of distributions, cannot be deﬁned as a function of L1loc (Ω). For this reason two diﬀerent approaches have been developed to solve (4.1.2) and the problems associated, the method of p-superharmonic functions and the method of renormalized solutions. The ﬁrst paragraphs are devoted to present some important aspects of the two theories and to prove some of them for the sake of completeness. 249

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In the sequel we use the term of p-harmonic, p-superharmonic and p-subharmonic functions when dealing with functions satisfying, in some sense, −div A(x, ∇u) = 0 ,

−div A(x, ∇u) ≥ 0 ,

−div A(x, ∇u) ≤ 0

(4.1.4)

with A as above. We will return to the mere p-Laplacian when necessary.

4.2 4.2.1

p-superharmonic functions The notion of p-superharmonicity

The following deﬁnition of p-superharmonic functions associated to classical potential theory, complements the notion of weakly p-superharmonic functions introduced in Chapter 1. Deﬁnition 4.1 A lower semicontinuous function u deﬁned in Ω is psuperharmonic with value in R ∪ {∞} and not identically equal to ∞, i.e. proper, if for any bounded subdomain D of Ω such that D ⊂ Ω and any h ∈ C(D) ∩ W 1,p (D), u ≥ h on ∂D implies u ≥ h in D. Similarly, An upper semicontinuous function u : Ω → R ∪ {−∞} is p-subharmonic if −u is p-superharmonic. Remark. I- If u is p-superharmonic (resp. p-subharmonic) and ﬁnite a.e. in Ω, then it is locally bounded from below (resp. from above) in Ω. As a consequence, for any k ∈ R, min{k, u} ∈ L1loc (Ω) (resp. max{k, u} ∈ L1loc (Ω)). This property, easy consequence of semicontinuity, is shared by weakly p-superharmonic functions (resp. weakly p-subharmonic functions) as a consequence of weak Harnack inequality (see Theorem 1.3.1). II- A measurable function u admits a lower semicontinuous representative, still denoted by u, if u(x) = ess lim inf u(y),

(4.2.1)

y→x

for almost all x ∈ Ω. An important property of p-superharmonic functions is that (4.2.1) holds for all x ∈ Ω. The cΩ 1,p -capacity (Sobolev capacity), has a key role in the study of equations with measure data. It has been introduced in Section 1-4.

Remark. We have seen in Section 1-4 that if μ ∈ W −1,p (Ω) = (W01,p (Ω))

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there exists a unique uμ ∈ W01,p (Ω) such that A(x∇u), ∇ζdx = μ(ζ) ∀ζ ∈ W01,p (Ω).

(4.2.2)

Ω

If μ ∈ W −1,p (Ω) ∩ M+ (Ω), then the mapping ∀ζ ∈ W01,p (Ω), ζ → ζdμ Ω

is an element of W −1,p (Ω). Furthermore uμ is nonnegative and it is the weak solution of the Dirichlet problem −div A(x, ∇u) = μ u=0

in Ω on Ω.

(4.2.3)

This allows to deﬁne a dual c˜Ω 1,p -capacity by −1,p c˜Ω (Ω) ∩ M+ (Ω), μ(K c ) = 0, uμ ≤ 1 . A (K) = sup μ(K) : μ ∈ W (4.2.4) This set function is extended to general set by the rules (1.2.58) and (1.2.59), and although we need not go into the abstract deﬁnition of capacities (see [Choquet (1953-54)]), it is a capacity. Furthermore it is equivalent to the cΩ 1,p -capacity. Proposition 4.2.1

For any Borel set E ⊂ Ω, there holds p Ω ˜Ω cΩ 1,p (E) ≤ c A (E) ≤ qλ c1,p (E).

(4.2.5)

Proof. If A = −Δp , we denote c˜Ω ˜Ω 1,p . The set function deﬁned for a −Δp = c compact set K ⊂ Ω by $ % ∞ (K) = inf A(x, ∇ζ), ∇ζdx : ζ ∈ C (Ω) : ζ ≥ 1 in a neigh. of K , cΩ A 0 Ω

is a capacity equivalent to the cΩ 1,p -capacity in the sense that Ω p Ω cΩ 1,p (K) ≤ cA (K) ≤ λ c1,p (K).

The equality between cΩ ˜Ω 1,p (K) and c −Δp (K) is a consequence of minmax [ theorem (see Adams, Hedberg (1999), Th 2.5.1]). The following result is fundamental insofar it connects the notions of p-superharmonicity and weak p-superharmonicity.

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Proposition 4.2.2 Let Ω ⊂ RN be a domain and u a ﬁnite a.e. measurable function deﬁned in Ω. Then u admits a p-superharmonic representative if and only if for any k > 0, uk = min{u, k} is weakly p-superharmonic, 1,p (Ω) and that is uk ∈ Wloc A(x, ∇uk ), ∇ζdx ≥ 0 ∀ζ ∈ C01 (Ω), ζ ≥ 0. (4.2.6) Ω

Proof. We just give a sketch of the proof the details of which can be found in [Heinonen, Kilpelainen and Martio (2006)]. If uk is a supersolution, it is locally bounded from below by Theorem 1.3.1. Let D ⊂ D ⊂ Ω be a bounded subdomain of Ω and h ∈ C(D) ∩ W 1,p (D) such that uk ≥ h on ∂D. Then for any > 0, (h−uk −)+ belongs to W01,p (D) and by the maximum principle it is identi1,p (Ω), cally 0. Letting → 0 yields h ≤ uk ≤ u. Furthermore, since uk ∈ Wloc it admits a quasicontinuous representative, which means that uk coincides with a continuous function, up to a set Ek of zero cΩ 1,p -capacity. The key point is to prove that uk coincides a.e. with a function, still denoted by uk , which satisﬁes uk (x) = ess lim inf uk (y) y→x

∀x ∈ Ω.

(4.2.7)

This property of supersolutions of (4.1.4) is a consequence of the weak Harnack inequality (see [Heinonen, Kilpelainen and Martio (2006), Th 3.63]). Since uk ↑ u, u admits a semicontinuous representative, up to a set with zero cΩ 1,p -capacity and it is therefore p-superharmonic. Assume now that u is p-superharmonic. It is easy to see that for any k, uk is p-superharmonic and locally bounded. For k > 0 ﬁxed and a bounded smooth domain D ⊂ D ⊂ Ω, there exists an increasing sequence {φj,k } ⊂ C0∞ (RN ) converging to uk in D. Deﬁne the nonempty and closed convex set of constraints Kφj,k := {w ∈ W 1,p (D) : w ≥ φj,k in D, w − φj,k ∈ W01,p (D)}, and consider the following obstacle problem A(x, ∇v), ∇w − ∇v)dx ≥ 0 Find v ∈ Kφj,k s.t.

(4.2.8)

∀w ∈ Kφj,k .

D

(4.2.9) The solution uj,k exists and it is unique thanks to (4.1.1) (iii). Moreover it is a continuous function in D as the obstacle is; this property, far from obvious, is proved in [Heinonen, Kilpelainen and Martio (2006), Th 3.67]).

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Furthermore since uj,k + ζ ∈ Kφj,k for any ζ ∈ C0∞ (D), ζ ≥ 0, uj,k is weakly p-superharmonic in D and thus p-superharmonic. Next let Uj be a connected component of the open set O := {x ∈ D : uj,k (x) > φj,k (x)}. Then uj,k is p-harmonic in Uj and coincides with φj,k on ∂Uj . Hence, if y ∈ ∂Uj , we have lim inf uk (x) ≥ uk (y) ≥ φj,k (y) = lim uj,k (x). x→y

(4.2.10)

x→y

By the property of p-superharmonic functions, it implies uk ≥ uj,k in Uj and thus in D. Therefore uk = lim φj,k ≤ lim uj,k ≤ uk . j→∞

(4.2.11)

j→∞

Since the uj,k are continuous, uk is lower semicontinuous. Since uj,k is weakly p-superharmonic, we take ζ = η p (k − uj,k ) as a test function where η ∈ C01 (D) is such that 0 ≤ η ≤ 1. Because ζ ≥ 0 and uj,k ≤ uk ≤ k, then A(∇uj,k ), ∇uj,k η p dx. p A(∇uj,k ), ∇ηη p−1 (k − uj,k )dx ≥ D

D

The function u is locally bounded from below since it is lower semicontinuous, therefore uk is also locally bounded from below and thus the functions φj,k , which are smooth and form an increasing sequence, are bounded from below on D, uniformly with respect to j. Thus uj,k D ≥ c∗ . Using (4.1.1) (i)-(ii) we obtain |∇uj,k |p η p dx ≤ λp |∇uj,k |p−1 η p−1 |∇η| (k − uj,k ) D

D

≤ λp(k − c∗ )

1− p1 p1 p p |∇uj,k | η p dx |∇η| dx .

D

D p

This implies that η∇uj,k is bounded in L (D) independently of j. Therefore 1,p (Ω). η∇uk is also bounded in Lp (D). As a consequence uk ∈ Wloc Deﬁnition 4.2 The very weak gradient of a p-superharmonic function u in Ω is deﬁned by ∇u = lim ∇ (min{u, k}) . k→∞

(4.2.12)

The fact that for almost all x ∈ Ω there holds ∇ (min{u, k}) = ∇ (min{u, j}) a.e. on the set {z ∈ Ω : u(z) ≤ j} for any k ≥ j, implies that for almost all x ∈ Ω the sequence {∇ (min{u(x), k})}k is eventually

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constant. However this very weak gradient presents some important integrability properties proved in [Heinonen, Kilpelainen and Martio (2006), Th 7.46] and [Kilpel¨ainen, Li (2000)]. Proposition 4.2.3 Ω ⊂ RN . Then

Let u be a p-superharmonic function in a domain N (p−1)

(i) If 1 < p < N , u ∈ LlocN −p (ii) If p > 2 −

1 N,

,∞

(Ω) and |∇u|

N

N −1 ∈ Lloc

,∞

(Ω).

∇u is the distributional gradient.

(iii) If p = N , u ∈ Lsloc (Ω) for s > 0 and |∇u| (iv) If p > N , u ∈

p−1

N −1

N

N −1 ∈ Lloc

,∞

(Ω).

1,p Wloc (Ω).

This result implies in particular that A(., ∇u) ∈ L1loc (Ω). Hence −div A(∇u) is deﬁned as a distribution in Ω by the identity −div A(∇u)(ζ) = A(x, ∇u), ∇ζdx ∀ζ ∈ C0∞ (Ω). (4.2.13) D

4.2.2

The p-harmonic measure

The next statement is at the core of the theory of p-superharmonic function. Proposition 4.2.4 Let u be a p-superharmonic function in Ω. Then there exists a unique nonnegative Radon measure μ := μ[u] such that in D (Ω).

−div A(∇u) = μ

(4.2.14)

1,p Proof. By Proposition 4.2.2 uk ∈ Wloc (Ω) is weakly p-superharmonic. Let ∞ ζ ∈ C0 (Ω), ζ ≥ 0, then A(x, ∇uk ), ∇ζdx ≥ 0. D

Since ∇uk (x) → ∇u(x) a.e. in Ω when k → ∞ and |∇uk | ≤ |∇u|, it follows from the estimates of ∇u stated in Proposition 4.2.3 and Lebesgue dominated convergence theorem that A(x, ∇u), ∇ζdx = lim A(x, ∇uk ), ∇ζdx ≥ 0. D

k→∞

D

Thus −div A(∇u) is a positive distribution which can be extended by a nonnegative Radon measure by Riesz representation theorem. This measure, denoted by μ[u], is unique and called the p-harmonic measure of u.

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The following kernel plays a key role comparable to the Green kernel in the study of Poisson equation. Deﬁnition 4.3 Let μ be a nonnegative Borel measure on the domain Ω ⊂ RN and 1 < p < N . Then the Wolﬀ potential of the measure μ denoted by W1,p [μ] is the function deﬁned in {(x, r) ∈ Ω × (0, ∞) : B r (x) ⊂ Ω} by W1,p [μ](x, r) = 0

r

μ(Bs (x)) sN −p

1 p−1

ds . s

(4.2.15)

The fundamental estimate, which is the deus ex machina of the study of degenerate quasilinear equations with measure data, is the following estimate [Heinonen, Kilpelainen and Martio (2006)], [Kilpel¨ainen, Mal` y ] (1994) . Theorem 4.2.5 Assume 1 < p ≤ N and u is a p-superharmonic function in some domain Ω, with the p-harmonic measure μ, then there exists c = c(N, p, λ) such that for any x ∈ Ω and any r > 0 such that B r (x) ⊂ Ω, 1 W1,p [μ](x, r) ≤ u(x) ≤ c Wμ1,p (x, r) + ess inf u(z) . (4.2.16) c z∈Br (x) The following statement is a particular case of very general decomposition theorems of Radon measures with respect to a capacity (see [Fukushima, Sato, Taniguchi (1991)], [Boccardo, Gallou¨et, Orsina (1996)]). Proposition 4.2.6

1- Let μ be a Radon measure on RN . Then − μ = μ0 + μ+ s − μs ,

(4.2.17)

− + − where μ0 , μ+ s and μs are Radon measures, μs , μs ≥ 0, μ0 satisfying N RN μ0 (E) = 0 for every Borel set E ⊂ R with c1,p (E) = 0 and μ± s with RN support on a Borel set F such that c1,p (F ) = 0. The measure μ0 is the N Ω absolutely continuous part of μ with respect to cR 1,p and we write μ0 0. This implies that div Af is coercive in W01,p (Ω). By (4.1.1) (iii), it is strictly monotone. It follows from Theorem 1.4.1 that there exists a unique v˜ ∈ W01,p (Ω) such that div Af (u) = 0. Then u = v˜ + f is the unique solution of (4.2.22) in W01,p (Ω). The statements below are proved in [Heinonen, Kilpelainen and Martio (2006), Chap 9]. For the sake of completeness we present here the main ideas. Deﬁnition 4.4 Let f : ∂Ω → R ∪ {−∞} ∪ {∞}. The upper class Uf (resp. lower class Lf ) is the set of all functions u deﬁned in Ω such that (i) u is p-superharmonic in Ω (resp. u is p-subharmonic in Ω), (ii) u is bounded from below (u is bounded from above), (iii) there holds lim inf u(x) ≥ f (y) (resp. lim sup u(x) ≤ f (y)) ∀y ∈ ∂Ω. x→y

x→y

The Perron solutions of (4.2.22) can be deﬁned as follows. Deﬁnition 4.5

The function

H f = inf{u ∈ Uf }

(resp. H f = sup{u ∈ Lf }),

(4.2.25)

is the upper Perron solution (resp. the lower Perron solution) of (4.2.22). The following result is fundamental in the construction of a solution to the Dirichlet problem (4.2.22). Theorem 4.2.9 Let f = ∂Ω → R ∪ {−∞} ∪ {∞} and p > 1. Then the following dichotomy occurs: (i) either H f is p-harmonic in Ω, (ii) either H f ≡ ∞ in Ω, (iii) either H f ≡ −∞ in Ω.

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The main question is whether H f (or H f ) achieves the value f on ∂Ω. Here comes the notion of regular boundary points. Deﬁnition 4.6 Let Ω be a bounded domain and 1 < p ≤ N . If θ ∈ C(Ω) ∩ W 1,p (Ω) we denote by uθ the unique p-harmonic function in Ω such that uθ − θ ∈ W01,p (Ω), which exists by Proposition 4.2.8. A boundary point y is called regular if lim uθ (x) = θ(y).

x→y

(4.2.26)

The regularity is a local property since there holds, Lemma 4.2.10

A boundary point y ∈ ∂Ω is regular if and only if lim H f (x) = f (y),

x→y

(4.2.27)

for each f : Ω → R, bounded and continuous at y. Proof. If (4.2.27) holds then uθ = H θ (x), for any θ ∈ W 1,p (Ω) ∩ C(Ω), thus (4.2.25) holds and y is regular. Conversely, if y is regular and f : Ω → R, is bounded and continuous at y, then for given > 0 there exists δ > 0 such that |f (x) − f (y)| ≤ for all x ∈ ∂Ω verifying |x − y| ≤ δ. Let g ∈ C(Ω) ∩ W 1,p (Ω) with values in [f (y) + , sup |f | + ] and such that g(y) = f (y) + and g(x) ≡ sup |f | + for all x ∈ Ω ∩ Bδc (y). Since f ≤ g∂Ω , then H f ≤ H g∂Ω and in particular lim sup H f (x) ≤ lim sup H g∂Ω (x) = lim ug∂Ω (x) = g(y) = f (y) + . x→y

x→y

x→y

In the same way lim inf H f (x) ≥ lim sup H g˜∂Ω (x) = lim ug˜∂Ω (x) = g˜(y) = f (y) − , x→y

x→y

x→y

where g˜ ∈ C(Ω) ∩ W 1,p (Ω) has values in [inf |f | − , f (y) − ] and is constant with value inf |f | − on Ω ∩ Bδc (y). This implies (4.2.27). A natural way to prove that (4.2.27) holds at a boundary point y is to use a special form of p-superharmonic function. Deﬁnition 4.7 Let Ω be a domain and 1 < p ≤ N . A function u deﬁned in Ω is called a barrier relative to Ω at the boundary point y if it satisﬁes: (i) u is p-superharmonic in Ω, (ii) lim inf u(x) > 0 for all z ∈ ∂Ω \ {y}, x→z

(iii) lim u(x) = 0. x→y

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The role of barrier is brought to light by the following result [Heinonen, Kilpelainen and Martio (2006), Theorem 9.8]. Theorem 4.2.11 Let Ω be a domain and 1 < p ≤ N . A boundary point y is regular if and only if there exists a barrier relative to Ω at y. Proof. We just present the suﬃciency of the condition since the necessity needs to use the notion of balayage that we have not introduced and which plays an important role in abstract potential theory [Brelot (1967)]. For simplicity we assume that Ω is bounded; actually this is not a restriction since the notion of regularity of a boundary point is a local notion. Let u be a barrier; since u is p-superharmonic in Ω, it cannot achieve its minimum in Ω, therefore it is positive. Let f be a bounded function deﬁned on ∂Ω and continuous at y. We can assume that f (y) = 0. For > 0 there exists δ > 0 such that |f (x)| ≤ for any x ∈ ∂Ω ∩ Bδ (y). Let Λ > sup{|f (z)| : z ∈ ∂Ω} and v be deﬁned by $ Λ in Ω ∩ Bδc (y) v= Λ in Ω ∩ Bδ (y), m min{u, m} where m = inf{u(z) : z ∈ Ω ∩ ∂Bδ (y)}. First, observe that v in psuperharmonic in Ω. Indeed, if D is a bounded domain such that D ⊂ Ω and h ∈ C(D) ∩ W 1,p (D) is p-harmonic and smaller than v on ∂D, then, for all x ∈ ∂Bδ (y) ∩ D, lim h(z) ≤ Λ = v(x).

z→x z ∈ Bδ (y) ∩ D

Next, if x ∈ Bδ (y) ∩ ∂D, lim h(z) = h(x) ≤ v(x) ≤

z→x z ∈ Bδ (y) ∩ D

lim inf

v(z).

lim inf

v(z).

z→x z ∈ Bδ (y) ∩ D

Hence, for all x ∈ ∂(Bδ (y) ∩ D), lim h(z) = h(x) ≤ v(x) ≤

z→x z ∈ Bδ (y) ∩ D

z→x z ∈ Bδ (y) ∩ D

Since v is p-superharmonic in D, h ≤ v in Bδ (y) ∩ D and because h ≤ Λ on ∂D, the maximum principle implies that the inequality holds in D. Therefore h ≤ v in D. Furthermore, by the choice of Λ and δ, lim (v(z) + ) ≥ f (z) ∀z ∈ ∂Ω.

z→x z∈Ω

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Thus v + ∈ Uf . Since v(z) → 0 when z → y, it follows that lim sup H f (z) ≤ lim (v(z) + ) = . z→y

z→y

Similarly −v − ∈ Lf , hence lim inf H f (z) ≥ lim inf H f (z) ≥ lim (−v(z) − ) = −. z→y

z→y

z→y

Combining the last two inequalities yields lim H f (z) = 0 = f (y).

z→y

The characterization of regular boundary points is expressed by a Wiener type test involving the c1,p -capacity. Deﬁnition 4.8 at y if 0

1

B (y)

c1,p2t

Assume 1 < p ≤ N . A set E ⊂ RN is thick (resp. thin) (E ∩ Bt (y)) tN −p

1 p−1

dt =∞ t

(resp. < ∞).

(4.2.28)

The next result characterizes regular points in terms of the geometry of ∂Ω (see [Maz’ya (1970)], [Kilpel¨ainen, Mal` y (1994)]). Theorem 4.2.12 Assume 1 < p ≤ N and Ω ⊂ RN is a domain. A boundary point y is regular if and only if Ωc is thick at y. Remark. A classical result of the theory of ﬁne topology associated to a capacity [Adams, Hedberg (1999), Chap. 6] asserts that the set e1,p (E) of thin points of a set E has zero capacity. Since the set of irregular boundary N points points of Ω is included in e1,p (Ωc ), it has zero cR 1,p -capacity. 4.3 4.3.1

Renormalized solutions Locally renormalized solutions

The concept of renormalized solutions is the major step for studying general degenerate elliptic equations with measure data. The initial deﬁnition is given in [Dal Maso, Murat, Orsina, Prignet (1999)] and extended in a local and very useful form in [Bidaut-V´eron (2003)]. If k > 0 we set Tk (u) = min{k, |u|}sgn(u).

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Deﬁnition 4.9 Let Ω ⊂ RN be a domain and μ a Radon measure in Ω and 1 < p ≤ N . A measurable function u : Ω → R ∪ {∞, −∞} and a.e. ﬁnite is a locally renormalized solution of (4.1.2) if (i)

1,p (Ω) Tk (u) ∈ Wloc

∀k > 0

(ii)

u∈

Lsloc (Ω)

∀1 ≤ s

0 and ω + , ω − ∈ W 1,r (Ω) ∩ L∞ (Ω) with r > N such that ω = ω + on {x ∈ Ω : u(x) > k} and ω = ω − on {x ∈ Ω : u(x) < −k}, then A(x, ∇u), ∇ωdx = ωdμ0 + ω + dμ+ − ω − dμ− (4.3.3) s s . Ω

D

Ω

D

D3-loc: for any k > 0 there exist two nonnegative measures αk and βk in Ω absolutely continuous with respect to cΩ 1,p , with support in the sets {x ∈ Ω : u(x) = k} and {x ∈ Ω : u(x) = −k} respectively, converging in the − weak topology of measures respectively to μ+ s and μs , and such that

{|u| R > 0 and let {φn }n∈N∗ ⊂ C01 (Ω), φn ≥ 0 such that $ φn (x) =

1 if |x| ≤ Rn+1 := (1 − 2−n−1 )R + 2−n−1 R 0 if |x| ≥ Rn := (1 − 2−n )R + 2−n R ,

and |∇φn | ≤ 2n+1 (R − R)−1 χBR

n

\BR n+1

. We replace φ by φpn in (4.3.8) for

homogeneity reason, set vk = min{v, k} and take h(v) = vkq for some q > 0, then ˜ ∇v), ∇φn φp−1 v q dx q |∇vk |p vkq−1 φpn dx ≤ −p A(x, n k Ω

Ω

≤ λp

|∇vk | Ω

which yields

|∇vk |

Ω

p

vkq−1 φpn dx

≤

p

vkq−1 φpn

pλ q

1 p

|∇φn |

Ω

p

p1

vkq+p−1 dx

p p

Ω

|∇φn | vkq+p−1 dx.

Next

|∇vk | Ω

p

vkq−1 φpn dx

= ≥

and ﬁnally

p+q−1 p

v p

k

W 1,p (BRn+1 )

p q+p−1 p q+p−1

p p+q−1 p φn ∇v p dx k Ω

p

BRn+1

p+q−1 p

p

≤ 2np λp cp1

vk

p+q−1 p ∇v p dx, k

Lp (BRn )

,

(4.3.9)

,

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where c1 is bounded provided q ≥ q0 > 0 and R − R ≥ 0 > 0. Assume p < N , then using Gagliardo-Nirenberg inequality, we deduce from (4.3.9),

p+q−1 p

p p n

vk N (p+q−1) ≤ 2 λc1 c2 , (4.3.10)

vk

L

N −p

(BRn+1 )

Lp (BRn )

and c2 is independent of n since R ≤ Rn+1 ≤ 2R. Therefore vk

p

N (q+p−1) L N −p

(BRn+1 )

≤ (2n λc1 c2 ) p+q−1 vk Lq+p−1 (BRn ) .

(4.3.11)

N (p−1) We choose q < p(p−1) N −p so that s = p + q − 1 < N −p and put τ = Then the n-th iterate of the previous estimate yields

vk Lτ n s (BR

p

n+1

)

≤ 2s

n k=1

k+1 τk

p

(λc1 c2 ) s

n

1 k=1 τ k

vk Ls (BR ) . 1

N N −p .

(4.3.12)

All the series in exponent are convergent, Rn ↓ R and R1 < R , hence we derive from (4.3.1) (ii), vk L∞ (BR ) ≤ C vk Ls (B2R ) ≤ C v Ls (BR ) = m.

(4.3.13)

This implies that vk L∞ (BR ) ≤ m. In the case p = N we use H¨ older’s inequality on the right-hand side of (4.3.9) and the proof follows by the same method. Proof of Theorem 4.3.2. For k, > 0 we set hk, (t) =

1 min{(k + − t)+ , }.

Let φ ∈ C01 (Ω) be nonnegative. Since hk, (t) ≤ 0 and hk, vanishes on (−∞, k) ∪ (k + , ∞) , we obtain 0 ≤ hk, (u)φdμ0 + h(∞) φdμs

Ω

Ω

=

Ω

A(x, ∇u), ∇uhk, (u)φdx +

A(x, ∇u), ∇φhk, (u)dx Ω

A(x, ∇u), ∇φhk, (u)dx.

≤ Ω

If we let → 0, then hk, (u)(x) → χ(−∞,k) (u(x)) and we derive A(x, ∇u), ∇φhk, (u)φ → A(x, ∇uk ), ∇φ,

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where uk = min{u, k}, hence A(x, ∇uk ), ∇φdx ≥ 0.

(4.3.14)

Ω 1,p Since uk is bounded from below it belongs to Wloc (Ω). This means that uk is weakly p-superharmonic. By Proposition 4.2.2 it is p-superharmonic. ˜, u˜ is p-superharmonic and it coincides with u a.e. in Ω. As uk ↑ u

The converse of Theorem 4.3.2 holds true. Theorem 4.3.4 Let p, Ω be as in Theorem 4.3.2 and A : Ω × RN → RN a Caratheodory vector ﬁeld satisfying (4.1.1). If u is a p-superharmonic function with Riesz measure μ, then it is a locally renormalized solution of the corresponding equation (4.1.2). The proof which requires in particular the existence of renormalized solutions of (4.1.1) with measure data, is long and delicate and contains several intermediate results which are interesting by themselves (see [Kilpel¨ainen, Kuusi, Tuhola-Kujanp¨aa¨ (2011)]).

If μ ∈ M+ (Ω), L > 0 and Ω is a bounded domain such that Ω ⊂ Ω we deﬁne the class Sμ,r,L (Ω ) of functions u : Ω → R by % $ 1 Sμ,r,L (Ω ) = u : W1,p [μ](x, r) ≤ u(x) ≤ cW1,p [μ](x, r) + L ∀x ∈ Ω . c (4.3.15) By Theorem 4.2.5 this class is naturally associated with p-superharmonic functions u which have μ for Riesz measure and with L = c ess inf u(z). z∈Br (x)

The main step is the following identity Proposition 4.3.5 Let 1 < p ≤ N , Ω and A : Ω × RN → RN be as in T heorem 4.3.4. Let u and v be p-superharmonic functions and assume that for all bounded domains Ω verifying Ω ⊂ Ω and all r > 0 small enough, there exists L < ∞ such that u, v ∈ Sμ,r,L where μ = μ0 + μs is the Riesz measure of u. Then for any h ∈ W 1,∞ (R2 ) such that ∇h has compact support, there holds A(x, ∇u), ∇(h(u, v)φ)dx = h(u, v)φdμ0 Ω Ω (4.3.16) + h(∞, ∞) φdμs ∀φ ∈ C01 (Ω). Ω

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The proof of Theorem 4.3.4 follows by taking v = u in Proposition 4.3.5. Remark. Because of Proposition 4.2.7 the support of μs is a subset of {x ∈ Ω : u(x) = ∞} and, since u and v belong to Sμ,r,L , there holds {x ∈ Ω : u(x) = ∞} = {x ∈ Ω : v(x) = ∞} = {x ∈ Ω : W1,p [μ](x, r) = ∞}. This justiﬁes the expression of the second integral on the right-hand side of (4.3.16). Furthermore, since h(u, v) is lower semicontinuous, there holds

h(u, v)φdμ0 + h(∞, ∞) φdμs = h(u, v)φdμ,

Ω

Ω

Ω

where the right-hand side is the integral of h(u, v)φ with respect to the measure μ (see [Bourbaki (1965)]). Similarly, if u is a lower semicontinuous locally renormalized solution of (4.3.7), identity (4.3.7) can be rewritten as

A(x, ∇u), ∇(h(u)φ)dx = Ω

h(u)φdμ− ,

h(u)φdμ+ − Ω

(4.3.17)

Ω

where the integrals are taken in the Bourbaki sense (and due to A. Weil). An important application of this equivalence is the proof of the removability of polar sets. Theorem 4.3.6 Let Ω ⊂ RN be a domain, 1 < p ≤ N and u a nonnegative p-superharmonic function in Ω \ K where K ⊂ Ω is a polar set, i.e. a ˜ deﬁned in whole Ω by set such that cΩ 1,p (K) = 0. Then the function u u ˜(x) = ess lim u(y), y→x

(4.3.18)

is p-superharmonic in Ω. Proof. By replacing Ω by a smaller domain, we can assume that it is bounded and smooth. For k > 0, uk = max{u, k} is p-superharmonic and ∞ bounded. Since cΩ 1,p (K) = 0 there exists a sequence {ηn } ⊂ C0 (Ω) of nonnegative functions such that ηn ≥ 1 in a neighborhood of K, ∇ηn Lp → 0 and ηn → 0 a.e. as n → ∞. Replacing ηn by max{1, min{η1 , η2 , ..., ηn }} we can suppose that the sequence is decreasing, with value 1 in a neighborhood of K, and we set η˜n = 1 − ηn . Let φ ∈ C0∞ (Ω), φ ≥ 0 and ψn = η˜n φ, then

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for α > 0, p A(x, ∇uk ), ∇ψn (1 + uk )−α ψnp−1 dx Ω

≥ α A(x, ∇uk ), ∇uk (1 + uk )−α−1 ψnp dx. Ω

Using (4.1.1) and H¨older’s inequality p λp p p −α p |∇uk | (1 + uk ) ψn dx ≤ |∇ψn | (1 + uk )p−1−α dx. α Ω Ω Letting n → ∞ yields p λp p −α p |∇uk | (1 + uk ) φ dx ≤ |∇φ|p (1 + uk )p−1−α dx. (4.3.19) α Ω Ω If we ﬁx α such that p − 1 > α > 0, we obtain p λp p p |∇uk | φ dx ≤ (1 + k)p−1 |∇φ|p dx. α Ω Ω

(4.3.20)

1,p (Ω). Taking ψn as a test function, we have Hence uk ∈ Wloc A(x, ∇uk ), ∇φ˜ ηn dx ≥ A(x, ∇uk ), ∇ηn φdx. Ω

Ω

By H¨ older’s inequality and (4.3.20) the right-hand side of the above inequality converges to 0 as n → ∞. Therefore A(x, ∇uk ), ∇φdx ≥ 0, (4.3.21) Ω Ω

which proves that uk is weakly p-superharmonic in Ω. By Proposition 4.2.2 uk admits a lower semicontinuous p-superharmonic representative in Ω which is given by (4.2.7). Finally, since uk ↑ u when k → ∞ and since an increasing limit of p-superharmonic functions is p-superharmonic, it follows that u, given by (4.3.18), is p-superharmonic in Ω. 4.3.2

The stability theorem

Renormalized solutions were initially introduced for solving the Poisson equation with measure (4.2.3). The next statement is by now a classical result in the theory of renormalized solutions [Dal Maso, Murat, Orsina, Prignet (1999)].

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Theorem 4.3.7 Let Ω be a bounded domain and μ ∈ Mb (Ω) with decomposition (4.2.17). A measureable function u deﬁned in Ω, ﬁnite a.e. and satisfying Tk (u) ∈ W01,p (Ω)

(i)

|∇u|

(ii)

p−1

∈ L (Ω) r

∀k > 0 ∀1 ≤ r

N with the property that h(u)φ ∈ W01,p (Ω), the identity (4.3.2) is valid. D2: if ω ∈ W01,p (Ω) ∩ L∞ (Ω) and if there exist k > 0 and ω + , ω − ∈ W 1,r (Ω) ∩ L∞ (Ω) with r > N such that ω = ω + on {x ∈ Ω : u(x) > k} and ω = ω − on {x ∈ Ω : u(x) < −k}, then (4.3.3) holds. D3: for any k > 0 there exist two nonnegative bounded measures αk and βk in Ω, absolutely continuous with respect to cΩ 1,p , with support in the sets {x ∈ Ω : u(x) = k} and {x ∈ Ω : u(x) = −k} respectively and converging in the narrow topology (i.e. with test functions in C(Ω) ∩ L∞ (Ω)) to μ+ s and 1,p ∞ respectively and such that (4.3.4) holds for all ψ ∈ W (Ω) ∩ L (Ω). μ− s 0 D4: for any h ∈ W 1,∞ (R) with compact support and any φ ∈ W 1,p (Ω) ∩ L∞ (Ω) with compact support in Ω such that h(u)φ ∈ W01,p (Ω), identities (4.3.5) and (4.3.6) hold. Remark. It is a useful observation in the previous results to point out that the truncation of a renormalized solution satisﬁes an equation. Following − 1 Proposition 4.2.6 we can write μ = f − div g + μ+ s − μs where f ∈ L (Ω), p ± RN g ∈ (L (Ω) and μs singular with respect to the c1,p -capacity, then (4.3.4) becomes A(x, Tk (u)) − g, ∇φdx = f φdx Ω

{|u| 0 Tk (u ) converges to Tk (u) in W01,p (Ω). The proof is long and necessitates several intermediate results which present their own interest. Proposition 4.3.9 Assume Ω and p are as in Theorem 4.3.8. Let μ ∈ Mb (Ω) such that there exists a renormalized solution u to 4.2.3. Then we have 1 A(x, u), ∇udx ≤ c μ Mb ∀n, k ≥ 0. (4.3.25) k {n≤|u|≤n+k} Furthermore I- If 1 < p < N , then for any k > 0, we have for some c = c(N, p) > 0, N

meas({|u| > k}) ≤ c

N −p μ M b

k

N (p−1) N −p

∀k ≥ 0,

(4.3.26)

and N

meas({|∇u| > k}) ≤ c

N −1 μ M b

k

N (p−1) N −1

∀k ≥ 0.

(4.3.27)

I- If p = N , then for any r > 0 there exists c = c(N, r) > 0 such that, r

meas({|u| > k}) ≤ c

μ Mb k r(N −1)

∀k ≥ 0.

(4.3.28)

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Furthermore, for any 0 < s < N there exists c = c(N, s) > 0 such that N

N −1 μ M b meas({|∇u| > k}) ≤ c ks

∀k ≥ 0.

(4.3.29)

Proof. We introduce the following truncation type functions ⎧ 0 if s < −n − k ⎪ ⎪ ⎪ ⎪ ⎪ s+n+k ⎪ if − n − k ≤ s < −n ⎪ k ⎪ ⎨ Hn,k (s) = 1 if − n ≤ s ≤ n ⎪ ⎪ ⎪ n+k−s ⎪ ⎪ if n ≤ s ≤ n + k ⎪ k ⎪ ⎪ ⎩ 0 if s > n + k, and Bn,k (s) = 1 − Hn,k (s). Then using D2 with ω = Bn,k (u+ ), we have 1 A(x, ∇u), ∇udx = Bn,k (u+ )dμ0 + Bn,k (u+ )dμ+ s k {n }) p∗ Tk (u) Lp∗ ≤ ∗ ∗ ≤ c2 k p μ pMb , which yields (4.3.25) by taking = k. Next, for k, > 0, we set Φ(, κ) = meas({|u| > } ∪ {|Du|p > κ}). Since Φ

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271

is nonincreasing in the second variable, we have 1 κ 1 κ Φ(0, κ) ≤ Φ(0, s)ds ≤ Φ(, 0) + (Φ(0, s) − Φ(, s)) ds. κ 0 κ 0 Notice that p

p

Φ(0, s)−Φ(, s) = meas({|u| > 0}∪{|Du| > s})−meas({|u| > } ∪ {|Du| > s}) p = meas({0 < |u| ≤ }∪{|Du| > s}). Since by (4.3.25), p |∇u| dx = {0 s})ds

0

≤ c μ Mb , we derive N

Φ(0, κ) ≤ c

N −p μ M b

N (p−1) N −p

+ cc

μ Mb . κ

N

Choosing such that c κ μ Mb =

N −p μ M

b N (p−1) N −p

and putting κ = k p , we obtain

(4.3.27). Case p = N . We combine H¨older and Gagliardo-Nirenberg inequalities and q , derive, with 1 < q < N and q ∗ = NN−q N Tk (u) N Lq∗ ≤ cN,q ∇Tk (u) Lq

≤ cN,q (meas({u < k}))

N −q q

≤ cN,q c1 k(meas({u < k}))

{|u| }) = meas({|Tk (u)| > }) q∗ Tk (u) Lq∗ ≤ ∗ ∗ ≤ c2 k q μ qMb , we infer, with = k, meas({|u| > k}) ≤ cN,q meas({|u| < k})k −

(N −1)q N −q

q

N −q μ M , b

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which implies (4.3.28) by setting r = same as the one of (4.3.27).

q N −q .

The proof of (4.3.29) is the

Remark. Inequality (4.3.29) is not optimal in the case p = 2 since in that case, it is possible to obtain by using estimates on the Poisson kernel the more general one, N

N −1 μ M b meas({|∇u| > k}) ≤ c kN

(4.3.32)

∀k ≥ 0.

Actually an optimal regularity result, useful in many applications, has been proved in [Dolzmann, Hungerb¨ uhler, M¨ uller (2000)]. Its expression needs to introduce the space BM O functions of bounded mean oscillation. A function u ∈ LN (RN ) belongs to BM O(RN ) if

Deﬁnition 4.11 [u]BMO(RN ) =

sup sup r

−N

N1

a∈RN r>0

N

|u − (u)a,r | dx

< ∞,

(4.3.33)

Qr (a)

where Qr (a) is the cube {x = (x1 , ..., xN : |xj − aj | ≤ This space is equipped with the norm

r 2

, ∀j = 1, ..., N }.

u BMO(RN ) = u LN (RN ) + [u]BMO(RN ) .

(4.3.34)

All the functions deﬁned on a domain Ω are extended by 0 in Ωc . If u = 0 on Ωc and a ∈ Ωc then the mean value of u on Qr (a) is estimated in the following way |u − (u)a,r | dx ≤ meas(Qr (a))[u]BMO(RN ) . |(u)a,r |meas(Qr (a)∩Ωc ) ≤ Qr (a)

If Ωc is of type A, that is there exists K > 0 such that meas(Br (x) ∩ Ωc ) ≥ KrN

∀x ∈ Ωc ,

(4.3.35)

then r−N

|u|N dx Qr (a)

N1 ≤ cN,K

r−N

N1

N

|u − (u)a,r | dx

,

Qr (a)

(4.3.36) and |(u)a,r | ≤ cK [u]BMO(RN ) .

(4.3.37)

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One of the important property of BMO functions is the John and Nirenberg inequality [John, Nirenberg (1961)] Proposition 4.3.10 Assume Ω ⊂ RN is a bounded domain. Then there exist constants c1 , c2 > 0 depending on N such that for any cube Qr (a) ⊂ Ω and any u ∈ BM O(Ω), meas ({x ∈ Q : |u(x) − (u)a,r | > σ}) ≤ c1 rN e

− [u]

c2 σ BM O(Qr (a))

.

(4.3.38)

If Ωc is of type A and u vanishes on Ωc there exists a constant c3 depending on Ω such that meas ({x ∈ Ω : |u(x) − uΩ | > σ}) ≤ c3 e

− [u]

c2 σ BM O(Ω)

,

(4.3.39)

where uΩ is the average of u in Ω. Proposition 4.3.11 Assume Ω ⊂ RN is a bounded domain and Ωc is of type A. If μ ∈ Mb (Ω) is such that there exists a renormalized solution u to (4.2.3) with p = N , then 1

N −1 , u BMO(Ω) ≤ c1 μ M b

(4.3.40)

and N

N −1 μ M b meas({|∇u| > k}) ≤ c kN

∀k ≥ 0.

(4.3.41)

Remark. Estimate (4.3.41) states that ∇u belongs to the Lorentz space LN,∞ (Ω) and there holds N

N −1 . ∇u LN,∞ (Ω) ≤ c μ M b

(4.3.42)

Actually the two estimates are not independent since any u such that ∇u ∈ LN,∞ (Ω) belongs to BM O(Ω) and there holds φ BMO(Ω) ≤ c ∇φ LN,∞ (Ω)

∀φ ∈ C01 (Ω),

(4.3.43)

where c = c(N ) > 0. The properties of functions which gradient belong to a Lorentz space Lp,q (RN ) for 1 < p < ∞ and 1 ≤ q ≤ ∞ are clearly presented in [Tartar (1998)]. Proof of Theorem 4.3.8. For > 0, we consider an approximation μ = f − − + − + − div g +μ+ s, −μs, of μ = f −div g +μs −μs by convolution of f , μs and μs . + − 1 + Then f μs, and μs, are smooth functions and f → f in L (Ω), μs, → μ+ s − −1,p and μ− (Ω), there exists s, → μs in the narrow topology. Since μ ∈ W

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a renormalized solution u of (4.2.3) with μ = μ . Therefore the existence part of the theorem is a particular case of the stability statement. From now we study the stability and assume that the measure μ admits 1 the decomposition μ = f − div g + λ⊕ − λ where f ∈ L (Ω) converges weakly to f , g ∈ (Lp (Ω))N converges to g in (Lp (Ω))N and div g remains + + bounded in Mb (Ω), and λ⊕ and λ , belong to Mb (Ω) and converge to μs − and μs respectively in the narrow topology. Notice that it is not assumed RN that λ⊕ or λ are singular with respect to c1,p . Actually we can write ⊕ ⊕ λ⊕ = λ,0 + λ,s

and

λ = λ,0 + λ,s ,

⊕ where λ⊕ ,0 , λ,0 (resp. λ,s , λ,s ) are absolutely continuous (resp. singular)

with respect to the cR 1,p -capacity, all the above measures being nonnegative. Therefore N

⊕ μ = f − divg + λ⊕ ,0 − λ,0 + λ,s − λ,s = μ,0 + μ,s .

(4.3.44)

R + ⊕ − where μ,0 0; it follows by Proposition 4.3.9 that u satisﬁes the same estimates (4.3.25), (4.3.26), (4.3.27) in the case 1 < p < N , and (4.3.28) and (4.3.41) in the case p = N . In particular (4.3.25) holds with u replaced by Tn+k (u ), 1 k

{n≤|u |≤n+k}

A(x, Tn+k (u )), ∇Tn+k (u )dx ≤ c∗

∀n, k ≥ 0, (4.3.45)

which yields, by (4.1.2) (i) and the deﬁnition of Bn,k , 1 k

|∇Bn,k (u ))| dx ≤ c∗ p

{n≤|u |≤n+k}

∀n, k ≥ 0.

(4.3.46)

With n = 0 we obtain the bound

|∇Tk (u ))| dx ≤ c∗ k p

Ω

∀k ≥ 0.

(4.3.47)

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Let q

1} ∞

|∇u |

q(p−1)

dx

tq(p−1)−1 meas ({|∇u | > t}) dt

1

≤ meas(Ω) + since meas ({|∇u | > t}) ≤ c∗ t− in Lq(p−1) for every q < NN−1 .

c∗ q , N N −1 − q N (p−1) N −1

. This implies that ∇u is bounded

Step 2: There exists a subsequence of u which is a Cauchy sequence in measure. Let σ, η > 0 be ﬁxed. For k > 0 and , δ > 0, we have {|u −uδ | > σ} ⊂ {|u | > k}∪{|uδ | > k}∪{|Tk (u )−Tk (uδ )| > σ}. (4.3.48) From (4.3.26) there exists k0 such that for every k ≥ k0 and every , δ > 0, meas({|u | > k}) + meas({|uδ | > k}) ≤

η . 2

If we choose k = k0 , then {Tk0 (u )} is bounded in W01,p (Ω) in (4.3.48). By Rellich-Kondrachov theorem there exists a subsequence, still denoted by {u }, such that {Tk0 (u )} is convergent in Lp (Ω). Furthermore {∇Tk0 (u )} converges weakly in Lp (Ω). This implies that {Tk0 (u )} is a Cauchy sequence in measure. Thus there exists 0 > 0 such that 0 < , δ ≤ 0 =⇒ meas({|Tk (u ) − Tk (uδ )| > σ}) ≤

η . 2

Then there exists a measurable function u such that u converges to u in measure. Up to an extracted subsequence u converges also to u a.e. in Ω, but for the sake of simplicity this new subsequence is still denoted by u . Step 3: Weak convergence of the truncates. Since u → u a.e. and Tk is continuous, Tk (u ) → Tk (u) a.e. in Ω. In Step 2 we have used the fact that {Tk (u )} converges to some function vk in Lp (Ω) and ∇Tk (u ) ∇vk in Lp (Ω)-weak. Then vk = Tk (u) and ∇Tk (u ) converges weakly in Lp (Ω) to ∇Tk (u). Consequently Tk (u) ∈ W01,p (Ω) and ∇u is well deﬁned a.e. in Ω. Step 4: Equation satisﬁed by Tk (u ). Since u is a renormalized solution − of (4.2.3) , for every k there exist two nonnegative measures λ+ ,k and λ,k ,

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absolutely continuous with respect to the cR 1,p -capacity, concentrated respectively on the sets {u = k} and {u = −k}, such that A(x, ∇Tk (u )), ∇φdx = φdμ,0 N

{|u | γ} , / E3 = {sup{|∇Tk (u )| , |∇Tk (uδ )|} ≤ κ} {|Tk (u ) − Tk (uδ )| ≤ γ} / {|∇Tk (u ) − ∇Tk (uδ )| > η} .

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Then {x ∈ Ω : |∇Tk (u )(x) − ∇Tk (uδ )(x)| > η} ⊂ E1

"

E2

"

E3 .

(4.3.53)

Since Tk (u ) and Tk (uδ ) are bounded in W01,p (Ω) we ﬁx κ > 0 such that meas(E1 ) ≤ η/3. Since A is a Caratheodory function, there exists a null set C ⊂ Ω such that ξ → A(x, ξ) is continuous for all x ∈ Ω \ C. We deﬁne ! K = (ξ, ξ ) ∈ RN × RN : |ξ| ≤ κ, |ξ | ≤ κ, |ξ − ξ | ≥ η . and, for x ∈ Ω \ C, Γ(x) = inf{A(x, ξ) − A(x, ξ ), ξ − ξ : (ξ, ξ ) ∈ K}.

(4.3.54)

By the assumption (4.1.1) (iii) the function Γ is positive, hence for any θ > 0 there exists τ > 0 such that Γ(x)dx ≤ τ =⇒ meas (A) ≤ θ. (4.3.55) A

Next for E2 , we take Tγ (Tk (u ) − Tk (uδ )) for test function in (4.3.52) and derive A(x, ∇Tk (u )) − A(x, ∇Tk (uδ )), ∇Tk (u ) − ∇Tk (uδ )dx {Tk (u )−Tk (uδ )≤γ} − ≤ γ λ+ ,k (Ω) + λ,k (Ω) + |μ,0 |(Ω) ≤ γM, − since λ+ ,k , λ,k and |μ,0 | are uniformly bounded for k ﬁxed. Hence γM ≥ A(x, ∇Tk (u )) − A(x, ∇Tk (uδ )), ∇Tk (u ) − ∇Tk (uδ )dx, E3

≥

Γ(x)dx. E3

(4.3.56) We ﬁx θ = η/3. Then there exists τ > 0 such that (4.3.55) holds. Next we choose γ = τ /M in (4.3.56) and get meas (E3 ) ≤ η/3. At end, we use Step 2 and derive that there exists 0 such that, if 0 < , δ < 0 , there holds meas (E2 ) ≤ η/3, which implies the claim. Step 6: The sequence {∇u } is a Cauchy sequence in measure. Let σ > 0, then " " {|∇u −∇uδ | > σ} ⊂ {|∇Tk (u )−∇Tk (u )| > σ} {|u | > k} {|uδ | > k} .

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Using (4.3.26) or (4.3.28), we can ﬁx k so that meas ({Tk (u ) > k}) ≤ η/3 uniformly with respect to . By Step 5 there exists 0 such that, for 0 < , δ < 0 , there holds meas ({|∇Tk (u ) − ∇Tk (u )| > σ}) ≤ η/3, which implies the claim. This implies in particular that, up to extracting a new p−1 remains subsequence, ∇u (x) → ∇u(x) a.e. in Ω. By Step 1 |∇u | N q bounded in L (Ω) for all 1 ≤ q < N −1 . By Vitali’s convergence theorem p−1 p−1 p−1 ∈ Lq (Ω) and |∇u | → |∇u| in Lq (Ω) in the we infer that |∇u| same range of values of q. Actually ∇u satisﬁes estimates (4.3.29) and (4.3.41). Furthermore, it follows from (4.1.1) (ii) and the continuity of A(x, .) that A(x, ∇u ) → A(x, ∇u) in (Lq (Ω))N . This implies that u is a solution of (4.2.3) in the sense of distributions in Ω. Step 7: Equation satisﬁed by Tk (u). We analyze the diﬀerent terms in (4.3.49) where the test function φ belongs to φ ∈ L∞ (Ω)∩W01,p (Ω). Writing μ,0 = f − div g + λ⊕ ,0 − λ,0 we develop (4.3.49), A(x, ∇Tk (u )) − g , ∇φdx = f φdx + φdλ⊕ ,0 {|u | p − 1, |∇u| ∈ 1 − − μ . Let φ, ξ and φ be as in Lloc (Ω). As usually we write μ = μ0 + μ+ n s s Step 1, then q q −1 A(x, ∇u), ∇uk φnp dx + qp A(x, ∇u), ∇φn uk φnp dx Ω Ω (5.2.46) qp q q + g(x, u, ∇u)Tk (u)φn dx = Tk (u)φnp dμ0 + k φnp d |μs | . Ω

Ω

Ω

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Using (4.1.1) (i)-(ii) we get qp −1 q p qp p−1 |∇uk | φn dx ≤ kqp λ |∇u| |∇φn | φn dx + k g(x, u, ∇u)φnp dx Ω

Ω

Ω

φqp d |μ| .

+k Ω

The claim follows by using the estimates on g(., u, ∇u) and |∇u| tained in Step 1.

p−1

ob-

Step 3: The function u is a locally renormalized solution in Ω. As usually − 1,∞ (R) with h compactly supported, we write μ = μ0 +μ+ s −μs . Let h ∈ W 1 φ ∈ C0 (Ω) (not necessarily nonnegative) and ξn and φn as in Step 1. Then (5.2.44) reads with φ = φn

A(x, ∇u), ∇(h(u)φn )dx +

Ω

g(x, u, ∇u)h(u)φn dx Ω

(5.2.47) − h(u)φn dμ0 + h(∞) φn dμ+ − h(−∞) φ dμ . n s s

= Ω

Ω

Ω

Since A(x, ∇u), ∇(h(u)φn ) = A(x, ∇u), ∇φn h(u) + A(x, ∇u), ∇uh (u)φn , the integral of the two terms on the right-hand side converges by dominated convergence and steps 1-2, thus A(x, ∇u), ∇(h(u)φn )dx = A(x, ∇u), ∇(h(u)φ)dx. lim n→∞

Ω

Ω

Similarly, using Step 1 and dominated convergence g(x, u, ∇u)h(u)φn dx = g(x, u, ∇u)h(u)φdxdx. lim n→∞

Ω

Ω

h(u)φdμ0 + Since μ(K) = 0, the right-hand of (5.2.47) converges to Ω φdμ− h(∞) φdμ+ s − h(−∞) s . This implies that (5.2.44) holds with arΩ

Ω

bitrary test function φ, which is the claim.

A counter part of this result is a necessary condition for solving (5.2.1). Theorem 5.2.8 Assume g satisﬁes the assumptions (5.2.40) and μ ∈ M+ (Ω). If there exists a locally renormalized solution u to (5.2.1) in Ω, N then μ p − 1. This problem is called supercritical if not every bounded measure is eligible, in particular no Dirac measure is eligible. When 1 < p < N , the critical value for q is q1 = NN(p−1) −p already deﬁned in (3.1.4). 5.3.1

Estimates on potentials

We have already seen the role of Wolﬀ potential W1,p [μ] for the representation of positive p-harmonic functions in RN . We introduce below a more general class of Wolﬀ potentials. Let α > 0, 1 < s < α−1 N , R > 0 and μ ∈ M+ (Ω). The R-truncated (α, s)-Wolﬀ potential of μ is R Wα,s [μ](x)

= 0

R

μ(Bt (x)) tN −αs

1 s−1

dt . t

(5.3.1)

∞ If R = ∞ we note Wα,s [μ] = Wα,s [μ]. We also deﬁne the R-truncated fractional maximal operator

μ(Bt (x)) , tN −α 0 1 and 0 < αp < N , we set ⎧ −αp N −αp N − αp − Np−1 ⎪ ⎪ if R < ∞ − R− p−1 ⎨ p − 1 (min{r, R}) (r, R) = ⎪ N −αp ⎪ ⎩ N − αp r− p−1 if R = ∞. p−1 (5.3.6) The next technical result is called a good λ-inequality. Proposition 5.3.1 Let 0 ≤ η < p − 1, 0 < αp < N and r > 0. Then exist c0 > 0 depending on N , α, p, η and 0 > 0 depending on N , α, p, η and r such that for all μ ∈ M+ (RN ) with diam(supp μ) ≤ r, R ∈ (0, ∞], 1 ∈ (0, 0 ] and λ > (μ(RN )) p−1 (r, R), there holds, ! ! WR [μ] > 3λ ∩ Mη,R [μ] ≤ (λ)p−1 α,p αp ≤ c0 exp

p−1−η 4(p−1)

p−1 p−1−η

! p−1 R αp− p−1−η ln 2 Wα,p [μ] > λ . (5.3.7)

Proof. Case 1: R = ∞. Since Wα,p [μ] is lower semi continuous, for any λ > 0 the set ! ! R R Dλ = x ∈ RN : Wα,p [μ](x) > λ := Wα,p [μ] > λ ,

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is open. By Whitney decomposition [Grafakos (2010)] there exists a counto

o

able set Q of closed cubes Qj such that Dλ = ∪j Qj , Qj ∩ Qk = ∅ if j = k and diam Qj ≤ dist (Qj , Dλc ) ≤ 4diam Qj .

! For > 0 we set F,λ = {Wα,p [μ] > 3λ} ∩ Mηαp [μ] ≤ (λ)p−1 . The key of the proof is to show that there exist c0 = c0 (N, α, p, η) > 0 and 0 = 0 (N, α, p, η, r) > 0 such that for any Q ∈ Q, ∈ (0, 0 ] and 1 λ > (μ(RN )) p−1 (r, ∞), there holds p−1 p−1 p − 1 − η p−1−η − p−1−η (5.3.8) αp ln 2 |Q| . |F,λ ∩ Q| ≤ exp 4(p − 1) We denote

! 5 diam(Q) [μ] > λ ∩ Mηαp [μ] ≤ (λ)p−1 ∩ Q. E,λ = Wα,p

Step 1. We claim that there exists c1 = c1 (N, α, p, η) > 0 such that for any Q ∈ Q, there holds F,λ ∩ Q ⊂ E,λ

∀ ∈ (0, c1 ] , ∀λ > 0.

(5.3.9)

Let Q ∈ Q verifying Q ∩ F,λ = ∅ and xQ ∈ Dλc be such that dist (xQ , Q) ≤ R [μ](xQ ) ≤ λ (by deﬁnition of Dλ ). If k ∈ N, r0 = 4diam(Q) and Wα,p 5diam (Q) and x ∈ F,λ ∩ Q, we have 1 2k+1 r0 μ(Bt (x)) p−1 dt = A + B, tN −αp t 2k r0 where A=

k+1 1+2k

2k 1+2

2k r0

r0

μ(Bt (x)) tN −αpq

1 p−1

dt , B= t

2k+1 r0

k+1 2k 1+2 k 1+2

r0

μ(Bt (x)) tN −αp

1 p−1

dt . t

By the deﬁnition of Mηαp [μ], μ(Bt (x)) ≤ tN −αp hη (t)Mηαp [μ](x) ≤ tN −αp hη (t)(λ)p−1 .

(5.3.10)

Plugging this inequality into the expression B and using the value of hη yields 2k+1 r0 1 dt ≤ c2 λ2−k , B ≤ λ (hη (t)) p−1 1+2k+1 t k 2 k r0 1+2

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where c2 = c2 (p, η) > 0. Set δ = N −αp p−1 .

c3 > 0 depends only on

(1 − δ)A ≤ c3 2

2k 2k +1

−αp Np−1

323

, then 1 − δ ≤ c3 2−k where

Therefore

−k

2k+1 r0

2k r

≤ c3 2−k ≤ c4 λ2

0

μ(Bt (x)) tN −αp

2k+1 r0

1

(hη (t)) p−1 2k r0

−k

1 p−1

dt t

dt t

,

where c4 = c4 (N, α, p, η) > 0. Since for any x ∈ F,λ ∩ Q and any t ∈ [r0 (1 + 2k ), r0 (1 + 2k+1 )] we have B 2k t (x) ⊂ Bt (xQ ), we get, by a suitable 1+2k

change of variable and using the explicit value of δ, ⎛

(1+2k+1 )r0

⎝

δA = (1+2k )r0

μ(B

2k t 1+2k

(x))

tN −αp

1 ⎞ p−1 1 p−1 (1+2k+1 )r0 dt (x )) dt μ(B t Q ⎠ ≤ . N −αp t t t (1+2k )r0

Summing the two last inequalities combined with the estimate on B, we obtain

2k+1 r0

2k r

0

μ(Bt (x)) tN −αp

1 p−1

dt ≤ c5 λ2−k + t

(1+2k+1 )r0

(1+2k )r

0

μ(Bt (xQ )) tN −αp

1 p−1

dt , t

where c5 = c5 (N, α, p, η) > 0. If we sum over k we infer

∞ r0

μ(Bt (x)) tN −αp

1 p−1

dt ≤ 2c5 λ + t

∞

2r0

μ(Bt (xQ )) tN −αp

1 p−1

dt t

(5.3.11)

≤ (1 + 2c5 )λ, since Wα,p [μ](xQ ) ≤ λ. Set c1 = (2c5 )−1 , then if ∈ (0, c1 ] there holds

∞

r0

μ(Bt (x)) tN −αp

1 p−1

dt ≤ 2λ. t

(5.3.12)

Since r0 = 5 diam(Q) and Wα,p [μ] > 3λ because x ∈ F,λ , we derive r0 [μ] > 3λ which yields (5.3.9). Wα,p

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1

Step 2. Proof of (5.3.8). Let λ > (μ(RN )) p−1 (r, ∞). There exists a ball / B2r (a), Br (a) containing the support of μ. If x ∈

∞

Wα,p [μ](x) = r

μ(Bt (x)) tN −αp

1 p−1

1 dt ≤ (μ(RN )) p−1 (r, ∞) < λ, t

therefore x ∈ Dλc . Equivalently Dλ ⊂ B2r (a). Put m0 = max{1,ln(40r)} so that 2−m r0 ≤ 2−1 for m ≥ m0 (remember ln 2 that r0 = 5diam (Q) ≤ 20r). Then, for any x ∈ E,λ ,

r0

2−m r0

μ(Bt (x)) tN −αp

1 p−1

r0 1 dt dt ≤ λ (hη (t)) p−1 t t 2−m r0 r0 2−m0 r0 η dt η dt − p−1 + λ (− ln(t)) (ln 2)− p−1 ≤ λ t −m t −m 0 r0 2 r0 2 η − p−1

≤ λm0 (ln 2)

η 1− p−1

(p − 1) ((m − m0 ) ln 2) + λ p−1−η

.

Notice that, in the last line, we have used inequality η

η

η

a1− p−1 − b1− p−1 ≤ (a − b)1− p−1

∀a ≥ b ≥ 0.

Therefore

r0

2−m r0

μ(Bt (x)) tN −αp

1 p−1

p−1 η η dt 2(p − 1) 1− p−1 ≤ m λ, ∀m ≥ (ln 2)− p−1 m0p−1−η. t p−1−η (5.3.13)

Set gi (x) = η

2−i r0 2−i+1 r0

μ(Bt (x)) tN −αp

1 p−1

dt . t

p−1

Then, for m ≥ (ln 2)− p−1 m0p−1−η , r0 [μ](x) ≤ Wα,p

≤

η 2(p − 1) 1− p−1 m λ + p−1−η

2(p − 1) m p−1−η

η 1− p−1

λ +

r0

2−m r

0

∞ i=m+1

μ(Bt (x)) tN −αp

gi (x).

1 p−1

dt t

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325

Consequently, for any β > 0, # ∞ ) * η 2(p − 1) 1− p−1 m gi > 1 − λ |E,λ | ≤ Q p − 1 − η # ∞ i=m+1 ∞ ) * η 2(p−1) 1− p−1 −β(i−m−1) −β m ≤ Q gi > (2 (1−2 ) 1− λ p−1−η i=m+1 i=m+1 $ % ∞ * η 2(p − 1) 1− p−1 Q gi > (2−β(i−m−1) (1 − 2−β ) 1 − m λ . ≤ p−1−η i=m+1 (5.3.14) We claim that | Q ∩ {gi > s}| ≤

c6 (N, η)2iαp | Q| (λ)p−1 . sp−1

(5.3.15)

To see that, we pick y ∈ E,λ and get, using Chebyshev’s inequality, |Q ∩ {gi > s}| ≤ = ≤

1 sp−1 1 sp−1 1 sp−1

(gi (x))p−1 dx Q

2−i r0

μ(Bt (x)) tN −αp

Q 2−i+1 r0

1 p−1

dt t

p−1 dx

μ(B2−i+1 r0 (x)) dx := A. −i N −αp Q (2 r0 )

By Fubini’s theorem, A= = ≤

1 sp−1 1 sp−1

1 (2−i r0 )N −αp 1 (2−i r0 )N −αp

1

1

sp−1

(2−i r0 )N −αp

Q RN

χB

2−i+1 r0

(z)dμ(z)dx

χB Q+B2−i+1 r

0

Q+B2−i+1 r

0

Q

Q

2−i+1 r0

|B2−i+1 r0 (z)| dμ(z)

≤

c7 (N ) −iαp αp 2 r0 μ(Q + B2−i+1 r0 ) sp−1

≤

c7 (N ) −iαp αp 2 r0 μ(Q + B(1+2−i+1 )r0 (y)), sp−1

since Q + B2−i+1 r0 ⊂ Q + B1+2−i+1 r0 (y).

(z) (x)dxdμ(z)

Since r0 = 5diam (Q) and

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μ(Bt (y)) ≤ (ln 2)−η tN −αp (λ)p−1 , we obtain N −αp c8 (N, η) −iαp αp 2 r0 (1 + 2−i+1 )r0 (λ)p−1 p−1 s c6 (N, η) −iαp ≤ 2 |Q|(λ)p−1 , sp−1

A ≤ c8 (N, η)

which is (5.3.15). Therefore, (5.3.14) can be rewritten as ∞

c6 2−iαp (λ)p−1 |Q| p−1 η 2(p−1) 1− p−1 i=m+1 2−β(i−m−1) (1 − 2−β ) 1 − m λ p−1−η p−1 ≤ c6 2−(m+1)αp η 2(p−1) 1− p−1 1 − p−1−η m ∞ × |Q|(1 − 2−β )1−p 2(β(p−1)−αp)(i−m−1) .

|E,λ | ≤

i=m+1

(5.3.16) If we choose β < αp we obtain p−1 2−mα |Q| , |E,λ | ≤ c9 η 2(p−1) 1− p−1 1 − p−1−η m

$ where c9 = c9 (N α, pη) > 0. We put 0 = min c1 , any ∈ (0, 0 ] we choose m ∈ N such that

p−1−η 2(p−1)

p−1 p−1−η

1 −1

p−1 p−1−η

p−1

η

∀m > (ln 2)− p−1−η m0p−1−η ,

p−1−η − 1 and m ∈ N we have if x ∈ E,λ , r0 [μ](x) ≤ mλ + Wα,p

∞

gi (x).

i=m+1

Inequality (5.3.17) reads |E,λ | ≤ c9 2−mpα

1 − m

p−1 |Q| ,

∀m > m0 .

! If we set 0 = min c1 , 12 , then for any ∈ (0, 0 ] and m ∈ N such that −1 − 2 < m ≤ −1 − 1 we infer from (5.3.10), |Q ∩ E,λ | ≤ |E,λ | ≤ c10 2αp exp(−αp ln 2−1 )|Q|,

(5.3.18)

which is the claim. Case 2: R < ∞. Since for λ > 0, Dλ is open, we use again Whitney covering Q = {Qk }k∈N . If Q ∈ Q with diam (Q) > R8 , there exists a ﬁnite nQ such that number nQ of closed dyadic cubes {Pj,Q }j=1 Q=

nQ

o

o

Pj,Q , Pj,Q ∩ Pi,Q = ∅ if i = j and

j=1

R R ≤ diam (Pj,Q ) ≤ . 16 8

We set F = Q ∪ Q where % $ R , Q = Q ∈ Q : diam (Q) ≤ 8 % $ R Q = Pj,Q , 1 ≤ j ≤ nQ , Q ∈ Q, diam (Q) > , 8

R p−1 For > 0 we denote again F,λ = {Wα,p [μ] > 3λ} ∩ {Mη,R }. αp [μ] ≤ (λ) Let Q ∈ F such that F,λ ∩ Q = ∅ and r0 = 5diam (Q). If dist (Dλc , Q) ≤ 4diam (Q), there exists xQ ∈ Dλc such that R [μ](xQ ) ≤ λ, we ﬁnd, by the same argudist (xQ , Q ≤ 4diam (Q) and Wα,p ment that in the case R = ∞ and (5.3.11), that for any x ∈ F,λ ∩ Q, there holds 1 R μ(Bt (x)) p−1 dt (5.3.19) ≤ (1 + c11 )λ, tN −αp t r0

with c11 = c11 (N, α, p, η) > 0.

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328

R 16

If dist (Dλc , Q) > 4diam (Q), we have Then, for all x ∈ F,λ ∩ Q, there holds

R

r0

μ(Bt (x)) tN −αp

1 p−1

< diam (Q) ≤

tN −αp ln 2−η (λ)p−1 5r tN −αp 16 η ≤ (ln 2)− p−1 λ ln 16 5

dt ≤ t

R

R 8

1 p−1

since Q ∈ Q .

dt t (5.3.20)

≤ 2λ. If we take ∈ (0, c12 ] where c12 = min{1, c−1 11 }, we derive F,λ ∩ Q ⊂ E,λ ,

(5.3.21)

where, as in case 1, !* η R Mαp,R [μ] ≤ (λ)p−1 . E,λ = Wα,p [μ] > λ Furthermore, if x ∈ / B2 ,

R

R Wα,p [μ](x) = min{r,R}

μ(Bt (x)) tN −αp

1 p−1

1 dt ≤ μ(RN ) p−1 (r, R). t

1 Hence, if λ > μ(RN ) p−1 (r, R), we infer Dλ ⊂ B2 , which implies again r0 = 5diam (Q) ≤ 20r. The remaining of the proof is as in Case 1. The following estimates link the Wolﬀ potential of a measure with its Bessel potential and the fractional maximal operator. Theorem 5.3.2 Assume α > 0, 0 < p − 1 < q < ∞, 0 < αp < N and 0 < s ≤ ∞. Then there exists a constant c13 = c13 (N, α, p, q, s) > 0 such that for any R ∈ (0, ∞] and μ ∈ M+ (RN ) there holds 1

R

p−1

c−1 q 13 Wα,p [μ] Lq,s (RN ) ≤ Mαp,R [μ]

s L p−1 p−1

For any R > 0 there exists c14 μ ∈ M+ (RN ) there holds

,

(RN )

R

≤ c13 Wα,p [μ] Lq,s (RN ) .

(5.3.22) = c14 (N, α, p, q, s, R) > 0 such that for

1

R

p−1

c−1 q 14 Wα,p [μ] Lq,s (RN ) ≤ Gαp [μ]

s L p−1 p−1 ,

(RN )

R

≤ c14 Wα,p [μ] Lq,s (RN ) . (5.3.23)

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329

Proof. We denote by μn = χBn μ. Step 1. We claim that 1

R

Wα,p [μ] q,s N ≤ c13 Mαp,R [μ] p−1q L (R )

L p−1

,

s p−1

(RN )

.

(5.3.24)

From Proposition 5.3.1 there exist three positive constants c0 = c0 (N, α, p), a = a(N, α) and 0 = 0 (N, α, p) such that for all n ∈ N∗ , t > 0, 0 < R ≤ ∞ and ∈ (0, 0 ], there holds ! !* ! a R p−1 R [μn ] > 3t MR,η [μ ] ≤ (t) [μ] > t . Wα,p ≤ c0 e− Wα,p n αp (5.3.25) When 0 < s < ∞ and 0 < q < ∞ we have ! s ! s sa R R Wα,p [μn ] > 3t q ≤ c15 e− q Wα,p [μn ] > t q ! s p−1 q + c15 MR,η , αp [μn ] ≤ (t) with c15 = c15 (N, α, p, q, s) > 0. Multiplying by ts−1 and integrating over (0, ∞) yields ∞ ∞ ! qs dt ! s dt − sa s R R q ≤e Wα,p [μn ] > 3t t ts Wα,p [μn ] > t q t t 0 0 ∞ s ! p−1 q dt . + c15 ts MR,η αp [μn ] ≤ (t) t 0 By a change of variable, we derive ∞ ! s dt − sa −s R q 3 − c15 e ts Wα,p [μn ] > t q t 0 ∞ ! s p−1 q dt . ts MR,η ≤ c15 αp [μn ] ≤ (t) t 0 We choose > 0 small enough so that 3−s − c15 e− q > 0. Using (1.2.8) and identity

1

1 1

s1 ∗ s2 s1

∀f ∈ Ls1 ,s2 (RN ),

t f s + dt = s1

tλf sa

L

2 (R

,

t

)

Ls2 (R+ , dt t )

valid for 0 < s1 < ∞, 0 < s2 ≤ ∞ (we recall that from Chap 1-2, Sf (t) = {|f | > t} and λf (t) = |Sf (t)|), we obtain 1

R

Wα,p [μn ] q,s N ≤ c13 Mαp,R [μn ] p−1q L (R )

L p−1

,

s p−1

(RN )

.

(5.3.26)

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Letting n → ∞, we derive (5.3.24) by Fatou’s lemma. Step 2. We claim that 1

R

Wα,p [μ] q,s N ≥ c13 Mαp,R [μ] p−1q L (R )

L p−1

,

s p−1

(RN )

.

(5.3.27)

For R > 0 we have 1 μn (Bt (x)) p−1 dt = + tN −αp t R 1 p−1 μn (B2R (x)) R [μn ](x) + . ≤ Wα,p RN −αp

2R [μn ](x) Wα,p

2R

R Wα,p [μn ](x)

(5.3.28)

Thus $ % μn (B2R (x)) 2R R p−1 {Wα,p [μn ] > 2t} ≤ {Wα,p [μn ] > 2t} + x : >t . RN −αp Consider a set of points {zj }m j=1 ⊂ B2 such that B2 ⊂ ∪j B 12 (zj ), thus B2R (x) ⊂ ∪j B R (x + Rzj ) for all x ∈ RN and all R > 0. Then 2 ⎧ ⎫ $ % ⎨ ⎬ μ (B R (x + Rzj )) n μ (B (x)) p−1 p−1 2 x : n 2R >t >t ≤ ⎩x : ⎭ RN −αp RN −αp j # ) μn (B R (x + Rzj )) tp−1 2 ≤ > x: RN −αp m j # ) μn (B R (x)) tp−1 2 ≤ > x − Rzj : RN −αp m j # ) μn (B R (x)) tp−1 2 = m x : > m . N −αp R Furthermore, from (5.3.28), 1 μn (B R (x)) p−1 2

RN −αp hence

R ≤ 2Wα,p [μn ](x),

$ $ % % t R x : μn (B2R (x)) > tp−1 ≤ m Wα,p . [μn ] > RN −αp 2mp−1

We deduce ! ! W2R [μn ] > 2t ≤ (m + 1) WR [μn ] > 2t . α,p α,p

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page 331

331

This implies

2R

R

Wα,p [μn ] q,s N ≤ c16 Wα,p

q,s N , [μ ] n L (R ) L (R ) and by monotone convergence,

2R

R

Wα,p [μ] q,s N ≤ c16 Wα,p [μ] Lq,s (RN ) , L (R )

(5.3.29)

with c16 = c16 (N, α, p, q, s) > 0. On the other hand, by (5.3.28), we have for any 0 < ρ < R, 2R 2ρ [μ](x) ≥ Wα,p [μ](x) ≥ c17 sup Wα,p

0 0. This yields 1

2R Wα,p [μ](x) ≥ c17 (Mαp,R [μ](x)) p−1 .

(5.3.30)

Combining (5.3.30) and (5.3.29), we obtain (5.3.27) and ﬁnally (5.3.22). Step 3. We claim that (5.3.23) holds. By (5.3.22) we have also

1

R

R

p−1 c−1 [μ] ≤ M [μ] ≤ c [μ] αp αp q

W

W

s αp,R 18 18 , ,2 ,2 q,s N N L

2

L p−1

(R )

p−1

2

(R )

(5.3.31) where c18 = c18 (N, α, p, q, s) > 0. Estimate (1.2.49) on Bessel kernel (with α, R replaced by αp, R2 ) takes the form c−1 19

χB R (x) χBR (x) |x| 2 ≤ G (x) ≤ c + c19 e− 2 αp 19 N −αp N −αp |x| |x|

∀x ∈ RN \ {0}.

Therefore c−1 19

χB R χBR 2 − |.| ∗ μ ≤ G [μ] ≤ c αp 19 N −αp ∗ μ + c19 e 2 ∗ μ. N −αp |.| |.|

By Fubini’s theorem χBR |.|N −αp

∗ μ(x) = (N − αp)WRαp ,2 [μ](x) + 2

μ(BR (x)) RN −αp

≥ (N − αp)WRαp ,2 [μ](x), 2

which implies

c20 WRαp ,2 [μ] 2

q

L p−1

,

s p−1

(RN )

≤ Gαp [μ]

q

L p−1

,

s p−1

(RN )

,

,

Lq,s (RN )

(5.3.32)

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where c20 = c20 (N, α, p, q, s) > 0. Since e−

|x| 2

|.|

≤ c21 e− 2 ∗ χB R with 2

c21 = c21 (N, R) > 0, we get |. |. |. e− 2 ∗ μ ≤ c21 e− 2 ∗ χB R ∗ μ = c21 e− 2 ∗ χB R ∗ μ . 2

2

Because χB R ∗ μ(x) = μ(B R (x)) ≤ c22 WRαp ,2 [μ](x), 2

2

2

where c22 = c22 (N, α, p, R) > 0, we derive, with c23 = c21 c22 , |.

|.

e− 2 ∗ μ ≤ c23 e− 2 ∗ WRαp ,2 [μ]. 2

Using Young’s inequality, we obtain

|.

|.

−2

∗ μ q , s ≤ c23 e− 2 ∗ WR [μ] q , s αp

e ,2 2 L p−1 p−1 (RN ) L p−1 p−1 (RN )

|.

R

−2

≤ c24 W αp ,2 [μ] q , s

e 1,∞ N 2 L p−1 p−1 (RN ) L (R )

R

≤ c25 W αp ,2 [μ] q , s N , p−1 p−1 L

2

(R )

(5.3.33) where c25 = c25 (N, α, p, R) > 0. Since by Fubini’s theorem there holds as above, χ BR 2

R

|.|N −αp

N −αp 2 ∗ μ(x) = (N − αp)W αp ,2 [μ](x) + 2

χB R (x)

2

2

RN −αp

≤ c26 WRαp ,2 [μ](x), 2

we infer

χ

BR

2 ∗ μ

|.|N −αp

q , s L p−1 p−1

≤ c27 WRαp ,2 [μ]

where c27 = c27 (N, α, p, R, s) > 0. This implies,

R

q ≤ c αp [μ] Gαp [μ] p−1

W

, s 28 ,2 N p−1 L

(R )

2

q

L p−1

2

(RN )

q

L p−1

,

s p−1

,

s p−1

(RN )

(RN )

,

, (5.3.34)

(5.3.35)

by combining (5.3.31), (5.3.33) and (5.3.34). We deduce that (5.3.23) holds from (5.3.32) and (5.3.35), and using (5.3.30) and (5.3.22).

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The next result provides an exponential estimate of the Wolﬀ potential of a measure under a constraint over its fractional potential. Theorem 5.3.3 Assume α > 0, 0 ≤ η < p − 1, 0 < αp < N and r > 0 p−1 p−1−η p−1−η . Then there exists c29 > 0, depending on N , and set δ0 = 12(p−1) α, p, η and r such that for any R ∈ (0, ∞], δ ∈ (0, δ0 ], μ ∈ M+ , any couple of balls B = Bκ (a) with κ ≤ r and B = B2κ (a), there holds ⎛ ⎞ p−1 R p−1−η ⎜ Wα,p [μB ](x) ⎟ 1 ⎟ dx ≤ c29 , δ exp ⎜ (5.3.36) 1

⎝ |B | B δ0 − δ

η

p−1−η ⎠

Mαp,R [μB ] ∞ L

(B)

where μB = χB μ. Furthermore, if η = 0, c29 is independent of r.

< ∞. By applying ProposiProof. Assume M := Mηαp,R [μB ] L∞ (B)

tion 5.3.1 with μ = μB , there exists c0 depending on N , α, p, η and 0 depending on N , α, p, η and r such that, for all R ∈ (0, ∞], ∈ (0, 0 ] and 1 t > (μB (B)) p−1 (κ, R), there holds ! ! R p−1 Wα,p [μB ] > 3t ∩ Mη,R αp [μB ] ≤ (t) ≤ c0 exp

p−1−η 4(p−1)

1

p−1 p−1−η

Because (μB (B)) p−1 (κ, R) ≤

αp

p−1 − p−1−η

! R ln 2 Wα,p [μB ] > t .

η N −αp − p−1 M p−1 (ln 2)

1

p−1

= t−1 Mηαp,R [μB ]

L∞ (B)

1 p−1

1

(5.3.37) , in (5.3.7) we choose ∀t > t0 ,

= M p−1

η 1 N −αp − p−1 where t0 = max −1 M p−1 . As in the proof of Proposi0 , p−a (ln 2 ! R [μB ] > t ⊂ B . Then tion 5.3.1, we have that Wα,p ! R Wα,p [μB ] > 3t ∩ B ≤ c0 exp

p−1−η 4(p − 1)

p−1 p−1−η

αp ln 2M

1 − p−1−η

t

p−1 p−1−η

|B | .

(5.3.38)

This can be re-written under the form |{F > t} ∩ B | ≤ |B | χ(0,t0 ] (t) + e−δ0 t |B | χ(t0 ,∞) (t),

(5.3.39)

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where p−1

R [μB ]) p−1−η , F = M − p−1−η (Wα,p 1

and p−1 p−1−η η − p−1 −1 N − αp (ln 2 } . t0 = 3 max{0 , p−a

Take δ ∈ (0, δ0 ], then, by Fubini’s theorem ∞ exp(δF (x))dx = δ e−δt |{F > t} ∩ B | dt B

0

δ |B |

t0

e−δt dt + c0 δ |B |

0

∞

e−(δ0 −δ)t dt

t0

c0 δ δt0 |B | , ≤ e −1+ δ0 − δ

which is (5.3.36). 5.3.2

Approximation of measures

The following result is an improvement of Proposition 1.2.7 (iii). Theorem 5.3.4 Let Ω ⊂ RN be a domain. Assume p − 1 < s1 < ∞, p − 1 < s2 ≤ ∞, 0 < αp < N , R > 0 and μ ∈ M+ (Ω). If μ is N , there absolutely continuous with respect to the capacity cR s1 s2 αp, , s1 +1−p s2 +1−p

exists a nondecreasing sequence {μn } ⊂ M+ (Ω) with compact support in Ω which converges to μ in the weak sense of measures and such that R [μn ] ∈ Ls1 ,s2 (RN ) for all n. Furthermore, if μ ∈ M+ Wα,p b (Ω), the convergence of {μn } towards μ holds in the narrow topology. Proof. By Proposition 1.2.7 (iii) there exists a nondecreasing sequence of s1 s2 αp, , nonnegative measures {μn } ⊂ (L s1 +1−p s2 +1−p (RN )) with compact support in Ω, which converges weakly to μ. If μ ∈ M+ b (Ω), the convergence holds in the narrow topology. Furthermore, if μ is a nonnegative measure in RN , s1

s2

Gαp [μ] ∈ L p−1 , p−1 (RN ) ⇐⇒ μ ∈ (L s1

s2

s1

αp, s

s2

s1 s2 , 1 +1−p s2 +1−p

(RN ))

and (Lαp, s1 +1−p , s2 +1−p (RN )) = L−αp, p−1 , p−1 (RsN ))s if s2 < ∞ (see (1.2.54) 1 2 and references therein). Then Gαp [μn ] ∈ L p−1 , p−1 (RN ) and by TheoR [μn ] ∈ Ls1 ,s2 (RN ). rem 5.3.2, Wα,p

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335

Solvability with measure data revisited

Throughout this section Ω ⊂ RN is a bounded domain, A : Ω × RN → RN satisﬁes the assumptions (4.1.1). We consider a Caratheodory function g : Ω × R → R subject to part of the following set of assumptions, (i)

sg(x, s) ≥ 0

a.e. x ∈ Ω, ∀s ∈ R,

(ii)

s → g(x, s)

is nondecreasing for a.e. x ∈ Ω

(iii)

|g(x, s)| ≤ g˜(|s|)

a.e. x ∈ Ω, ∀s, ∈ R

(iv)

s → g˜(s)

is nondecreasing on R+ , g˜(0) = 0.

(5.4.1)

We intend to prove existence of renormalized solutions to −div A(x, ∇u) + g(x, u) = μ u=0

in Ω (5.4.2) on ∂Ω,

when μ is a bounded measure and the function g does not satisﬁes the subcriticality assumptions (5.1.19) and (5.1.20). 5.4.1

The general case

The next two-side estimate for renormalized solutions is a consequence of Theorem 4.3.8 and Corollary 4.3.14. Theorem 5.4.1 There exists a constant c = c(N, p) > 0 such that for any bounded measure and any renormalized solution u of problem (4.2.3), there holds 2 diam(Ω)

−cW1,p

[μ− ] ≤ u ≤ cW1,p

2 diam(Ω)

[μ+ ]

a.e. in Ω.

(5.4.3)

− + − Proof. Let μ = μ+ − μ− = μ+ 0 + μs − (μ0 + μs ) the Jordan and Lebesgue decompositions of measures. By Theorem 4.3.7 the truncate uk = Tk (u) is a renormalized solution of − −div A(x, ∇uk ) = χ{|uk | 0. Then, for any bounded measure μ absolutely continuN -capacity, problem (5.4.2) admits a ous with respect to the cR p, q , q q+1−p q+1−p

renormalized solution u. Furthermore (5.4.8) holds. Proof. Similarly as in Theorem 5.4.5, μ± p−1, 0 ≤ β < N and μ is a bounded Radon measure in Ω. Then problem (5.4.29) admits a renormalized solution u and inequalities (5.4.8) hold if one of the next two conditions is satisﬁed. 1- The function g satisﬁes (5.4.1) (i)-(ii)-(iii) and (5.4.27), and μ is absoN -capacity. lutely continuous with respect to the cR Nq p, ,1 N q−(p−1)(N −β)

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2- The function g satisﬁes (5.4.28) and μ is absolutely continuous with N -capacity. respect to the cR Nq p, , q N q−(p−1)(N −β) q+1−p

5.4.3

Exponential type nonlinearities

The potential method can be used to prove existence of solutions of problem (5.4.2) when g satisﬁes |g(x, r)| ≤ g˜(|r|) ≤ eτ |r| − 1 λ

a.e. in Ω, ∀r ∈ R.

(5.4.31)

Theorem 5.4.8 Assume 1 < p < N , τ > 0, λ ≥ 1 and g satisﬁes (5.4.1) (i)-(ii)-(iii) and (5.4.31). There exists M > 0 depending on N , p, τ and λ such that for any bounded Radon measure in Ω which can be decomposed as μ = f1 − f2 + ν1 − ν1 , where fi ∈ L1+ (Ω), νi ∈ M+ b (Ω) and

(p−1)(λ−1)

λ

M

p,2 diam (Ω) [νi ]

L∞ (Ω)

≤ M,

(5.4.32)

for i=1, 2, there exists a renormalized solution to problem (5.4.2) and (5.4.8) holds. Proof. Let {Ωn } be an increasing sequence of subdomains of Ω such that Ωn ⊂ Ω and ∪n Ωn = Ω. We deﬁne μi,n = Tn (χΩn fi ) + χΩn νi . Then {μ1,n } and {μ2,n } are nondecreasing sequence of nonnegative measures with compact support which converge to μ1 = f1 + ν1 and μ2 = f2 + ν2 respectively. Clearly for any > 0 there exists c > 0 such that

2 diam (Ω)

W1,p

λ [μi,n ]

λ λ 2 diam (Ω) ≤ c n p−1 + (1 + ) W1,p [νi ] ,

(5.4.33)

which implies λ λ 2 diam (Ω) 2 diam (Ω) ≤ c,n,c exp τ (1 + ) cW1,p . [μi,n ] [νi ] exp τ cW1,p (5.4.34) If there holds

(p−1)(λ−1)

λ

M

p,2 diam (Ω) [νi ]

L∞ (Ω)

≤

p ln 2 τ (12cλ)λ

p−1 λ ,

(5.4.35)

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we can choose small enough such that τ (1 + )cλ ≤

p ln 2 λ

p−1

(p−1)(λ−1)

λ λ

(12λ) Mp,2 diam (Ω) [νi ]

. (5.4.36)

L∞ (Ω)

we derive By Theorem 5.3.3 with η = (p−1)(λ−1) λ λ 2 diam (Ω) ∈ L1 (Ω). [νi ] exp τ (1 + ) cW1,p This implies

λ 2 diam (Ω) exp τ cW1,p ∈ L1 (Ω), [μi,n ]

and we conclude by Theorem 5.4.2. Remark. If λ = 1, the condition on νi is

νi (Bt (x))

Mp,2 diam (Ω) [νi ] ∞ ≤ M ⇐⇒ sup ess sup ≤ M. L (Ω) tN −p x∈Ω t>0 5.5

(5.4.37)

Large solutions revisited

Throughout this section Ω ⊂ RN is a bounded domain and A : Ω × RN → RN satisﬁes the assumptions (4.1.1) with 1 < p ≤ N . We assume that g : Ω×R → R is a Caratheodory function on which precise growth estimates will be made later on. A solution u of −div A(x, ∇u) + g(x, u) = 0

in Ω

(5.5.1)

is called a large solution u if it satisﬁes lim u(x) = ∞

(5.5.2)

ρ(x)→0

where ρ(x) = dist (x, Ωc ). The model problems for large solutions are −Δp u + |u|q−1 u = 0 lim u(x) = ∞,

in Ω

−Δp u + eu − 1 = 0 lim u(x) = ∞.

in Ω

(5.5.3)

ρ(x)→0

with q > p − 1 and

ρ(x)→0

(5.5.4)

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5.5.1

The maximal solution

In any domain there exists a maximal solution to (5.5.3) and (5.5.4). When Ω is smooth this maximal solution is a large solution. When Ω is no longer smooth, the maximal solution uΩ of (5.5.3) may not blow-up at the boundary. However, if 1 < q < q1 := NN(p−1) −p , it is easy to prove that this maximal solution satisﬁes uΩ (x) ≥ ωc (ρ(x))

p − q+1−p

,

(5.5.5)

where ωc = ωc (N, p, q) is some explicit positive constant which vanishes when q = q1 . As for (5.5.4) a large solution always exists when p = N . The existence of a large solutions is based upon the construction of local radial super solutions and the proof that this maximal solution is a large solution is based upon the existence of explicit radial singular subsolutions. Concerning equation (5.5.2), this approach is no longer possible. The maximal solution is constructed in proving a local universal energy estimate and the fact that this maximal solution is a large solution by potential estimates. Proposition 5.5.1 satisfy (4.1.1) and

Assume N ≥ 2, 1 < p ≤ N , A and the function g

rg(x, r) ≥ θ|r|β+1 − τ

a.e. in Ω, ∀r ∈ R,

(5.5.6)

respectively for some β > p − 1, θ > 0 and τ ≥ 0. Then for any compact K ⊂ Ω there exists cK = c(N, p, β, θ, τ, λ, K) > 0, such that any locally upper bounded solution of (5.5.2) satisﬁes sup ess u(x) ≤ cK . x∈K

(5.5.7)

Proof. Let B := Br (a) ⊂ B r (a) ⊂ Br (a) := B ⊂ Ω and ζ ∈ C01 (B ), with value 1 on B, such that 0 ≤ ζ ≤ 1 and |∇ζ| ≤ 2(r − r)−1 χB \B . We deﬁne m by 1 1 1 β+1−p 1 + = 1 ⇐⇒ = . + p β+1 m m p(β + 1) Taking u+ ζ m as test function in (5.5.1), we have

p

|∇u+ | ζ m − λm |∇u+ |

p−1

m |∇ζ| u+ ζ m−1 + θuβ+1 − τ u+ ζ m dx ≤ 0, + ζ

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and we obtain by H¨older’s inequality, |∇u+ |p−1 |∇ζ| u+ ζ m−1 dx ≤

1 1 β+1 p β+1 m |∇u+ | ζ dx u+ ζ dx

p

B

B

1r |∇ζ| dx . r

m

B

We have also θ m m m uβ+1 θuβ+1 dx ≥ ζ − τ u ζ + + + ζ dx − c2 |B |. 2 Therefore p m β+1 m |∇u+ | ζ + θu+ ζ dx ≤ 2c2 |B | + c3

≤ c4 r

B

N

+

m

|∇ζ| dx

r N −1 (r −r)m−1

(5.5.8)

,

where c2 = c2 (q, θ) > 0, c3 = c3 (p, β, λ) > 0, c4 = (N, p, β, λ) > 0. This is a local energy estimate independent of the solution. It follows by the method of proof of Lemma 4.3.3 that u is bounded from above on any compact subset of Ω (in this lemma a local bound from below for super solution is proved). Corollary 5.5.2 Assume A satisﬁes (4.1.1) and g : Ω × R → R is a Caratheodory function which satisﬁes (5.5.6), is locally bounded in Ω × R and (i)

r → g(x, r)is nondecreasing on R+ a.e. in Ω

(ii)

g(x, 0) = 0

(5.5.9)

a.e. in Ω.

Then equation (5.5.9) admits a nonnegative maximal solution in Ω. Proof. We need not suppose that Ω is bounded. The construction is standard. We consider a sequence of smooth bounded domains {Ωn } such that Ωn ⊂ Ωn ⊂ Ωn+1 and ∪n Ωn = Ω. For each k > 0 there exists a solution u = un,k of −div A(x, ∇u) + g(x, u) = 0 u=k

in Ωn

(5.5.10)

on ∂Ωn .

This fact follows easily from Theorem 1.4.4. Furthermore un,k is unique and 0 ≤ un,k ≤ k. By Proposition 5.5.1, for any compact set K ⊂ Ωn0 , un,k is bounded from above in K by a constant independent of n ≥ n0 + 1.

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Furthermore k → un,k is nondecreasing. When k → ∞, un,k converges to a solution uΩn of −div A(x, ∇u) + g(x, u) = 0

in Ωn

u=∞

(5.5.11)

on ∂Ωn .

Since un+1,k ≤ un,k , the sequence {uΩn } is nonincreasing. Its limit uΩ is a locally bounded solution of (5.5.1) in Ω. Since g is locally bounded in Ω × R, any solution u of the same equation in Ω is H¨ older continuous and thus also locally bounded. As a consequence u ≤ un,k for n ﬁxed and k large enough, and ﬁnally u ≤ uΩ . 5.5.2

Potential estimates

The following space of functions is a little more general than the Marcinkiewicz space Ls,∞ Deﬁnition 5.2 Let Ω be a bounded domain and 1 ≤ s ≤ ∞. A function f ∈ L1loc (Ω) belongs to the Morrey class Ms (Ω) if N |f | dy ≤ Cr s ∀x ∈ Ω, ∀r > 0. (5.5.12) Ω∩Br (x)

The smallest constant such that this inequality holds is denoted by f Ms (Ω) and it is a norm. Note that Ls (Ω) ⊂ Ls,∞ (Ω) ⊂ Ms (Ω) ⊂ L1 (Ω). The following variant of (5.2.43) in Theorem 4.2.5 (see [Mal` y and Ziemer (1997), Th 4.35]) can be veriﬁed by a modiﬁcation of the proof of Theorem 4.3.8. Proposition 5.5.3 Let Ω be a bounded domain, 1 < p ≤ N and 0 < γ < N p. If μ ∈ Mb (Ω) and f ∈ M p−γ (Ω), any locally renormalized solution of −div A(x, ∇u) + f = μ

(5.5.13)

in Ω,

satisﬁes u(x) + f

N M p−γ

ρ(x)

≥ inf ess u + c11 W1,p4 [μ] (x) Ω

a.e. in Ω.

(5.5.14)

As in the case p = 2 potential estimates lead to suﬃcient conditions for existence of large solutions. We recall that when 1 < p ≤ N , the

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R HN −p -Hausdorﬀ capacity of a compact set K is deﬁned by

R HN −p (K) = inf

⎧ ⎨ ⎩

rjN −p : K ⊂

j

Brj (aj ), rj ≤ R

j

⎫ ⎬ ⎭

,

(5.5.15)

(see (1.2.14) for the general deﬁnition). Then this set function is extended to general sets using the usual rules developed in Section 1-2-4. In the same way as for the Bessel-Lorentz capacity (see Proposition 1.2.6), it is possible to give a dual deﬁnition of the HN −p -capacity linking it to the truncated fractional maximal operator deﬁned in (5.3.2). This result due to [Frostman (1935)] is also proved in [Turesson (2000), Th 3.4.27]. Proposition 5.5.4 Let 1 < p < N and R > 0. There exists c > 0 such that for Borel set E ⊂ RN there holds

R R R

cHN −p (E) = sup μ(E) : μ ∈ M+ (E), Mp [μ] L∞ ≤ 1 ≤ HN −p (E), (5.5.16) N c where M+ (E) = {μ ∈ M+ (R ) : μ(E ) = 0}. The main estimate is the following. Theorem 5.5.5 Let 1 < p < N . Assume A and g are Caratheodory functions which verify (4.1.1) and (5.5.6) and (5.5.9) respectively with B1 = Ω. Let K ⊂ B 41 \ {0} be a compact subset and U the maximal solution of (5.5.1) in B1 \ K. (i) If g satisﬁes g(x, r) ≤ er − 1

a.e. in B1 , ∀r ≥ 0,

(5.5.17)

there exist positive constants c1 , c2 , depending on N , p and q, such that U (0) ≥ −c1 + c2

1

0

1 HN −p (K ∩ Br ) rN −p

1 p−1

dr . r

(5.5.18)

a.e. in B1 , ∀r ∈ R+ ,

(5.5.19)

(ii) If q ∗ > q1 , g satisﬁes g(x, r) ≤ arq

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for some p − 1 < q

0. Step 1. We claim that there exist = (N, p) > 0 small enough and c2 > 0, independent of J, such that

A := B1

R exp W1,p

0 J

1

μk (x) dx ≤ c2 .

(5.5.22)

k=3

We decompose this sum as follows

A=

j=0

0

∞

exp

Sj

R W1,p

J

1

μk (x) dx.

k=3

For any j ≥ 0 0 R W1,p

J

k=3

1

⎡

R ⎣ μk ≤ c∗ W1,p

J

k≥j+2

⎤

⎡

⎤

j−2

R ⎣ μk ⎦ + c∗ W1,p

μk ⎦

k=3

⎡

R ⎣ + c∗ W1,p

j+1

k=max{j−1,3}

⎤ μk ⎦ ,

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⎛ where c∗ = max{1, 5

2−p p−1

}. Since exp ⎝

5

⎞ aj ⎠ ≤

j=1

j=0

0

∞ Sj

R exp W1,p

J

1 ≤

μk

∞ Sj

j=0

k=3

+

+

∞ Sj

j=0

exp (5aj ), we deduce

j=1

⎛

⎡

R ⎣ exp ⎝c5 W1,p

⎤⎞

J

μk ⎦⎠

k≥j+2

⎛

∞ j=0

5

349

Sj

⎡

⎤⎞

R ⎣ exp ⎝c5 W1,p

μk ⎦ ⎠

j−2 k=3

⎛

⎡

j+1

R ⎣ exp ⎝c5 W1,p

⎤⎞ μk ⎦⎠

k=max{j−1,3}

≤ A1 + A2 + A3 . Estimate of A3 . By applying Theorem 5.3.3 with η = 0, μ = μk and B = Bk−1 , we get R N exp c3 W1,p [μk ] dx ≤ c4 rk−1 , Bk−2

1 , which is the constant δ0 in Theorem 5.3.3. For k = where c3 ≤ 12 j − 1, j, j + 1 this inequality is valid provided the domain of integration Bk−2 be replaced by Sj , thus

A3 ≤ c

∞

rjN = c5 < ∞.

(5.5.23)

j=0

Estimate of A1 . If t ∈ (0, rj+1 ), x ∈ Sj there holds

μk (Sj ) = 0, thus

k≥j+2

A1 =

∞ j=1

Sj

⎛ ⎜ exp ⎜ ⎝ c3 ⎛

≤

∞

⎛ 1

μk (Bt (x))

⎜ k≥j+2 ⎝ tN −p rj+1

1 ⎞ p−1

⎛

p−1 ⎝ ⎜ exp ⎝c3 μk (Sk )⎠ N −p j=1 k≥j+2

1 ⎞ p−1

⎟ ⎠

⎞ dt ⎟ ⎟ dx t⎠ ⎞

p−1 −N −p ⎟ rj+1 ⎠ |Sj | .

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N −p Since μk (Sk ) ≤ μk (Brk−1 ) ≤ rk−1 , we are led to

⎛ ⎝

∞

1 ⎞ p−1

⎛ p−1 −N −p rj+1

μk (Sk )⎠

≤⎝

k=j+2

1 ⎞ p−1

∞

k=j+2

≤

∞

−

N −p ⎠ rk−1

p−1

N −p rj+1

1 p−1

N −p rk−1

k=0

≤ 1 − 2p−N

1 1−p

.

Therefore

1 p−1 1 − 2p−N 1−p |B1 | := c6 . A1 ≤ exp c3 N −p

(5.5.24)

Estimate of A2 . For x ∈ Sj ,

R W1,p

0 j−2

1

μk (x) =

k=3

⎛ 1

μk (Bt (x)

⎜ k≤j−2 ⎝ tN −p rj−1

=

j−1

⎛ ri−1

ri

i=1

1 ⎞ p−1

⎟ ⎠

dt t

μk (Bt (x))

⎜ k≤j−2 ⎝ tN −p

1 ⎞ p−1

⎟ ⎠

For all i = 1, ..., j − 1 and ri < t < ri−1 , there holds

dt . t

μk (Bt (x) = 0,

k≤i−2

hence R W1,p

0 j−2

1 μk (x) =

j−1 ri−1 i=1

k=3

≤

j−1 i=1

≤

ri

≤ c7 j

ri

μk (Bt (x)

⎜ i−1≤k≤j−2 ⎝ tN −p ⎛

ri−1

j−2 j−1 i=1

⎛

μk (Sk

⎜ i−1≤k≤j−2 ⎝ tN −p 1 p−1

N −p rk−1

i−1

with c7 =

1 ⎞ p−1

⎟ ⎠

1 ⎞ p−1

⎟ ⎠

dt t

N −p

rip−1

4N −p 1 − 2p−N

dt t

1 p−1

.

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Therefore A2 ≤

∞

ec3 c7 j dx = |S1 |

j=1 Sj ∞

≤ |S1 |

∞

rjN ec3 c7 j

j=1

rjN ec3 c7 j−jN ln 2

(5.5.25)

= c8 < ∞,

j=1 ln 2 provided < N 2c3 c7 . Combining (5.5.23), (5.5.24) and (5.5.25) we obtain that A is bounded by some constant c2 > 0 depending on N and p provided ≤ 0 (N, p).

Step 2. Proof of (5.5.18). Using (5.5.12) with the Morrey space M derive from (5.5.22),

0 J 1

p

R W1,p μk

exp

2N 2N k=3 M p (B1 ) ≤ c9 B1

0

R exp W1,p

J

1

p 2N

2N p

we

(5.5.26)

μk (x) dx

k=3

≤ c10 . Set B = B 14 . By Theorem 5.4.2 and (5.4.8), any renormalized solution of −div A(x, ∇u) + g(x, u) = μ u=0

in B on ∂B,

(5.5.27)

where g satisﬁes rg(., r) ≥ 0 and μ is a positive bounded measure satisﬁes 1

2 [μ] (x) u(x) ≤ c0 W1,p

a.e. in B,

(5.5.28)

and c0 depends only on N and p. We ﬁx ≤ 0 (N, p) as above and put 1 p−1 . 1 = 2Npc0 It follows from Theorem 5.4.8 and estimate (5.4.37) in the remark here after that there exists a renormalized solution to −div A(x, ∇u) + g(x, u) = 2

J

μj

in B

j=3

u=0

on ∂B,

(5.5.29)

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for 2 = 2 (N, p, c0 ) > 0 independent of J since the μj have disjoint supports. By Proposition 5.5.3 and (5.5.26) 0 J 1 1 2 (5.5.30) u(0) ≥ −c11 + c12 W1,p μk (0). k=3

Therefore

u(0) ≥ −c11 + c12

∞

J−2 ri

J−2 ri ri+1

i=2

≥ −c11 + c13 ≥ −c11 + c13

J−2

i=2 ∞

μk (Bt (0))

⎜ 3≤k≤J ⎜ ⎝ tN −p

ri+1

i=2

≥ −c11 + c12

ri

ri+1

i=2

≥ −c11 + c12

⎛

μi+2 (Bt (0)) tN −p μi+2 (Si+2 ) tN −p

1 ⎞ p−1

⎟ ⎟ ⎠

1 p−1

dt t

1 p−1

(5.5.31) ,

1 p−1 − N −p 1 HN ri p−1 , −p (K ∩ Si+2 )

1 p−1 − N −p 1 HN ri p−1 . −p (K ∩ Si )

i=4

Since Si = Bi−1 \ Bi , there holds

1 HN −p (K

1 1 1 1 1 p−1 HN −p (K ∩ Bi−1 ) p−1 p−1 − HN ∩ Si ) ≥ , −p (K ∩ Bi ) 2−p max 1, 2 p−1

from which inequality we infer, ⎡ ⎤ 1 p−1 ∞ 1 −p 1 (K ∩ B ) HN −N i−1 −p p−1 1 p−1⎦ − H (K ∩ B ) u(0) ≥ −c11 + c13 ⎣ r i N −p 2−p i max 1, 2 p−1 i=4 ⎞ ⎛ N −p ∞ −p p−1 1 1 p−1 −N 2 p−1 ⎠ − 1 H ≥ −c11 + c13 ⎝ (K ∩ B ) r i N −p 2−p i max 1, 2 p−1 i=4 1 1 1 HN −p (K ∩ Bt ) p−1 dt . ≥ −c14 + c15 tN −p t 0

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Since the maximal solution U1 = U of (5.5.1) in K c satisﬁes the same equation in B \ K and dominates u on ∂B, it is larger than u. This implies (5.5.18). Step 3. Proof of 5.5.20. By Proposition 1.2.6 there exists μj ∈ M+ (K ∩ Sj ) such that q∗ N ∗ μj (K ∩ Sj ) = (Gp [μj ]) p−1 dx = cR (K ∩ Sj ). p, q q∗ +1−p

RN

For θ > 0 and k ∈ N we deﬁne θk,s by ⎧ ⎨1 θk,s = ⎩ 1+θ s−1 (θ + 1)k(s+1) θ

if 0 < s ≤ 1 if s > 1.

If {ak } is a sequence of nonnegative real numbers, we have by Jensen inequality, ∞ s ∞ ak ≤ θk,s ask . k=0

k=0

Hence, using (5.3.23) and Proposition 1.2.6, we derive B1

0 1 W1,p

J

1

q∗

μk (x)

dx ≤ B1

k=3

≤

≤ c17

q∗ θk, 1 θk,q ∗ p−1

k=3

≤ c16

q∗ 1 1 W θk, p−1 1,p [μk ] (x)

k=3

J

≤ c16

J

J k=3 J k=3 J k=3

∗

q θk,

1 p−1

∗

q θk,

B1

q ∗ 1 W1,p [μk ] (x) dx

θk,q∗

q∗

(Gp [μk ] (x)) p−1 dx

RN

θk,q∗ cR p,

N

1 p−1

dx

q∗ q∗ +1−p

(K ∩ Sj )

∗ −k N − q∗pq q∗ +1−p θk, 1 θk,q∗ 2 p−1

≤ c18 . Therefore

0 J 1q

1 μk

W1,p

k=3

q∗ M q

0 J 1 q

1 ≤ c19 W1,p μk

k=3

Lq ∗

≤ c20 , (5.5.32)

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where c20 is independent of J. Since the measure Jk=3 μk is absolutely conN tinuous with respect to the cR -capacity, it follows from Theorem 5.4.6 q p, q+1−p that there exists a renormalized solution to −div A(x, ∇u) + g(x, u) =

J

μk

in B

(5.5.33)

k=3

u=0

on ∂B.

Thus, using lower estimate Proposition 5.5.3 with 0 ≤ f = g(x, u) ≤ auq , and (5.5.33), we obtain as in Step 2, 0 J 1 1 4 μk (0) u(0) ≥ −c21 + c22 W1,p ≥ −c23 + c24 0

⎛ 1

⎜ ⎝

k=3

cR

1 ⎞ p−1

N

q∗ p, q∗ +1−p

rN −p

⎟ (K ∩ Br )⎠

(5.5.34) dr . r

As in Step 2, the maximal solution U2 = U of (5.5.1) in B1 \ K dominates u. This implies (5.5.20). 5.5.3

Applications to large solutions

In order to obtain estimates from below of the maximal solution of (5.5.1) near the boundary, we have to compare it with the restriction to Ω of solutions deﬁned in a larger domain than Ω. Therefore, in this section, we assume that A and g are Caratheodory functions respectively deﬁned in Ω × RN and Ω × R where Ω is a domain which contains Ω and that the assumptions (4.1.1) on A, and the diﬀerent assumptions on g are veriﬁed for x ∈ Ω . For the sake of simplicity we assume that {x ∈ RN : dist (x, Ω) ≤ 1} ⊂ Ω . Theorem 5.5.6 Assume N ≥ 2, 1 < p < N and the function g satisﬁes (5.5.6) (i), (5.5.9) and (5.5.15). If 1 1 1 HN −p (Ωc ∩ Br (x)) p−1 dr (5.5.35) = ∞ ∀x ∈ ∂Ω, rN −p r 0 then uΩ is a large solution. 1 Proof. Let x0 ∈ ∂Ω, and for simplicity we assume x0 = 0. Let 0 < δ < 12 and z0 ∈ Ω ∩ B δ . We put K = Ωc ∩ B 14 (z0 ) and denote by U1 the maximal

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solution of (5.5.1) in B1 \ K. Then uΩ ≥ U1 in Ω. Furthermore U1 (z0 ) ≥ −c1 + c2

1

δ

1 c HN −p (Ω ∩ Br (z0 )) rN −p

1 p−1

dr r

1 p−1 1 c HN dr −p (Ω ∩ Br−|z0 | ) ≥ −c1 + c2 N −p r r δ 1 1 p−1 1 c r HN dr −p (Ω ∩ B 2 ) ≥ −c1 + c2 N −p r r 2δ 1 12 p−1 1 c HN dr −p (Ω ∩ Br ) . ≥ −c1 + c3 N −p r r δ

1

The proof follows from the next estimate 1 2

inf uΩ ≥ inf U 1 ≥ −c1 + c3

Bδ ∩Ω

Bδ ∩Ω

δ

as Br−|z0 | ⊂ Br (z0 )

1 c HN −p (Ω ∩ Br ) rN −p

1 p−1

and the assumption (5.5.35), by letting δ → 0.

dr , r

Remark. In the case p = N the maximal solution is a large solution. This follows from the fact that for any z0 ∈ ∂Ω and any k > 0, there holds uΩ ≥ ukδz0 , where ukδz0 is a renormalized solution of −div A(x, ∇u) + eu − 1 = kδz0 u=0

in B on ∂B,

(5.5.36)

and k < M deﬁned in (5.4.37). Using Serrin’s characterization of isolated singularities Theorem 1.3.7 (ii) we derive that ukδz0 (x) ≥ ln(|x|−1 ) −

∀x ∈ B 12 .

In the power case, the proof is similar and left to the reader. Theorem 5.5.7 Assume N ≥ 2, 1 < p < N , q ∗ > q1 , g satisﬁes (5.5.6) ∗ (i), (5.5.9) and (5.5.19) for some p − 1 < q < pq N . If 0

⎛ 1

⎜ ⎝

cR p,

N q∗ q∗ +1−p

(K ∩ Br (x))

rN −p

then uΩ is a large solution.

1 ⎞ p−1

⎟ ⎠

dr =∞ r

∀x ∈ ∂Ω,

(5.5.37)

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5.5.4

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The case p = 2

When p = 2 and q > 1 the previous result is not optimal since there exists a necessary and suﬃcient condition [Labutin (2003)] for the maximal solution of −Δu + uq = 0

(5.5.38)

in Ω to be a large solution. A local version of Labutin’s condition [Marcus, V´eron (2009)] provides a necessary and suﬃcient condition in order a positive solution of (5.5.38) tend to inﬁnity at a particular boundary point. Theorem 5.5.8 Assume N ≥ 2, q > 1, Ω ⊂ RN is a bounded domain and z ∈ ∂Ω. There exists a positive solution u of (5.5.38) which tends to inﬁnity at z if and only if 0

1

cR 2,q (K ∩ Br (z)) N

dr = ∞. rN −1

(5.5.39)

Furthermore, if the above condition holds for any boundary point z, the maximal solution is the unique large solution of (5.5.38) in Ω. The proof of a universal capacitary estimate from above of any solution of (5.5.38) in Ω necessitates some notations and intermediate results. Lemma 5.5.9 Assume N ≥ 3, q ≥ q1 := q1,2 = NN−2 , F ⊂ RN is compact and u is nonnegative nontrivial solution of (5.5.38) in RN \ F . Then ∂u dS := ˜u ≤ 0. (5.5.40) lim r→∞ ∂B ∂n r Proof. We assume and F ⊂ BR for some R > 0. Since u is nonnegative and nontrivial, it is positive in F c by the strong maximum principle. By the Keller-Osserman estimate (3.1.7), 0 ≤ u(x) ≤ cN,q (|x| − R)− q−1 2

c ∀x ∈ BR .

(5.5.41)

Writing (5.5.38) under the form −Δu + V u = 0 where V = uq−1 we deduce from Harnack inequality (see Chapter 1) that sup u(x) ≤ cN,q inf u(x)

|x|=r

|x|=r

∀r ≥ 2R.

(5.5.42)

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Let u ¯(r) be the spherical average of u(r, σ) (in spherical coordinates). Hence, by convexity, u ¯ +

N −1 u ¯ −u ¯q ≥ 0 r

in [R, ∞).

(5.5.43)

y − aN,q s−q− N −2 y q ≥ 0 in [S, ∞).

(5.5.44)

¯(r) with s = Set y(s) = rN −2 u

r N −2 N −2 ,

then

N −4

If y is unbounded, then y(s) ≥ cs for s ≥ s0 by convexity, therefore N −4

y ≥ aN,q cq s− N −2 =⇒ y(s) ≥ c s N −2 N

for s ≥ s1 .

Iterating this process, implies that there exist sequences {αn } ⊂ R+ tending to ∞ and {cn } ⊂ R+ such that y(s) ≥ cn sαn for s large enough which contradicts (5.5.41). Consequently y is bounded, which implies by Harnack inequality (5.5.42), 0 ≤ u(x) ≤ c|x|2−N

c ∀x ∈ BR ,

(5.5.45)

where c may depends on u and R. In the case q = q1 this estimate can be improved and (5.5.44) becomes y − aN s−2 y q1 ≥ 0 in [S, ∞).

(5.5.46)

This implies that y(s) → 0 as s → ∞ and y is decreasing. We set z(t) = y(s) with s = et , then z(t) → 0 as t → ∞ and z − z − aN z q1 ≥ 0 in [T, ∞).

(5.5.47)

By the maximum principle, z is dominated on [T, ∞) by any solution of equation Z − Z − Z q1 = 0 which tends to 0 at inﬁnity and dominates z(t) for t = T . Such a solution Z is easily constructed by approximation on (T, n) and letting n → ∞. It is decreasing and satisﬁes ∞ e−τ Z q1 (τ )dτ = 0 in [T, ∞). (5.5.48) Z (t) − aN et t

Therefore Z (t) − aN Z (t) ≤ 0. By integration q 1−1 1 2−N aN =⇒ u ¯(r) ≤ cr2−N (ln r) 2 Z(t) ≤ t(q − 1) q

in [R, ∞).

This implies ﬁnally 0 ≤ u(x) ≤ c|x|2−N (ln(|x|))

2−N 2

c ∀x ∈ BR ,

(5.5.49)

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with a constant c depending on N, p, u and R. For r > R there holds ∂u ∂u q dS = − dS. u dx − ∂n ∂n Br \BR ∂Br ∂BR c By estimates (5.5.45) if q > q1 and (5.5.49) if q = q1 , uq ∈ L1 (BR ), therefore there exists ∂u ˜ dS. (5.5.50) u = lim r→∞ ∂B ∂n r

We write u(r, σ) = r2−N v(t, σ) with r = et , t > t0 := ln R. Then v is bounded and satisﬁes vtt + (2 − N )vt + Δ v − e(N −2)(q−q1 )t v q = 0 in (t0 , ∞) × S N −1. (5.5.51) By linear elliptic equations local regularity theory [Gilbarg, Trudinger], vt , vtt , ∇ v, ∇ vt and D2 v are uniformly bounded in some H¨older norm on (t0 + 1, ∞) × S N −1 . If φ denotes the spherical average of φ, then v tt + (2 − N )v t − e(N −2)(q1 −q)t v q = 0, which in turn implies v t (t) = e(N −2)t

∞

e(N −2)(q1 −q−1)τ v q (τ )dτ.

(5.5.52)

t

Case 1: q > q1 . From the previous estimate, |v t (t)| ≤ ce(N −2)(q1 −q)t for some c > 0, consequently v(t) admits a ﬁnite nonnegative limit γ when t → ∞. From (5.5.51) there holds d vt2 − |∇ v|2 dσ = 2 (N − 2)vt2 + e(N −2)(q−q1 )t v q vt dσ. dt S N −1 S N −1 (5.5.53) 2 N −1 ). Diﬀerentiating the equation with This implies that vt ∈ L ((t0 , ∞)×S respect to t, multiplying by vtt and integrating on S N −1 implies that vtt ∈ L2 ((t0 , ∞) × S N −1 ). Moreover the bounds of vt and vtt implies that these functions are uniformly continuous in (t0 , ∞) × S N −1 ), therefore vt (t, .) and vtt (t, .) converge to 0 in L2 (S N −1 ) as t → ∞. The uniform bounds on v and its covariant derivatives implies that the set of functions T [v] := {v(t, .) : t ≥ t0 + 1} is relatively compact in the C 2 (S N −1 )-topology. Thus the ω-limit set Ω[T [v]] (see Theorem 3.1.16 in Chapter 3) is a nonempty connected compact subset of the set of solutions of Δ ω = 0. Therefore this limit set is composed of constant functions. Because v(t) → γ we infer that

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v(t, .) → γ in the C 1 (S N −1 ) topology. Applying l’Hospital rule to (5.5.52) yields v t (t) =

e(N −2)(q1 −q)t q γ (1 + ◦(1)) as t → ∞. N −2

Since rN −1 ur (r, σ) = (2 − N )v(t, σ) + vt (t, σ) we have ∂u dS = rN −1 ur (r) = (2 − N )v(t) + v t (t) → (2 − N )γ ≤ 0. (5.5.54) ∂Br ∂n Case 2: q = q1 . Equation (5.5.51) becomes vtt + (2 − N )vt + Δ v − v q1 = 0

in (t0 , ∞) × S N −1 .

Integrating on S N −1 yields v tt + (2 − N )v t − v q1 = 0, thus ∞ v t (t) = e(N −2)t e(2−N )τ v q1 (τ )dτ.

(5.5.55)

(5.5.56)

t

Since vt (t, .) ≤ ct

2−N 2

by (5.5.49), we derive from (5.5.55)

v t (t) ≤

N c t− 2 N −2

for t ≥ t1 > t0 .

The relation rN −1 ur (r, σ) = (2−N )v(t, σ)+vt(t, σ) still holds, thus (5.5.54) is also valid which implies ∂u dS → 0 as r → ∞. (5.5.57) ∂n ∂Br In order to prove the estimate from above in Theorem 5.5.8, we introduce some notations. Deﬁnition 5.3 Let 1 < q < ∞ such that q > q1 . If K is a compact subset of RN the Riesz capacity c˙2,q (K) is deﬁned by

q c˙2,q (K) = inf D2 ψ q N : ψ ∈ C0∞ (RN ), 0 ≤ ψ ≤ 1, ψ = 1 on K . L (R )

(5.5.58) The function c˙2,q is actually a Riesz capacity (see Section 1.2.4 and [Adams, Hedberg (1999)] for details). It is extended to general sets by the usual rules of capacities. The main reason for introducing this capacity

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are the Gagliardo-Nirenberg interpolation inequality (see below) and the homogeneity property (which is not shared by the Bessel capacity),

c˙2,q (aK) = aN −2q c˙2,q (K)

∀a > 0,

(5.5.59)

and there holds c˙2,q (K) ≤ cR 2,q (K). N

(5.5.60)

Furthermore [Adams, Hedberg (1999), Prop 5.1.4, 5.6.1], for all m > 0 there exists Am = A(m, N, q) > 0 such that N cR s.t. diam(E) ≤ m, 2,q (K) ≤ Am c˙ 2,q (K) ∀E ⊂ R N

and there exists M = M (N, q) > 0 such that N N N −2q cR 2,q (K) ≤ M c˙ 2,q (K) + (c˙ 2,q (K))

∀E ⊂ RN .

(5.5.61)

(5.5.62)

Notice also this link between the Bessel capacities relative to RN and relative to a domain G ⊂ RN . Let Θ ⊂ Θ ⊂ G be a bounded domain, then there exists B = A(G, Θ, N, q) such that G R cR 2,q (E) ≤ c2,q (E) ≤ Bc2,q (E) N

Deﬁnition 5.4 denote

N

∀E ⊂ Θ, E Borel.

(5.5.63)

If F ⊂ RN is a closed set, x ∈ RN , δ > 0 and m ∈ Z, we

Tm,δ (x) = y ∈ RN : 2m δ ≤ |x − y| ≤ 2m+1 δ

!

∗ Fm,δ (x) = F ∩ B 2m+1 δ (x) Fm,δ (x) = F ∩ Tm,δ (x) ∞ 2(2−N )m c˙2,q (Fm,δ (x)) WF (x) = δ 2−N

(5.5.64)

m=−∞ ∞

∗ WF∗ (x) = δ 2−N 2(2−N )m c˙2,q (Fm,δ (x)) ∞ m=−∞ dr WF (x) = c˙2,q (F ∩ Br (x)) N −1 . r 0

The function WF is called the c˙2,q -Riesz capacitary potential of F . The following inequalities are easy to verify: WF (x) ≤ WF∗ (x) ≤ (1 − 22−N )−1 WF (x)

∀x ∈ RN ,

(5.5.65)

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and since ∞

WF (x) =

m=−∞

2m+1 δ

2m δ

c˙2,q (F ∩ Br (x))

dr , rN −1

there holds 1 − 2−N ∗ 2N − 1 ∗ WF (x) ≤ WF (x) ≤ WF (x) N N

∀x ∈ RN .

(5.5.66)

Lemma 5.5.10 Assume F ⊂ RN is compact and u satisﬁes (5.5.38) in RN \ F . Let η ∈ C0∞ (RN ) with value 1 in a neighborhood of F and 0 ≤ η ≤ 1. Then

q uq (1 − η)2q dx ≤ c1 D2 η Lq , (5.5.67) RN

for some c1 = c1 (N, q). 2q

Proof. Set ζ = (1 − η) . Assuming that supp(η) ⊂ B2R , we have for r > 2R ∂u q dS + u ζdx = uΔ(1 − η)2q dx ∂n Br ∂Br Br ∂u u −2q (1 − η)2q −1 Δη = dS + ∂Br ∂n RN +2q (2q − 1)(1 − η)2q −2 |∇η|2 dx

∂u ≤ dS + c2 ∂Br ∂n

1q u ζdx q

Br

q1 q 2q |Δη| + |∇η| dx ,

Br

where c2 = 2q (2q − 1). By the Gagliardo-Nirenberg interpolation inequality (1.2.34) (using the fact that η L∞ = 1), RN

1 q

|∇η|2q dx ≤ c3 D2 η Lq ,

where c3 depends on N and q. Using (5.5.40) we obtain, letting r → ∞,

u ζdx ≤ c4 q

RN

which implies (5.5.67).

q1

2

D η q , u ζdx L q

RN

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Corollary 5.5.11 Assume F ⊂ BR is compact and u satisﬁes (5.5.38) in RN \ F , then RN 2R uq dx ≤ c1 cB (5.5.68) 2,q (F ) ≤ c1 Bc2,q (F ), c B2R

where B is the constant appearing in (5.5.63). Proof. If we assume that supp(η) ⊂ B2R and take the inﬁmum over all η ∈ C0∞ (B2R ) we derive the ﬁrst inequality in (5.5.68). Using (5.5.63) we obtain the second one. Remark. The following technical form of the previous estimate will be used in the next lemma

R 2R uq dx − ˜u ≤ c1 cB 2,q (F ) ≤ c1 Bc2,q (F ), N

c B2R

Lemma 5.5.12

∀r ≥ 2R.

(5.5.69)

Under the assumptions of Corollary 5.5.11 there holds R 2R udS ≤ c2 RcB 2,q (F ) ≤ c2 BRc2,q (F ), N

∂B2R

(5.5.70)

and u(x) ≤ c4 |x|

2−N

2R cB 2,q (F ) ≤ c4 B |x|

2−N

cR 2,q (F ), N

(5.5.71)

where the cj > 0, j = 2, 3, 4, depend on N and q. Proof. Set

N −2 N −2 1 R − 2 |x|

1 ψR (x) = N −2

c ∀x ∈ B2R .

Then ψR is harmonic, vanishes on if |x| = 2R, satisﬁes 0 ≤ ψR < 1 in Br \ B2R for any r > 2R and ∂ψR −1 ∂B2R = N −1 ∂n 2 R

and

∂ψR RN −1 ∂Br = N −1 , ∂n r

if n stands for the outward unit vector of Br \ B2R . Multiplying (5.5.38) by ψR and integrating on Br \ B2R yields (since ψR is harmonic) q ψR u dx = ψR Δudx Br \B2R

Br \B2R

=

∂Br

∂ψR ∂u ∂ψR ψR − u dS − udS. ∂n ∂n ∂B2R ∂n

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By (5.5.40)

lim

r→∞

∂Br

and

˜u ∂u ∂ψR ψR − u dS = ≤ 0, ∂n ∂n (N − 2)2N −2

lim

r→∞

∂ψR −1 udS = N −1 ∂n 2 R ∂B2R

udS. ∂B2R

Therefore

˜u 1 ψR u dx − = N −1 udS, c (N − 2)2N −2 2 R ∂B2R B2R q

(5.5.72) c

which, combined with (5.5.69), implies (5.5.71) since ψR ≤ 1. Let P B2R c be the Poisson kernel in B2R . Its expression is easy to obtain by Kelvin transform. P B2R (x, y) = cN (2R)N −1 |x| c

|x|2 − 4R2

N

N

|4R2 x − |x|2 y|

c ∀(x, y) ∈ B2R × ∂B2R .

(5.5.73) Thus, for |x| ≥ 4R N −1

u(x) = cN (2R)

|x| (|x| − 4R ) N

2

u(y)

2

∂B2R

|4R2 x

|x| + 2R u(y)dS(y) R(|x| − 2R)N −1 ∂B2R cN 2N −1 ≤ u(y)dS(y). R|x|N −2 ∂B2R

− |x|2 y|

N

dS(y)

≤ cN

Jointly with (5.5.68) we infer (5.5.71).

Proof of Theorem 5.5.8: I- Estimate from above. If G ⊂ RN is a compact set, we denote by UG the maximal solution of (5.5.38) in Gc . If x ∈ F c , we set δ = dist (x, F ) and consider the slicing described in (5.5.64). Then Fm = ∅ if m < 1 or if m ≥ (x) for some integer (x). One can ﬁnd an integer K such that for each m ∈ N∗ there exist K points aj,m such that 2m−1 δ ≤ |x − aj,m | ≤ 2m+1 δ and Tm,δ (x) ⊂

K j=1

B 2m−3 δ (aj,m ).

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(x) K

We set Fm,j,δ (x) = B 2m−3 δ (aj,m ) ∩ F . Since

UFm,j,δ (x) is a super

m=1 j=1

solution of (5.5.38) we have that u ≤

(x) K

UFm,j,δ (x) . In particular

m=1 j=1 (x) K

u(x) ≤

UFm,j,δ (x) (x).

(5.5.74)

m=1 j=1

Using (5.5.74) we obtain

u(x) ≤ c4

(x)

(2m δ)2−N

K

m=1 (x)

≤ Kc4

2R cB 2,q (Fm,j,δ )

j=1 2R (2m δ)2−N cB 2,q (Fm,δ )

(5.5.75)

m=1

≤ c4 KWF (x). Hence, if (5.5.39) does not hold, u(x) remains bounded when x → x0 ∈ ∂Ω. Proof. II- Estimate from below. Thanks to the semilinearity, the estimate of U2 uses, in a sharper way than in Theorem 5.5.5 (ii), the construction of the solution u of −Δu + |u|

q−1

u=

J

μk

in B

k=3

u=0

(5.5.76)

on ∂B.

A solution u of (5.5.38) in Ω is called σ-moderate if there exists a bounded nonnegative measure μ in RN , such that μ(Ω) = 0 and −Δu + |u|q−1 u = μ

in RN .

(5.5.77)

We recall that the necessary and suﬃcient condition for a nonnegative meaN sure μ in order the above problem can be solved is μ 1, R < N and Borel set E ⊂ RN , R cR 1,R (K) ≤ Acp, R (K), N

N

(5.6.1)

p

where A depends on p, R, N and diam(E). In particular, if R > pqp N and cR (K) = 0, it follows from Theorem 5.2.4 that K is removable for p, R p

equation (5.2.1). 5.6.2.

2 diam (Ω) In Theorem 5.4.2 it is assumed that g˜ cW1,p [μi ] ∈ L1 (Ω),

which is a strong requirement which may not satisﬁed if μi ∈ L1 (Ω). In the case p = 2 and g˜ satisﬁes the Δ2 -condition i.e. g˜(a + b) ≤ k (˜ g (a) + g˜(b))

∀a, b ≥ 0,

(5.6.2)

for some k > 0 a much sharper assumption is to require that only the sin2 diam (Ω) gular parts of μi satisfy g˜ cW1,p [μi s ] ∈ L1 (Ω) (see [V´eron (1996), Th 4.2] for a similar problem in the case p = 2). We conjecture that this result can be extended to the case p = 2. 5.6.3. Theorem 5.4.6 provides a suﬃcient condition for the solvability of problem (5.4.2). We conjecture that this condition is also a necessary condition. 5.6.4. Theorem 5.2.7 gives only a suﬃcient condition for the removability of singularities for equation (5.2.1). We conjecture that this condition is N necessary. In the same way, we conjecture that the condition μ < cR 1,qp obtained in Theorem 5.2.8 is also a suﬃcient condition for solving (5.2.1).

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5.6.5. If Ω ⊂ RN is a smooth bounded domain, it could be interesting to study the following nonlinear boundary value problem − divA(x, ∇u) = 0 A(x, ∇u), n + g(x, u) = μ

in Ω in ∂Ω

where g ∈ C(∂Ω × R) and μ ∈ M(∂Ω). Weak solutions verify A(x, ∇u, ∇ζdx + g(., u)ζdS = ζdμ ∀ζ ∈ C 1 (Ω). Ω

∂Ω q−1

(5.6.3)

(5.6.4)

∂Ω

If g(x, u) = |u| u, q > p − 1, a clearly critical exponent is q1 = Note 4.4.4 could provide useful tools for this study.

N (p−1) N −p .

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Chapter 6

Quasilinear equations with source

In this Chapter we study properties of quasilinear equations of the type −divA(x, ∇u) = g(x, u, ∇u)

(6.0.1)

in a domain Ω and, if μ is a Radon measure in Ω, −divA(x, ∇u) = g(x, u, ∇u) + μ,

(6.0.2)

where A satisﬁes the structural assumptions (4.1.1) and g : Ω×R×RN → R is a Caratheodory function called a source type reaction, that is g(x, r, ξ)r ≥ 0 6.1

∀(x, r, ξ) ∈ Ω × R × RN .

(6.0.3)

Singularities of quasilinear Lane-Emden equations

The model equation considered here is the following Δp u + |u|q−1 u = 0,

(6.1.1)

in the range of exponents q > p − 1 > 0. When p = 2 it is called a Lane-Emden equation. 6.1.1

Separable solutions in RN

With spherical coordinates (r, σ), separable solutions of (6.1.1) under the p and ω satisﬁes form u(x) = u(r, σ) = r−β ω(σ) exist if β = βp,q = q+1−p 2 2 p−2 ω + |∇ ω|2 2 ∇ ω + |ω|q−1 ω div βp,q (6.1.2) p−2 2 2 2 2 N −1 + βp,q Λ(βp,q ) βp,q ω + |∇ ω| ω = 0 in S , 367

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where

pq . q+1−p (6.1.3) If q > q1 := NN(p−1) (1 < p < N ) then Λ(β ) < 0 and there exist radial p,q −p singular solutions under the form p βp,q Λ(βp,q ) = βp,q ((p − 1)βp,q + p − N ) = q+1−p

N−

u(r, σ) = r−βp,q ω ˜c

(6.1.4)

1 p−1 q+1−p ω ˜ c := ω ˜ c,p,q = −Λ(βp,q )βp,q .

(6.1.5)

where

If p − 1 < q ≤ q1 then Λ(βp,q ) ≥ 0; integrating equation (6.1.2) shows that there exists no nontrivial solution to (6.1.2). We will see in next section that there could exist signed solutions provided q is large enough. Existence of positive solutions can be obtained by a bifurcation method. We recall that S N −1 can be represented by the angle coordinates (θ1 , ..., θN −1 ), see (2.2.50). Proposition 6.1.1

Assume 1 < p < N − 1 and set qs :=

Np − N + 1 . N −1−p

(6.1.6)

Then there exist τ0 > 0 and a C 1 curve τ → (q(τ ), ω(τ )) from [0, τ0 ) ˜ c,p,qs ) such that ω(τ ) is a into R × C 1 (S N −1 ) with (q(0), ω(0)) = (qs , ω nonconstant solution of (6.1.2) satisfying q = q(τ ) and depending only on one spherical coordinates θj . −1)p Proof. We ﬁrst notice that qs + 1 = (N N −1−p is the exponent corresponding to the Gagliardo-Nirenberg inequality for W 1,p (S N −1 ). If we linearize the equation at the constant solution ω ˜ c we obtain

−Δ φ + (q + 1 − p)βp,q Λ(βp,q )φ = 0 in S N −1 .

(6.1.7)

The bifurcation following the ﬁrst eigenspace of −Δ with functions depending only on one variable occurs when (p − 1 − q)βp,q Λ(βp,q ) = N − 1 ⇐⇒ q = qs .

(6.1.8)

The fact that the branch of solutions (q(τ ), ω(τ )) is a C 1 curve follows from [Crandall, Rabinowitz (1971)].

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+1 [ ] [ Remark. If p = 2, qs = N N −3 ; it is proved in Gidas, Spruck (1980) , BidautV´eron, V´eron (1991)] that if 1 < q < qs the only positive solution is the constant ωc . If q = qs this equation becomes

Δ ω −

N +1 (N − 1)(N − 3) ω + ω N −3 = 0. 4

(6.1.9)

By a stereographic projection π of S N −1 \ {σS } onto RN −1 where σS is the N −3 2 2 ω ◦ π south pole, the equation becomes, with ψ(x) = ( 1+|x| 2) N +1

Δψ + ψ N −3 = 0 in RN −1 .

(6.1.10)

The solutions of (6.1.10) are the extremals of the Gaglardo-Nirenberg inequality in RN −1 with exponent p = 2 and they form a N-dimensional family of solutions since they are explicitly given by N2−3

(N − 1)(N − 3) . (6.1.11) ψ(x) := ψ,a (x) = 2 2 + |x − a| This implies that the set of solutions of (6.1.8) has the structure of a N dimensional manifold. When p = 2, the set of positive solutions of (6.1.1) is at least (N-1)-dimensional. It is a completely open problem to prove that all the positive solutions of (6.1.2) are constant when p − 1 < q < qs . Remark. Equation (6.1.2) admits a variational structure in the case q = p∗ − 1 = N (p−1)+p . In that case the solutions are critical points of the N −p functional p2 2 ∗ N −p 1 1 2 2 w + |∇ w| dσ − ∗ |w|p dσ, J(w) = p S N −1 p p S N −1 (6.1.12) which is well deﬁned in W 1,2 (S N −1 ).

6.1.2

Isolated singularities

The ﬁrst result on isolated singularities deals with the subcritical case and the following theorem is an extension of the previous ones due to [Guedda, V´eron (1988)] and [Bidaut-V´eron (1989)]. We assume that g : R × RN → R is a continuous function satisfying tg(t, ξ) ≥ 0

∀(t, ξ) ∈ R × RN ,

(6.1.13)

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and q

r

|g(t, ξ)| ≤ c1 |t| + c2 |ξ| + c3

∀(t, ξ) ∈ R × RN ,

(6.1.14)

for exponents q > p − 1, r > 0 and constants cj ≥ 0, j = 1, 2, 3, and we deﬁne the critical exponent qc =

N (p − 1) . N −1

(6.1.15)

Theorem 6.1.2 Let Ω be a domain containing 0, 1 < p ≤ N and g a continuous function satisfying the assumptions (6.1.13) and (6.1.14) with q < q1 and r < qc . If u ∈ C 1 (Ω \ {0}) is a nonnegative solution of −Δp u = g(u, ∇u),

(6.1.16)

in Ω \ {0}, then g(u, ∇u) ∈ L1loc (Ω) and there exists γ ≥ 0 such that −Δp u = g(u, ∇u) + γδ0

in Ω.

(6.1.17)

Furthermore u(x) =γ |x|→0 μp (x) lim

N −1

and lim r N −p ∇u(r, σ) = cN,p γσ,

(6.1.18)

r→0

where μp is the fundamental solution of the p-harmonic equation in RN deﬁned by (1.5.14). Proof. We assume B 1 ⊂ Ω. Since u ≥ 0 and (6.1.13) holds, the right-hand side of (6.1.16) is nonnegative, hence, by Theorem 1.5.14, the following statements hold: N N p−1 N −p ,∞ N −1 ,∞ (Ω), |∇u| ∈ Lloc (Ω) and there (i) If 1 < p < N , up−1 ∈ Lloc exist γ ≥ 0 and h ∈ L1loc (Ω) such that −Δp u = h + γδ0

in Ω.

(6.1.19)

(ii) If p = N , um ∈ L1loc (Ω) for all m > 0, |∇u|s ∈ L1loc (Ω) for all 0 < s < N . Furthermore, if u(x) → ∞ when x → 0, then the conclusion of (i) holds. In the two cases it implies in particular that g(u, ∇u) ∈ L1loc (Ω). In order to apply Theorem 1.3.7, we write (6.1.14) under the form |g(u, ∇u)| ≤ c1 |u|q+1−p |u|p−1 + c2 |∇u| N

(r+1−p)+

p−1

|∇u|

+ c3 .

+

p If p < N with q < q1 , then |u|q+1−p ∈ Lloc (Ω) for some > 0. If p = N , (p−1)(N −)

r+1−p (r+1−p)(N −1) ∈ Lloc (Ω). |u|q+1−p ∈ L1+ loc (Ω). If r + 1 − p > 0 then |∇u|

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(p−1)(N −) (r+1−p)(N −1)

Since r < rc , we can ﬁnd > 0 such that

> N + . If

(r+1−p)+

r + 1 − p ≤ 0, then |∇u| = 1. Therefore assumptions (1.3.6) (i)-(ii) are fulﬁlled. It follows that either there exists c > 0 such that c−1 μp (x) ≤ u(x) ≤ cμp (x)

∀x ∈ B 12 ,

(6.1.20)

or u ∈ L∞ loc (Ω). Step 1. If u ∈ L∞ loc (Ω) we claim that 0 is a removable singularity and (6.1.16) is satisﬁed in whole Ω. Let ζ ∈ C0∞ (RN ) with support in B1 , satisfying 0 ≤ ζ ≤ 1, ζ(x) = 0 if |x| ≤ and |∇ζ | ≤ 2−1 . Then p−2 p p |∇u| ∇u, ∇ζ ζp−1 udx + |∇u| ζp dx = g(u, ∇u)ζp udx. B1

B1

Hence

p

|∇u| ζp dx − p B1

B1

1 p p |∇u| ζp dx

B1

p1 p |∇ζ | up dx

B1

≤

g(u, ∇u)ζp udx. B1

Since

p1 N |∇ζ | u dx ≤ cN u L∞ (B1 ) p −1 , p

B1

and

p

B1

g(u, ∇u)ζp udx ≤ u L∞ (B1 )

g(u, ∇u)dx, B1

we obtain that ∇u ∈ Lploc (Ω). If φ ∈ C0∞ (Ω), then p−2 p−2 |∇u| ∇u, ∇φζ dx + p |∇u| ∇u, ∇ζ φdx = g(u, ∇u)ζ φdx. Ω

Ω

B1

By the previous estimate, the second term on the left-hand side tends to 0 when → 0, thus p−2 |∇u| ∇u, ∇φdx = g(u, ∇u)φdx. (6.1.21) Ω

B1

Step 2. Assume (6.1.20) holds for some c > 0. We deﬁne γ > 0 by γ = lim sup |x|→0

u(x) . μp (x)

(6.1.22)

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372

N −p

If 1 < p < N , we set uλ (x) = λ p−1 u(λx) for 0 < λ ≤ 1, then N −1

∇uλ (x) = λ p−1 ∇u(λx). If we deﬁne gλ by N −p

N −1

gλ (v, ξ) = λN g(λ− p−1 v, λ− p−1 ξ)

∀(v, ξ) ∈ R × RN ,

there holds N −p

N −1

|gλ (v, ξ)| ≤ c1 λN −q p−1 |v|q + c2 λN −r p−1 |ξ|r + λN c3 ≤ c1 |v|q + c2 |ξ|r + c3

(6.1.23)

p

≤ C(|v|)(1 + |ξ| ), for some nondecreasing function C, because 0 < λ ≤ 1, q ≤ q1 and r ≤ qc . Since uλ satisﬁes −Δp uλ = gλ (uλ , ∇uλ ),

(6.1.24)

in λ1 Ω \ {0}, we can apply local regularity estimates of Section I-3: for R > 0, uλ L∞ (B2R \B R ) is bounded by some constant independent of λ, 2

hence (i) (ii)

|∇uλ (x)| ≤ M |∇uλ (x) − ∇uλ (y)| ≤ M |x − y|α

∀x ∈ B 3R \ B 2R 2 3 (6.1.25) ∀x, y ∈ B 3R \ B 2R . 2

3

There exist sequences {rn } ⊂ R+ converging to 0 and {σn } ⊂ S N −1 converging to σ such that N −p

lim rnp−1 u(rn , σn ) = γ = lim urn (1, σn ).

rn →0

rn →0

(6.1.26)

1 By (6.1.25) the sequence {urn } is eventually relatively compact for the Cloc N topology of R \{0}. Up to a subsequence still denoted by {rn } it converges to some function U which is p-harmonic in RN because |grn (v, ξ)| → 0 by (6.1.23). Moreover U ≤ cμp by (6.1.20). Using Corollary 1.5.9, it follows that U is a multiple of μp and ﬁnally U = γμp because of (6.1.23). This implies (6.1.18) in the case 1 < p < N .

If N = p, we put uλ (x) = (− ln λ)−1 u(λx) for 0 < λ ≤ 12 ; hence ∇uλ (x) = λ(− ln λ)−1 ∇u(λx). We deﬁne gλ by gλ (v, ξ) = λN g((− ln λ)v, (− ln λ)λ−1 ξ)

∀(v, ξ) ∈ R × RN .

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Then |gλ (v, ξ)| ≤ (c1 (− ln λ)q λN |v|q + c2 (− ln λ)r λN −r |ξ| + c3 λN , r

which implies that |gλ (v, ξ)| → 0 when λ → 0, uniformly on compact subsets of R×RN . Since uλ satisﬁes (6.1.21), we can apply the same method as in the previous case: the set of functions urn is eventually relatively 1 topology of RN \ {0}, therefore it converges up to a compact for the Cloc subsequence to γμN (here we use again (6.1.22)). Thus (6.1.18) follows. The supercritical case q ≥ q1 is much more diﬃcult to handle and the results, although very deep, are only partial. Most of the results have been published for the ﬁrst time in [Serrin, Zou (2002)]. Theorem 6.1.3 Assume Ω ⊂ RN is a proper domain, 1 < p < N , g : R+ → R+ is a contiuous function and u ∈ C 1 (Ω) is a a weak nonnegative solution of Δp u + g(u) = 0

in Ω.

(6.1.27)

Then there exists a constant c = c(N, p, q, Λ) such that u(x) ≤ c (ρ(x))

p − q+1−p

∀x ∈ Ω,

(6.1.28)

where ρ(x) = dist (x, ∂Ω), if one of the next conditions is veriﬁed: (i) There exist C > 0 and p − 1 < q < q1 such that C −1 uq ≤ g(u) ≤ Cuq for all u > 0. (ii) g ∈ C 1 ((0, ∞)) and there exist C > 0 and q ∈ (p − 1, p∗ − 1) such that C −1 uq ≤ g(u) ≤ Cuq and qg(u) ≥ ug (u)

∀u > 0.

(6.1.29)

If (ii) holds, g is called subcritical (for this equation). The ﬁrst step of the proof of this result is based upon an integral estimate of the inﬁmum of a solution in a ball due to [Bidaut-V´eron, Pohozaev (2001), Lemma 2.5]. Proposition 6.1.4 Assume Ω ⊂ RN is a domain, 0 < p − 1 < q and 1,p (Ω) ∩ Lqloc (Ω) is a nonnegative and satisﬁes u ∈ Wloc −Δp u ≥ uq

in Ω.

(6.1.30)

Then for any γ ∈ (0, q − 1) there exists c = c(N, p, q, γ) > 0 such that for any y ∈ Ω and R > 0 such that B2R (y) ⊂ Ω there holds γp uγ dx ≤ cRN − q+1−p . (6.1.31) BR (y)

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Furthermore, for any σ ∈ (0, p − pq ), σ(q+1) |∇u|σ dx ≤ cRN − q+1−p .

(6.1.32)

BR (y)

These estimates are obtained by multiplying the inequality by u−δ Φ for some suitable δ > 0 and well chosen test function Φ in a similar way as in the proof of Theorem 6.1.8 below. As a consequence of (6.1.31) we derive, 1 p c γ − q+1−p R . (6.1.33) inf u(x) ≤ ωN x∈BR (y) The second step of the proof is an Harnack inequality which is obtained by using higly technical integral identities. Proposition 6.1.5 Under the structural assumptions (i) or (ii) in Theorem 6.1.3 any nonnegative solution u of (6.1.27) with Ω = B2R0 \ {0} satisﬁes sup

u(x) ≤ c

x∈B |y| (y)

inf x∈B |y| (y)

u(x)

∀y ∈ BR0 ,

(6.1.34)

2

2

for some c = c(N, p, q, C) > 0. Combining (6.1.33) and (6.1.34) we infer (6.1.28). As a consequence Corollary 6.1.6 If 1 < p < N , Ω ⊂ RN is a domain containing 0 and the structural assumptions (i) or (ii) in Theorem 6.1.3 hold. Then any nonnegative weak solution u ∈ C 1 (Ω \ {0}) of (6.1.27) with Ω replaced by Ω \ {0} satisﬁes u(x) ≤ cR− q+1−p p

∀x ∈ Ω \ {0},

(6.1.35)

where R = min{|x| , ρ(x)}. 6.1.3

Rigidity theorems

The following consequence follows Corollary 6.1.7 If 1 < p < N and p − 1 < q < p∗ − 1 the only weak nonnegative fonction satisfying (6.1.27) in RN is the zero function. Proof. It suﬃcies to apply Theorem 6.1.3 in BR and to let R → ∞.

We give below a simple proof of non-existence for a more general class of operators but with a reaction term with a weaker growth. The results are

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due to [Bidaut-V´eron (1989)] and [Bidaut-V´eron, Pohozaev (2001)]. They are based upon integral estimates and do not necessitate to diﬀerentiate the equation as Theorem 6.1.3 does. Theorem 6.1.8 Assume A : RN × RN → RN satisﬁes the structural assumptions (4.1.1) with 1 < p ≤ N and g : RN × R × RN → R is a Caratheodory function which satisﬁes (6.0.3) and ∀(x, r, ξ) ∈ RN × R × RN ,

g(x, r, ξ) ≥ c1 rq

(6.1.36)

for some c1 > 0 and p − 1 < q. (i) If 1 < p ≤ N and p − 1 < q < q1 , there exists no nonnegative nontrivial weak solution to inequality −divA(x, ∇u) ≥ g(x, u, ∇u),

(6.1.37)

in RN . (ii) If A(x, ξ) = a(|ξ|)ξ, 1 < p ≤ N and p − 1 < q ≤ q1 (any q > p − 1 in p = N ), there exists no nonnegative nontrivial weak solution to (6.1.37) in RN \ Bτ for τ > 0. Proof. Assume that the assumptions of (i) hold, let α ∈ (1 − p, 0), , > 0 c , |∇ζ| ≤ 2τ −1 and ζ := ζτ ∈ C0∞ (RN ), 0 ≤ τ ≤ 1, ζ = 1 on Bτ , ζ = 0 on B2τ α for some τ > 0. We set u = u + and take u ζ as test function, then α

RN

|∇u |

p

uα−1 ζ dx

+ RN

−1 |∇u |p−2 ∇u , ∇ζuα dx ζ

≥

Next

RN

|∇u |

p−2

≤

g(x, u, ∇u )uα ζ dx.

−1 ∇u , ∇ζuα dx ζ

≤

RN

RN

|α| 2

|∇u |

RN

p

1 p

uα−1 ζ dx

RN

|∇ζ|

p |∇u | uα−1 ζ dx + C(α, )

p

p1 uα−1+p ζ −p dx p

RN

|∇ζ| uα−1+p ζ −p dx.

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Therefore |α| p |∇u | uα−1 ζ dx + g(x, u, ∇u )uα ζ dx 2 RN RN p |∇ζ| uα−1+p ζ −p dx. ≤ C(α, )

(6.1.38)

RN

Next p

RN

|∇ζ| uα−1+p ζ −p dx α+p−1 α+q

≤

RN

≤ c2 ρ

uα+q ζ dx

N (q+1−p) −p α+q

RN

RN

|∇ζ|

p(α+q) q+1−p

ζ

− p(α+q) q+1−p

q+1−p α+q dx

(6.1.39)

α+p−1 α+q

uα+q ζ dx

where c2 = c2 (N ) > 0, provided ≥

, p(α+q) q+1−p .

Since we have assume p − 1

τ such that vr (r0 ) ≤ 0, hence r v q (s)sN −1 ds. −rN −1 (a(|vr |)vr ) (r) ≥ −r0N −1 (a(|vr |)vr ) (r0 ) + r0

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Furthermore, if v is not identically zero for r ≥ r1 > r0 , there exists γ > 0 such that −rN −1 (a(|vr |)vr ) (r) ≥ γ

∀r ≥ r1 .

Since (4.1.1) holds, this implies vr < 0. Because v(r) → 0 when r → ∞ by (6.1.35), we get ∞ 1 γ p−1 p−N 1−N s p−1 ds = γ1 r p−1 , v(r) ≥ λ r for some γ1 > 0 and thus v q ≥ γ2 r

q(p−N ) p−1

. Since q ≤ q1 , q q(p − N ) =N 1− N −1+ − 1 ≥ −1. p−1 q1

If q = q1 we get 1

1−N

p−N

1

−vr (r) ≥ γ3 r p−1 (ln r) p−1 =⇒ v(r) ≥ γ4 r p−1 (ln r) p−1

as r → ∞,

which contradicts (6.1.35). If p − 1 < q < q1 , we have 1−N

−vr (r) ≥ γ5 r p−1 + =⇒ v(r) ≥ γ6 rm

as r → ∞

where m=

q(N − p) p p − , > 2 p−1 (p − 1) q+1−p

which again contradicts (6.1.35), which is the claim. Step 2. End of the proof. We suppose by contradiction that there exists a nontrivial nonnegative solution u. By the strong maximum principle we can assume that ess min u = ν > 0 on {|x| = τ }. Let k > τ , we construct a sequence {vn,k } of radial functions by q −divA(x, ∇vn,k ) = c1 vn−1,k

in Bk \ Bτ

vn,k (x) = ν

on ∂Bτ

vn,k (x) = 0

on ∂Bk ,

(6.1.43)

with v0 = 0. The sequence {vn,k } is increasing both in n and k. Furthermore vn,k ≤ u in Bk \ Bτ . Letting successively n → ∞ and k → ∞

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and using regularity results of Theorem 1.3.9, we infer that there exists a positive radial solutions of −divA(x, ∇v) = c1 v q v(x) = ν

in RN \ Bτ (6.1.44)

on ∂Bτ

which satisﬁes v ≤ u in RN \ Bτ , which contradicts Step 1.

Remark. Corollary 6.1.7 is clearly not valid if q ≥ p∗ − 1. If one consider the diﬀerential equation rN −1 |ur |p−2 ur + rN −1 uq = 0 on (0, rmax ) (6.1.45) r ur (0) = 0 , u(0) = a > 0, then rmax = ∞, r → u(r) is decreasing from (0, ∞) to (a, 0) and u(r) → 0 when r → ∞. The fact that u cannot vanish follows from a Pohozaev type identity. In the particular case where q = p∗ − 1 the radial solutions are explicit, depending on two parameters, > 0 and x0 ∈ RN ⎛

⎞ p−1 p N ⎜ ⎟ ⎟ u(x) = u,x0 (x) = ⎜ p p ⎝ p−1 ⎠ p−1 + |x − x0 | 1 p−1

1 p

N −p p−1

N −p p

∀x ∈ RN .

(6.1.46)

It is remarkable that this class of solutions describes the whole set of positive solutions of ∗

−Δp u = up

−1

which belong to the space D1,p (RN ) $ 1,p N p∗ N D (R ) = v ∈ L (R ) :

RN

in RN ,

(6.1.47) %

p

|∇v| dx < ∞ .

(6.1.48)

The following result is proved in [Damascelli, Merch´an, Montoro, Sciunzi (2014)] when p ≥ 2 and [Sciunzi (2016)] when p < 2. Theorem 6.1.9 Let N ≥ 2 and N2N +2 < p < N . Then any positive solution u ∈ D1,p (RN ) of (6.1.47) is radial and given by (6.1.46). Their proofs are based upon a delicate adaptation of the moving plane method developed by [Gidas, Ni, Nirenberg (1979)]. In the case p = 2, the assumption u ∈ D1,2 (RN ) is unnecessary, see [Gidas, Spruck (1980)].

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The following nonexistence result of positive solution in a half-space is proven in [Gidas, Spruck (1980)] in the case p = 2 and in the general case 1 < p < N in [Zou (2008)]. Theorem 6.1.10 Let 1 < p < N , p − 1 < q < p∗ − 1 and g ∈ C(R+ ) ∩ C 1 ((0, ∞)) satisﬁes for some C > 0 and λ ∈ (0, p∗ − 1), C −1 uq ≤ g(u) ≤ Cuq and λg(u) ≥ ug (u)

∀u > 0.

(6.1.49)

Then there exists no nontrivial nonnegative solution to −Δp u = g(u) u=0

in RN + on ∂RN +.

(6.1.50)

The proof involves several intermediate results which have their own interest. For their proofs we introduce some notations. We set N N −1 RN = {x = (x , 0)}, + = {x = (x , xN ) = (x1 , ..., xN ) : xN > 0} , ∂R+ ∼ R N Br = Br ∩ ∂RN + , Sr = ∂Br ∩ ∂R+ = ∂ Br and Σr = Sr × (0, ∞),

and Dr,s = {x = (x , xN ) : |x | < r, xN = s} = Br × {s}. N

Lemma 6.1.11 Let u ∈ C 1,α (R+ ) be a solution of (6.1.50), then p p p−2 ∂u ∂u |∇u(x , 0)| dx = |∇u(x , xN )| dSdxN , (6.1.51) p − 1 ∂n ∂xN Br Σr where n is the normal outward unit vector to Σr . Proof. We ﬁrst notice that if there exists a nonnegative solution u of (6.1.50), there holds for xN > 0, p

−

u(x , xN ) ≤ c1 xN q+1−p ,

(6.1.52)

by (6.1.28) and −

q+1

|∇u(x , xN )| ≤ c2 xN q+1−p ,

(6.1.53)

by Lemma 3.3.2, for some c1 , c2 > 0 depending on N, p, q, C. Furthermore, we write (6.1.50) under the form −Δp u − V (x)up−1 = 0,

(6.1.54)

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380

where

$ V (x) =

up−1−q g(u) 0

if u > 0 if u = 0.

Then for any ξ ∈ RN + , ξN > 0, there holds max{|V (x)| : x ∈ B 2|ξ| (ξ)} ≤ c3 |ξ|−p , 3

where c3 = c3 (N, p, q, C) > 0. Therefore, by Harnack inequality (see Chapter 1), max

x∈B |ξ| (ξ)

u(x) ≤ c4

min

x∈B |ξ| (ξ)

2

where c4 = c4 (N, p, q, C) > 0. hence ∂u p−2 Δp udx = |∇u| Θ ∂xN ∂Θ = |∇u|p−2 ∂Θ

|∇u|

=

p−2

∂Θ

u(x),

(6.1.55)

2

For 0 < < s set Θ := Θr,,s = Br × (, s), ∂u ∂u dS − ∂n ∂xN

|∇u| Θ

p−2

∇u,

∂∇u dx ∂xN

∂u ∂u 1 ∂ |∇u|p dS − dx ∂n ∂xN p Θ ∂xN ∂u ∂u 1 p p dS + (|∇u| (x , ) − |∇u| (x , s)) dx . ∂n ∂xN p Br

The previous estimate needs to diﬀerentiate u twice, but the computation ∂u by h−1 (u(x , xN + h) − u(x , xN )) can be easily justiﬁed by replacing ∂x N for h > 0 and performing an Abel’s transform. By (6.1.53) 1 p |∇u| (x , s)dx = 0. lim s→∞ p B r Moreover lim |∇u| (x , ) = |∇u| (x , 0), p

→0

p−2

lim |∇u|

→0

p

uniformly in Br ,

∂u p (x , ) = |∇u| (x , 0), ∂n

uniformly in Br ,

∂u since ∇u = (0, ∂x ) and n = −eN on Br × {0}. In turn this yields N 1 p−2 ∂u ∂u p p |∇u| dS + (|∇u| (x , ) − |∇u| (x , s)) dx lim lim →0 s→∞ ∂Θ ∂n ∂xN p Br 1−p p p−2 ∂u ∂u = |∇u| (x , 0)dx + |∇u| dSdxN , p ∂n ∂xN Br Σr

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and

381

∂u Δp udx ∂x N Θ (6.1.56) 1−p p p−2 ∂u ∂u |∇u| (x , 0)dx + |∇u| dSdxN . = p ∂n ∂xN Br Σr

lim lim

→0 s→∞

Set

u

G(u) =

g(t)dt, 0

then lim G(u(x , )) = 0 uniformly on compact subsets of RN −1 . Because →0

∂u ∂u Δp udx = − g(u)dx Θ ∂xN Θ ∂xN (G(u(x , ) − G(u(x , s)) dx , = Br

which converges to 0 when → 0 and s → ∞, we infer (6.1.51).

Lemma 6.1.12 Under the assumptions of Lemma 6.1.11 there exists c = c(N, p, q, C) > 0 such that

u(x , xN ) ≤ c

p q+1 ∂u (x , 0) ∂xN

∀xN > 0,

(6.1.57)

and u(x , xN ) ≤ cxN

∂u (x , 0) ∂xN

∀xN > 0.

(6.1.58)

Proof. Let x0 = (x0 , x0,N ) ∈ RN + , x0,N > 0; put M = u(x0 ) > 0 and deﬁne w by w(x) = M −1 u (x0 , 0) + M θ x ∀x ∈ RN +, for some θ to be chosen later on. Then −Δp w − M 1−p+θp g(M w) = 0 w=0 We denote V˜ (x) = M 1−p+θp g(M w)w1−p , clearly 0 ≤ V˜ (x) ≤ Ccq+1−p x−p 1 N .

in RN + in ∂RN +.

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Set ξ = (0, M −θ x0,N ), B 1 = B|ξ| (ξ) and B 2 = B |ξ| (ξ), then w(ξ) = 1 and 2 by Harnack inequality as in (6.1.55), w(x) ≥ c5 w(ξ) = c5 Deﬁne w(x) ˜ = 2− c5 |ξ|

−

|x − ξ|

− |ξ|

∀x ∈ B 2 .

−

with =

N −p , p−1

then w ˜ is p-harmonic in RN \ {ξ}, it vanishes on ∂B 1 and − < c5 ≤ min2 w(z) ∀x ∈ ∂B 2 . w(x) ˜ = c5 1 − 2− ) |ξ| z∈∂B

Since in B 1 \ B 2 ,

˜ = M 1−p+θp g(M w) > 0 −Δp w + Δp w

it follows from the comparison principle that w ≥ w. ˜ Therefore M −1+θ

∂u ∂w (x0 , 0) = (0) ∂xN ∂xN ≥

∂w ˜ −1 −1 (0) = 2− c5 |ξ| = c6 |ξ| , ∂xN

which yields ∂u −1 (x , 0) ≥ c6 M 1−θ |ξ| = c6 M x−1 0,N , ∂xN 0

(6.1.59) q+1−p p

since |ξ| = M −θ x0,N . This implies (6.1.58). Because x−1 0,N ≥ c1 we derive ﬁnally q+1−p q+1 ∂u (x0 , 0) ≥ c6 c1 p M p , ∂xN

M

q+1−p p

(6.1.60)

which is (6.1.57).

The reverse pointwise estimate of ∇u(x) with respect to u(x) and the distance of x to ∂RN + is also the consequence of Harnack inequality combined with the local regularity theory Theorem 1.3.9. Lemma 6.1.13

Under the assumptions of Lemma 6.1.11 there holds

|∇u(x , xN )| ≤ cx−1 N u(x , xN )

for some c = c(N, p, q, C) > 0.

∀xN > 0,

(6.1.61)

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383

Proof. Let ξ = (ξ , ξ0 ) ∈ RN + , ξ0 > 0. For r > 0 we denote by M the supremum of u on B ξ0 (ξ). By (6.1.52) and Harnack inequality (6.1.55), 2

M ≤ c1 2

p q+1−p

− p ξ0 q+1−p

and u(ξ) ≥ inf u(x) ≥ c−1 4 M. B ξ0 (ξ)

(6.1.62)

2

If we set w(x) = M −1 u(ξ + ξ0 (x − ξ)), then ξ + ξ0 (x − ξ) ∈ Bξ0 (ξ) if and only if x ∈ B1 (ξ). Furthermore Δp w + M 1−p ξ0p g(M w) = 0

in B1 (ξ).

Clearly 0 < w ≤ 1 in B 12 (ξ), where, by (6.1.52), there also holds 0 < M 1−p ξ0p g(M w(x)) ≤ CM q+1−p ξ0p wq (x) ≤ c7 wq (x) ≤ c7 . By the gradient estimates Theorem 1.3.9, it implies sup

M −1 ξ0 |∇u(ξ + ξ0 (x − ξ))| =

|∇w(x)| = c8 .

sup

x∈B ξ0 (ξ)

x∈B 1 (ξ)

(6.1.63)

4

4

Combining (6.1.62) and (6.1.63) we derive (6.1.61).

Lemma 6.1.14 Under the assumptions of Lemma 6.1.11, for any γ ∈ (0, 1) and ∈ (0, p − 1), there exists c1 = c1 (N, p, q, C, γ, ) such that p p |∇u|p dSdxN ≤ c1 |∇u(x , 0)| γ dS + c1 |∇u(x , 0)| dS, Σr

Sr

Sr

(6.1.64) where pγ := p − 1 + γ +

(1 − γ)p (1 + )p > p := p − 1 − + . q+1 q+1

Proof. By Fubini’s theorem |∇u|p dSdxN = Σr

1

(6.1.65)

p

|∇u| dS dxN Sr ∞

0

+ 1

p |∇u| dS dxN

(6.1.66)

Sr

= I + II. By Lemma 6.1.13 1 I ≤ cp2 x−p up dSdxN N 0

Sr

and

II ≤ cp2

∞ 1

x−p N

up dSdxN . Sr

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384

Using (6.1.57) and (6.1.58) for any γ ∈ (0, 1),

p(1−γ) q+1 ∂u u (x , xN ) ≤ c3 (x , 0) , ∂xN p−1+γ ∂u p−1+γ (x , xN ) ≤ c4 xN (x , 0) . u ∂xN 1−γ

(i) (ii)

This yields

p x−p N u (x , xN )

p(1−γ) p−1+γ q+1 ∂u ∂u xN ≤ (x , 0) (x , 0) ∂xN ∂xN pγ ∂u ≤ c5 xγ−1 (x , 0) , N ∂xN c5 x−p N

and

|∇u(x , 0)|

pγ

I ≤ c6

dS.

(6.1.67)

|∇u(x , 0)| dS,

(6.1.68)

Sr

In the same way

p

II ≤ c7 Sr

which implies (6.1.64). Proof of Theorem 6.1.10. Since p − 1 < q < p∗ − 1, p>p−1+

p(N − 1) p(N − 2) p > > . q+1 N N −1

Therefore there exist γ ∈ (0, 1) and ∈ (0, p − 1) such that (i) (ii)

p > pγ := p − 1 + γ +

p(N − 2) p(1 − γ) > , q+1 N −1

p(N − 2) p(1 + ) p > p := p − 1 − + > . q+1 N −1

We ﬁrst assume that N > 2 and deﬁne, for r > 0, p Ψ(r) = |∇u(x , 0)| dx ; Br

(6.1.69)

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then Ψ (r) =

Sr

385

|∇u(x , 0)| dS. p

By H¨older’s inequality, pγ γ p |∇u(x , 0)| γ dS ≤ c1 rp (Ψ (r)) p , Sr

and

Sr

|∇u(x , 0)| dS ≤ c1 rp (Ψ (r)) p

p p

,

where pγ =

(N − 2)(p − pγ ) (N − 2)(p − p ) > 0 and p = > 0. p p

By (6.1.64) γ pγ p p |∇u| dSdxN ≤ c2 rp (Ψ (r)) p + rp (Ψ (r)) p .

(6.1.70)

Σr

Combined with (6.1.51) we get p p |∇u| dSdxN Ψ(r) ≤ p − 1 Σr γ pγ p ≤ c3 rp (Ψ (r)) p + rp (Ψ (r)) p .

(6.1.71)

Let r ≥ 1; since pγ < p we get, γ pγ p 1 ≤ c3 rp (Ψ (r)) p (Ψ(r))−1 + rp (Ψ (r)) p (Ψ(r))−1 ppγ pp p 1 1 − p + c4 r pγ Ψ (r)(Ψ(r)) pγ + + c5 r p Ψ (r)(Ψ(r))− p 4 4 pp p 1 − p ≤ + c5 r p (Ψ(r)) pγ + (Ψ(r))− p Ψ (r), 2

≤

which yields r−

pp p

p − p ≤ c6 (Ψ(r)) pγ + (Ψ(r))− p Ψ (r).

Now (N − 2)(p − p ) p pp = = (N − 2) p p p

p −1 p

< 1,

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by (6.1.69) (ii). Integrating on (1, r) we obtain p 1− ppγ 1− pp 1− pp p − 1 r (Ψ(1)) , ≤ c + (Ψ(1)) 7 p − pp which leads to an impossibility if r → ∞. In the case N = 2, x = (x1 , x2 ) x2 > 0, Br = (−r, r) and Sr = {−r, r}. Then p r r ∂u p |∇u(x , 0)| dx = (x , 0) Ψ(r) = ∂x2 1 dx1 , −r −r and

p p ∂u ∂u (r, 0) + (−r, 0) . Ψ (r) = ∂x2 ∂x2

Therefore (6.1.64) takes the form ∞ p p p |∇u| dx2 = (|∇u(r, x2 )| + |∇u(−r, x2 )| ) dx2 Σr

0

pγ pγ ∂u ∂u ≤ c8 (r, 0) + (−r, 0) ∂x2 ∂x2 p p ∂u ∂u (r, 0) + (−r, 0) + c8 ∂x2 ∂x2 p p ≤ c10 (Ψ (r)) γ + (Ψ (r)) .

Using Lemma 6.1.11 we arrive to pc10 p p (Ψ (r)) γ + (Ψ (r)) . Ψ(r) ≤ p−1 By Young’s inequality we infer that there holds, for r > 1, 1− p 1− p r ≤ 1 + c11 (Ψ(1)) pγ + (Ψ(1)) p , which yields a contradiction.

(6.1.72)

(6.1.73)

(6.1.74)

The subcriticality assumption on the reaction term can be removed provided some global regularity assumption is made. The next results which point out this fact have been proved in [Farina, Montoro, Sciunzi (2012)], [Farina, Montoro, Sciunzi (2017)], [Farina, Montoro, Riey, Sciunzi (2017)]. Theorem 6.1.15 Assume 1 < p < 2 and g is a locally Lipschitz continu1,∞ ous. If u ∈ C(RN (RN +)∩W + ) is a nonnegative solution of (6.1.50), then it is identically zero if one of the following holds:

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(i) N = 2 and sg(s) > 0 if s = 0. (N −1)p

(ii) N ≥ 3, sg(s) > 0 if s = 0 and g(s) ≤ cs N −1−p + d for some positive constants c, d. (iii) N ≥ 3, sg(s) > 0 if s = 0 and g(s) ≥ cs positive constants c, δ.

(N −1)(p−1) N −1−p

on [0, δ] for some

Theorem 6.1.16 Assume p > 2. Any nonnegative function u ∈ C(RN +) ∞ N such that ∇u ∈ L (R+ ) satisfying −Δp u = uq u=0

in RN + in ∂RN +,

(6.1.75)

is identically zero if one of the next conditions holds: (i) p − 1 < q < ∞ if N ≤ ∗ (ii) p − 1 < q < qN,p if N

∗ = qN,p

p(p+3) p−1 . > p(p+3) p−1 ,

where

2 ((p − 1)N − p) + p2 (p − 2) − N (p − 1) + 2 (p − 1)(N − 1) (N − p) ((p − 1)N − p(p + 3))

.

(6.1.76) Furthermore, if we suppose that u ∈ L∞ (RN + ), then the same conclusion holds if we only assume: p(p+3) p−1 . − 1 > p(p+3) p−1 ,

(i) p − 1 < q < ∞ if N − 1 ≤ ∗ (ii) p − 1 < q < qN −1,p if N ingly.

∗ where qN −1,p is deﬁned accord-

There are several results proving monotonicity properties of positive solutions of (6.1.50) under more general conditions on g, but with a restriction on the class of solutions u. Theorem 6.1.17 Assume g is locally Lipschitz continuous on R and satisﬁes sg(s) > 0 for s = 0. If p > 1 any positive solution of (6.1.50) such that ∇u is bounded is monotone increasing in the direction xN . Remark. The proofs of these results are based upon a delicate use of the moving planes method combined with weak comparison principles in a narrow strip. The assumption that ∇u ∈ L∞ implies that u remains bounded and arbitrarilly small whenever xN is small enough, which allows to use this comparison method. Notice that the boundedness of u implies the one of ∇u by extension of u by reﬂection to RN − and standard regularity results.

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6.1.4

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Separable solutions in RN +

The study of boundary isolated singularities of solutions of (6.1.1) in RN + , or more generally in a cone CS , generated by a smooth subdomain S ⊂ S N −1 , which vanish on ∂CS \ {0} leads to the following problem 2 2 p−2 2 2 div βp,q ω + |∇ ω| ∇ ω + |ω|q−1 ω p−2 2 2 (6.1.77) + βp,q Λ(βp,q ) βp,q ω + |∇ ω|2 2 ω = 0 in S ω = 0 in ∂S. To the problem we associate the spherical p-harmonic eigenvalue problem treated in Chapter 2-2 p−2 p−2 div β 2 φ2 + |∇ φ|2 2 ∇ φ + βΛ(β) β 2 φ2 + |∇ φ|2 2 φ = 0 in S φ = 0 in ∂S, (6.1.78) with Λ(β) = β(p − 1) + p − N . We denote by (βS , φS ) the unique couple solution with βS > 0 and φS > 0. The general nonexistence result holds Theorem 6.1.18 Assume 0 < p − 1, S ⊂ S N −1 is a C 2 domain and g : S × R × T S N −1 → R is a locally Lipschitz continuous function which satisﬁes sg(σ, s, ξ) > 0

∀(σ, s, ξ) ∈ S × R × T S N −1, s = 0.

(6.1.79)

Then for any β ≥ βS there exists no nontrivial nonnegative function ω ∈ C 2 (S) ∩ C(S) satisfying p−2 p−2 div β 2 ω 2 + |∇ ω|2 2 ∇ ω + βΛ(β) β 2 ω 2 + |∇ ω|2 2 ω (6.1.80) + g(., ω, ∇ω) = 0 in S ω=0 in ∂S. Proof. Assume ω is such a solution. By the strong maximum principle it is positive in S and Hopf boundary lemma holds on the relative boundary ∂S. If β = βS , we denote by φS the largest solution of the spherical p-harmonic eigenvalue problem (6.1.78) which is smaller than ω. Then w := ω − φS ≥ 0 in S. As in the proof of Theorem 3.2.1 we consider the linear elliptic equation satisﬁed by w near σ0 . Then (with the notations of the above

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mentioned theorem) ∂w −1 ∂ 2 w Lw = √ ai + Ci + Cw = g(., ω, ∇ω) ≥ 0. g ∂σi ∂σ ∂σi i

(6.1.81)

,i

Either the graphs of ω and φS are tangent at some σ0 ∈ S, or ω − φS > 0 in S and there exists some σ0 ∈ ∂S such that ω(σ0 ) = φS (σ0 ) = 0 and

∂ω ∂φS (σ0 ) = (σ0 ) < 0, ∂n ∂n

where n is the normal unit outward vector to ∂S in T S N −1 . In the ﬁrst case this is a contradiction with Proposition 1.3.11. In the second case it contradicts Proposition 1.3.12. If β > βS we introduce the operator 2 2 p−2 p−2 2 2 β η + |∇ η| ∇ η + βΛ(β) β 2 η 2 + |∇ η|2 2 η, T (η) = div and set θ =

β βS

> 1. Then

1 p−2 β θ−1 2 2 2 2 T (φS ) = βφS βS φS + |∇ φS | φ (Λ(β) − Λ(βS )) βS S p−1 2 2 p−2 β βS φS + |∇ φS |2 2 |∇ φS |2 > 0. + θ(θ − 1) βS θ

θ

Again we take the largest φS such that w := ω − φS ≥ 0, and since θ > 0 the tangency of the two graphs cannot occur on ∂S and there exists σ0 ∈ S such that w(σ0 ) = 0. Since Lw ≥ 0 we infer a contradiction from Proposition 1.3.11 as in the ﬁrst case. The following existence result is proved in [Porretta, V´eron (2013)]. Theorem 6.1.19 Assume 1 < p < N − 1, S ⊂ S N −1 is a C 2 domain and 0 < p − 1 < q < qs . Then for any 0 < β < βS there exists a positive function ω ∈ C 2 (S) ∩ C(S) satisfying (6.1.78). The problem is not variational, and since the reaction term is a source, it is diﬃcult to ﬁnd a supersolution of the equation. In such situations topological and deformations arguments associated to degree theory are often used. Such arguments were thoroughly studied in [Krasnosel’skii (1964)] and the following result is a variant proved in [de Figuereido, Lions, Nussbaum (1982), Prop. 2.1, Remark 2.1].

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Proposition 6.1.20 Let X be a Banach space with norm . X and K ⊂ X a closed cone with a non-empty interior. Let F : K × R+ → K be a compact mapping, and let Φ(u) = F (u, 0) (it is a compact mapping from K to K). Assume that there exist R1 < R2 and T > 0 such that (i) u = sΦ(u) for every s ∈ [0, 1] and every u such that u X = R1 , (ii) F (u, t) = u for every (u, t) such that u X ≤ R2 and t ≥ T , (iii) F (u, t) = u for every u such that u X = R2 and t ≥ 0. Then the mapping Φ has a ﬁxed point u such that R1 < u X < R2 . Another intermediate result is the following one which proof is a mere adaptation of the one of Proposition 2.1.3 ([Porretta, V´eron (2013), Appendix]). Proposition 6.1.21 $ λ1,β = inf S

Let p > 1, β > 0, S ⊂ S N −1 and denote

β 2 η 2 + |∇ η|

2

p2

% |η|p dS = 1 .

dS : η ∈ W01,p (S),

S

(6.1.82) Then λ1,β > 0 is achieved by some φ ∈ W01,p (S) which is positive in S and satisﬁes −div

p−2 β 2 φ2 + |∇ φ|2 2 ∇ φ p−2 p−2 φ + β 2 β 2 φ2 + |∇ φ|2 2 φ = λ1,β |φ| φ=0

in S

(6.1.83)

on ∂S.

Moreover φ is the unique nonnegative nontrivial solution of (6.1.83) and λ1,β is the smallest of all the λ ∈ R such that there exists a solution η ∈ W01,p (S) to −div

2 2 p−2 β η + |∇ η|2 2 ∇ η p−2 p−2 η + β 2 β 2 η 2 + |∇ η|2 2 η = λ |η| η=0

in S

(6.1.84)

on ∂S,

and it is isolated among the set of such λ. Remark. In the above statement one can replace the condition S ⊂ S N −1 by S bounded and S ⊂ M , where (M, g) is any Riemannian manifold, in particular RN .

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Proof of Theorem 6.1.19. The following functional p 1 2 2 J (η) = β η + |∇ η|2 2 dS, p S

391

(6.1.85)

deﬁned on W01,p (S) is strictly convex and coercive. Its Frechet derivative A deﬁned by 2 2 p−2 p−2 2 2 A(η) = −div β η + |∇ η| ∇ η + β 2 β 2 η 2 + |∇ η|2 2 η, (6.1.86) is a bijection from W01,p (S) into W −1,p (S) and its inverse A−1 is continuous ¯ endowed with the norm C 1 (S), ¯ a norm [Lions (1969)]. We set X = C01 (S) 1,p for which the imbedding into W0 (S) is continuous. We denote by K the cone of positive function in S; it has a non-empty interior. For t > 0 we set p−2 F (η, t) = A−1 β(β + t + Λ(β)) β 2 η 2 + |∇ η|2 2 η + (η + t)q , (6.1.87) and put

p−2 Φ(η) := F (η, 0) = A−1 β(β + Λ(β)) β 2 η 2 + |∇ η|2 2 η + η q . (6.1.88)

Step 1: The mapping F is compact. Actually, if F (η, t) = φ, it means that φ ∈ W01,p (S) satisﬁes p−2 p−2 −div β 2 φ2 + |∇ φ|2 2 ∇ φ + β 2 β 2 φ2 + |∇ φ|2 2 φ p−2 = β(β + t + Λ(β)) β 2 η 2 + |∇ η|2 2 η + (η + t)q in S (6.1.89) φ = 0 on ∂S. Therefore, if we assume that η belongs to a bounded subset of K ∩ X, then the right-hand side of (6.1.89) is bounded in C(S), hence by Theorem 1.3.10, φ remains bounded in C 1,γ (S) for some γ ∈ (0, 1) and therefore relatively compact in C 1 (S). Step 2: Condition (i) of Proposition 6.1.20 holds. Suppose by contradiction that there exists a sequence {tn } such that the problem 2 2 p−2 p−2 2 2 β η + |∇ η| ∇ η + β 2 β 2 η 2 + |∇ η|2 2 η −div p−2 2 2 2 2 q = tp−1 η + tp−1 in S (6.1.90) n β(β + Λ(β)) β η + |∇ η| n η η = 0 on ∂S,

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admits a positive solutions ηn for any n ∈ N and that ηn X → 0 as n → ∞. −1 Then wn = ηn X ηn satisﬁes p−2 p−2 −div β 2 wn2 + |∇ wn |2 2 ∇ wn + β 2 β 2 wn2 + |∇ wn |2 2 wn p−2 2 2 q+1−p q 2 2 = tp−1 wn + tp−1 ηn X wn in S n β(β + Λ(β)) β wn + |∇ wn | n wn = 0 on ∂S. Up to a subsequence we can assume that tn → t ∈ [0, 1], and by compactness, wn → w in C 1 (S) where w is positive and w X = 1. Then w satisﬁes p−2 −div β 2 w2 + |∇ w|2 2 ∇ w p−2 = β(tp−1 β − β + tp−1 Λ(β)) β 2 w2 + |∇ w|2 2 w in S w = 0 on ∂S. It follows from Theorem 2.2.3 with the notations therein applied with v = − β1 ln w that tp−1 β − β + tp−1 Λ(β) = λ(β), where λ(β) is the ergodic constant. Since β < βS , there holds by Proposition 2.2.5 tp−1 β − β + tp−1 Λ(β) > Λp (βS ). Since t ≤ 1 we get, Λp (βS ) < tp−1 β − β + tp−1 Λ(β) ≤ tp−1 Λ(β), which is impossible if we explicit the values of Λ(β) and Λp (βS ). Step 2: Condition (ii) holds. Since, for t large enough, Λ(β) + β + t > 0, using the fact that q > p − 1, for any δ > 0, one can ﬁnd T > 0 such that p−2 2 β (Λ(β) + β + t) ω β 2 ω 2 + |∇ ω|2 + (ω + t)q > (λ1,β + δ)ω p−1

∀t ≥ T, ∀ω ≥ 0,

where λ1,β has been deﬁned in (6.1.82). Therefore, if t ≥ T and F (ω, t) = ω for some ω ∈ K ⊂ X, ω = 0, one obtains p−2 p−2 2 2 2 2 β 2 ω 2 + |∇ ω| ∇ ω + β 2 β 2 ω 2 + |∇ ω| ω −div ≥ (λ1,β + δ)ω p−1 ω=0

in S on ∂S.

(6.1.91)

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Since the minimum φ of (6.1.82) is a subsolution of equation (6.1.84) with λ = λ1,β + δ and we may adjust it so that it is smaller than ω, it follows from Theorem 1.4.7 that there exists a solution ψ of the same equation such that φ ≤ ψ ≤ ω. Since δ is arbitrary, it would imply that λ1,β is not isolated, a contradiction. Notice that (ii) holds independently of the norm of ω for t ≥ T . Step 3: Condition (iii) holds. Since (ii) holds independently of the norm of ω, it is enough to show that (iii) holds for every t ≤ T . This is done if one can ﬁnd an a priori estimates, i.e. if we can prove the existence of a constant R2 such that for any t ≤ T every positive solution of 2 2 p−2 p−2 2 2 β ω + |∇ ω| −div ∇ ω + β 2 β 2 ω 2 + |∇ ω|2 2 ω p−2 (6.1.92) = β(Λ(β) + β + t) β 2 ω 2 + |∇ ω|2 2 ω + (ω + t)q in S ω=0

on ∂S,

satisﬁes ω X ≤ R2 . By the regularity estimates Theorem 1.3.9, it is suﬃcient to prove the existence of an a priori bound of ω W 1,p . We proceed by contradiction in assuming that there exist sequences {tn } and {ωn } such that 0 < tn ≤ T and ωn is a positive solution of (6.1.92) with t = tn satisfying ωn L∞ → ∞

as n → ∞,

and there exists a sequence of points σn ∈ S such that ωn L∞ = ωn (σn ). Furthermore, it can be assumed that σn → σ0 ∈ S. We set Mn = p−1−q ( ωn L∞ ) p . Using local coordinates in a neighborhood of σ0 , we set υn (y) =

p ωn (σn + Mn y) = Mnq+1−p ωn (σn + Mn y). ωn L∞

(6.1.93)

The sequence {υn } is uniformly bounded in the ball BδMn−1 (σ0 ) ⊂ RN −1 and thus it is also eventually locally uniformly bounded in the C 1,α topology of RN −1 and hence eventually relatively compact in the C 1 topology of RN −1 . Up to a subsequence, it converges to some nonnegative υ which satisﬁes (for some c0 > 0) −Δp υ = c0 uq

(6.1.94)

−1 −1 if σ0 ∈ ∂S and υ vanishes on ∂RN . either in RN −1 if σ0 ∈ S, or in RN + + In both case υ(0) = 1. Such a solution of (6.1.94) cannot exist, either by Corollary 6.1.7 or by Theorem 6.1.10.

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The exponent qs is critical for existence of positive solution of problem (6.1.77), provided S veriﬁes some geometric condition: a C 1 domain N −1 is starshaped, if there exists a nonzero eigenfunction φ of the S ⊂ S+ Laplacian on S N −1 with corresponding eigenvalue N − 1, positive on S, such that ∇φ, na ≤ 0

∀a ∈ ∂S,

(6.1.95)

where na is the normal outward unit vector at a in T S N −1. The following result is proved in [Porretta, V´eron (2013)]. Theorem 6.1.22 Assume 1 < p < N − 1, S ⊂ S N −1 is a C 2 starshaped domain and q = qs . Then there exists no positive function ω ∈ C 2 (S)∩C(S) satisfying (6.1.77). The proof is based upon a series of integral identities dealing with solutions of 2 2 p−2 p−2 2 2 β ω + |∇ ω| ∇ ω + βΛ(β) β 2 ω 2 + |∇ ω|2 2 ω div (6.1.96) in S + |ω|q−1 ω = 0 ω=0 in ∂S. N −1 and φ1 be the ﬁrst eigenfunction Proposition 6.1.23 Let S S+ 1,2 1,p N −1 of −Δ in W0 (S+ ). If ω ∈ W0 (S) ∩ C(S) is a positive solution of 1 (6.1.96) and if we set Q = β 2 ω 2 + |∇ ω|2 2 , then ∂ω p ∂φ1 1 q+1 1− ∂n dS = A ω φ1 dσ p ∂S ∂n S (6.1.97) p−2 2 p−2 2 |∇ ω| φ1 dσ + C Q ω φ1 dσ, +B Q S

S

where A = A(β) = −

B = B(β) = C = C(β) = β 2

N −1 − β(pβ + p − N ), q+1

N −1−p − β(pβ + p − N ), p

N −1 − (pβ + p − N )Λ(β) . p

(6.1.98)

(6.1.99)

(6.1.100)

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Proof. The key step is a general identity valid even for any S ⊂ S N −1 . We recall that by Theorem 1.3.9, ω ∈ C 1,α (S). Furthermore, since 1 2 2 β ω + |∇ ω|2 2 > 0 in S, ω is C 2 in S. Step 1: We claim that for any φ ∈ C 2 (S) there holds: ∂ω p ∂φ1 Δφ 1 − β(βp + p − N )φ ω q+1 dσ 1− dS = ∂n p q+1 ∂S ∂n S 1 p − Q Δ φdσ + Qp−2 D2 φ∇ ω, ∇ ωdσ p S S 2 + β(βp + p − N ) Qp−2 |∇ ω| φdσ S

+ β Λ(β)(βp + p − N ) Qp−2 ω 2 φdσ. 2

S

Multiplying (6.1.96) by ζφ where ζ ∈

Q S

p−2

C01 (S)

(6.1.101) and integrating yields

1 2 2 ∇ |∇ ω| , ∇ φ + D φ∇ ω, ∇ ω ζdσ 2 + Qp−2 ∇ ω, ∇ ζ∇ ω, ∇ φdσ S

= βΛ(β) Qp−2 ω∇ ω, ∇ φζdσ + S

1 q+1

∇ ω q+1 , ∇ φζdσ.

S

Since 1 p−2 2 1 Q ∇ |∇ ω| , ∇ φ = ∇ Qp , ∇ φ − β 2 Λ(β)Qp−2 ω∇ ω, ∇ φ, 2 p and Λ(β) = (p − 1)β + p − N , we obtain 1 p

∇ Qp , ∇ φζdσ +

S

Qp−2 D2 φ∇ ω, ∇ ωζdσ

S

+

Qp−2 ∇ ω, ∇ ζ∇ ω, ∇ φdσ

S

= β(βp + p − N ) Qp−2 ω∇ ω, ∇ φζdσ + S

1 q+1

∇ ω q+1 , ∇ φζdσ.

S

Integrating by parts the ﬁrst and the last terms leads to

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−

1 p

S

Qp ∇ ζ, ∇ φdσ +

+ S

Q

+

Qp ω q+1 − q+1 p

p−2

1 q+1

ω q+1 ∇ ζ, ∇ φdσ S

ζΔ φdσ +

Qp−2 D2 φ∇ ω, ∇ ωζdσ

S

(6.1.102)

∇ ω, ∇ ζ∇ ω, ∇ φdσ

S

= β(βp + p − N ) Qp−2 ω∇ ω, ∇ φζdσ. S

For δ > 0 small enough, we consider ζ := ζδ in the above equalities where the {ζδ } forms a sequence of functions in C01 (S) converging to 1 locally uniformly in S and such that |∇ ζδ | remains bounded in L1 (S). Since for any continuous vector ﬁeld F on S there holds lim F, ∇ ζδ dσ = − F, n dS, δ→0

S

∂S

∂ω we infer, by letting δ → 0 and since ∇ ω = − n on ∂S, ∂n q+1 ∂ω p ∂φ Qp ω 1 − Δ φdσ dS = 1− ∂n p q+1 p ∂S ∂n S + Qp−2 D2 φ∇ ω, ∇ ωdσ + β(βp + p − N ) Qp−2 ω∇ ω, ∇ φdσ.

S

S

(6.1.103) Next we multiply (6.1.96) by ωφ and integrate Qp−2 ω∇ ω, ∇ φdσ S

=−

Q

p−2

p−2 2 |∇ ω| φdσ + βΛ(β) Q ω φdσ + ω q+1 φdσ.

2

S

S

S

Replacing the last term in (6.1.102) yields q+1 ∂ω p ∂φ Qp ω 1 − Δ φdσ 1− ∂n dS = p q+1 p ∂S ∂n S 2 Qp−2 |∇ ω| − βΛ(β)ω 2 − ω q+1 φdσ + β(βp + p − N ) + S

S

Qp−2 D2 φ∇ ω, ∇ ωdσ,

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which is the claim. Step 2: End of the proof. We chose φ as ﬁrst eigenfunction of −Δ in N −1 ), thus −Δ φ = (N − 1)φ and D2 φ = −φg, hence W01,2 (S+ ∂ω p ∂φ N −1 1 + β(βp + p − N ) ω q+1 φdσ dS = − 1− ∂n p q+1 ∂S ∂n S 2 2 − Qp−2 |∇ ω| φdσ + β(βp + p − N ) Qp−2 |∇ ω| φdσ S

N −1 + p

S

Q φdσ − β (βp + p − N )Λ(β) Qp−2 ω 2 φdσ. p

2

S

S

(6.1.104) Replacing Q by its value we obtain (6.1.97) with A, B and C given by (6.1.98), (6.1.99) and (6.1.100) respectively. Proof of Theorem 6.1.22. First we compute the coeﬃcients A, B and C p and we express it by introducing in Proposition 6.1.23 with βp,q = q+1−p βs := βp,qs where βs =

N −1−p p = . qs + 1 − p p

Then (N − 1)βp,q − βp,q (pβp,q + p − N ) p(1 + βp,q ) N −1 βp,q =− + p(βp,q + 1)2 − N (βp,q + 1) βp,q + 1 βp,q pβp,q 1 βp,q + 1 − (βp,q − βs ), =− βp,q + 1 p

A=−

B = βs + βp,q (p(βp,q − βs ) − 1) = (βp,q − βs ) (pβp,q − 1) , and

2 C = βp,q

N −1 − (pβp,q + p − N ) ((p − 1)βp,q + p − N ) p

2 βs + 1 − (p(βp,q − βs ) − 1) (p(βp,q − βs ) − (βp,q + 1)) = βp,q p(N − p) 2 . = (1 − p)βp,q (βp,q − βs ) pβp,q − 1 − p−1

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Therefore A ≥ 0, B ≥ 0 and C ≥ 0 can be obtained only if q = qs i.e. βp,q = βs , in which case A = B = C = 0. Since ∂n φ ≤ 0 because S p is starshaped, we deduce from (6.1.97) that |∂n ω| ∂n φ = 0. Unless ω is identically zero, we have ∂n ω < 0 by Hopf boundary lemma. Then ∂n φ = 0 on ∂S. Using the equation satisﬁed by φ and Green formula we derive (N − 1) φdS = 0. S

Thus φ = 0 in S, a contradiction with φ > 0. Remark. It is conjectured that non-existence holds also for any q > qs . 6.1.5

Boundary singularities

In this section we prove the following universal a priori estimate for positive solutions of −Δp u = uq

in Ω in ∂Ω \ {0},

u=0

(6.1.105)

where Ω ⊂ RN is a C 2 bounded domain such that 0 ∈ ∂Ω and p − 1 < q < p∗ − 1. Theorem 6.1.24 Assume p − 1 < q < p∗ − 1, then any positive solution of (6.1.105) satisﬁes p

u(x) ≤ c |x|− q+1−p

∀x ∈ Ω,

(6.1.106)

for some c > 0 independent of the solution. The proof is based upon a method which connects Liouville type results and a priori estimate due to [Pol`acik, Quittner, Souplet (2007)]. In the case p = 2 the above estimate has been obtained in [Bidaut-V´eron, Ponce, V´eron (2011)]. The main ingredient is a topological result called the doubling lemma (see [Pol`acik, Quittner, Souplet (2007)]). Proposition 6.1.25 Let (X, d) be a complete metric space Γ X and γ : X \ Γ → (0, ∞). We assume that γ is bounded on all compact subset of X \ Γ. Given k > 0, let y ∈ X \ Γ such that γ(y)dist (y, Γ) > 2k. Then there exists x ∈ X \ Γ satisfying

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(i) γ(x)dist (x, Γ) > 2k, (ii) γ(x) > γ(y), (iii) 2γ(x) ≥ γ(z), for all z ∈ B

k γ(x)

(x).

Proposition 6.1.26 Assume p − 1 < q < p∗ − 1 and 0 < r < 12 diam (Ω). Then every nonnegative solution of −Δp u = uq

in Ω ∩ (B2r \ B r )

(6.1.107)

in ∂Ω ∩ (B2r \ B r ),

u=0 satisﬁes u(x) ≤ c (dist (x, Γr )

p − q+1−p

∀x ∈ Ω ∩ (B2r \ B r ),

(6.1.108)

where Γr = Ω ∩ (∂B2r ∪ ∂Br ) and c > 0 is independent of u and r. Proof. We proceed by contradiction, assuming that (6.1.108) is false. Then for every k ∈ N∗ , there exists rk ∈ (0, 12 diam (Ω)) and a solution uk of (6.1.107) with r = rk and yk ∈ Ω ∩ (B2rk \ B rk ) such that p

uk (yk ) > (2k) q+1−p (dist (yk , Γrk )− q+1−p , p

(6.1.109)

where Γrk = Ω ∩ (∂B2rk ∪ ∂Brk ). We apply Proposition 6.1.25 with X = Ω ∩ (B 2rk \ Brk ) , Γ = Γrk and γ(x) = (uk (x))

q+1−p p

.

One can ﬁnd xk ∈ X \ Γrk such that p − q+1−p

p

(i) uk (xk ) > (2k) q+1−p (dist (xk , Γrk ))

,

(ii) uk (xk ) > uk (yk ), p

(iii) 2 q+1−p uk (xk ) ≥ uk (z), for all z ∈ BRk (xk ) where we denote Rk = k (uk (xk ))

− q+1−p p

.

By (i) we have Rk < 12 dist (xk , Γrk ) and therefore BRk (xk ) ∩ Γrk = ∅. Furthermore dist (xk , Γrk ) ≤ 12 rk < 14 diam (Ω), hence, from (i), uk (xk ) ≥

8k diam (Ω)

p q+1−p

.

In particular, uk (xk ) → ∞ as k → ∞. For every k ∈ N∗ let − q+1−p p

tk = (uk (xk ))

! 1 Dk = ξ ∈ RN : |ξ| ≤ 2k and xk + tk ξ ∈ Ω = (Ω − xk ), tk

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and vk (ξ) =

uk (xk + tk ξ) uk (xk )

∀ξ ∈ Dk .

Then vk satisﬁes −Δp vk = vkq 0 ≤ vk ≤ 2

p q+1−p

in Dk in Dk

vk (0) = 1. Up to a subsequence, we encountered the following dichotomy: (A) either for every a > 0 there exists ka ≤ 1 such that Batk (xk ) ∩ ∂Ω = ∅. Since the sequence {vk } is locally uniformly bounded, it follows from Theorem 1.3.9 and Ascoli’s compactness theorem that it is eventually relatively 1 (RN ) topology and that there exist a subsequence, still compact in the Cloc denoted by {vk } and a function v ∈ C 1 (RN ) which satisﬁes (6.1.1) in RN , p 0 ≤ v ≤ 2 q+1−p and v(0) = 1, which is impossible by Corollary 6.1.7. (B) or there exists a0 > 0 such that Ba0 tk (xk ) ∩ ∂Ω = ∅ for all k ∈ N∗ . We set xk = P roj∂Ω (xk ), then |xk − xk | ≤ a0 tk . Since vk is bounded in Dk , it follows from Theorem 1.3.9 that ∇vk remains locally bounded therein, and because uk (xk ) = 0 it implies that |xk − xk | ≥ a1 tk for some a1 ∈ (0, a0 ). −1 Up to a subsequence, we can assume that t−1 k xk → x0 and thus tk xk → x0 , that −eN is the normal outward unit vector to ∂Ω at x0 and that Dk converges to the half-space H passing through x0 with normal outward unit vector −eN and that x0 − x0 = −aeN for some a ∈ [a1 , a0 ]. We denote ˜ the union of H and its reﬂection through ∂H. In order to analyze by H the convergence of {vk }, we perform the same reﬂexion through ∂( t1k Ω) as the one used in Lemma 3.3.2. The reﬂected function v˜k converges, up 1 ˜ topology to some v˜ which satisﬁes (6.1.1) (H) to a subsequence in the Cloc ˜ in H, vanishes on ∂H, is nonnegative in H and such that v(0) = 1. By Theorem 6.1.10 such a function cannot exist, which ends to proof. Proof of Theorem 6.1.24. Let x ∈ Ω such that |x| ≤ 34 diam (Ω). We apply Proposition 6.1.26 with r = 23 |x|. Since dist (x, Γr ) = 13 r we derive p r − q+1−p − p − p (6.1.110) ≤ c˜ |x| q+1−p . u(x) ≤ c (dist (x, Γr ) q+1−p = c 3 The following estimate is an immediate consequence of Theorem 6.1.24 and Lemma 3.3.2.

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Corollary 6.1.27 Under the assumptions of Theorem 6.1.24, any positive solution u of (6.1.105) satisﬁes u(x) ≤ cρ(x) |x|

q+1 − q+1−p

∀x ∈ Ω,

(6.1.111)

∀x ∈ Ω,

(6.1.112)

where ρ(x) = dist (x, ∂Ω), and |∇u(x)| ≤ c |x|

q+1 − q+1−p

for some c = c(Ω, p, q) > 0. The ﬁrst critical exponent q for boundary singularities is q = q˜1 deﬁned p . by relations (3.2.5), for which value β N −1 = q˜1 +1−p S

+

Theorem 6.1.28 Let Ω be a bounded C 2 domain such that 0 ∈ ∂Ω and ˜1 . Ω ∩ Br0 = Br+0 := Br0 ∩ RN + for some r0 > 0. Assume 0 < p − 1 < q < q If u is a positive solution of (6.1.105) there exists c > 0, depending on u such that u(x) ≤ c |x|

−β

S

N −1 +

(6.1.113)

∀x ∈ Ω.

Proof. We write the equation under the form −Δp u − V (x)up−1 = 0,

(6.1.114)

where V (x) = (u(x))q+1−p ≤ c|x|−p by Theorem 6.1.24. Using boundary Harnack inequality Corollary 2.3.2, there exists c1 > 0 such that c−1 1

u(x) u(x) u(y) ≤ ≤ c1 ρ(x) ρ(y) ρ(x)

|x| = |y| = 0. (6.1.115) ∀x, y ∈ B + r0 s.t. 2

Let us suppose that (6.1.113) does not hold. Then there exists a sequence {xk } ⊂ Ω such that u(xk ) ≥ k |xk |

−β

S

N −1 +

∀k ∈ N∗ .

(6.1.116)

By Theorem 6.1.24 the sequence {xk } converges to 0, and because of (6.1.115) we can assume that ρ(xk ) = |xk | = rk ≤ r20 . We recall that −β

VRN (x) = VRN (r, σ) = r + +

S

N −1 +

φ

S

N −1 +

(σ),

is the singular spherical p-harmonic function deﬁned in RN + constructed in Theorem 2.2.1 and VBr+ is the singular p-harmonic function deﬁned in 0

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Br+0 constructed in Proposition 3.3.5 which vanishes on ∂Br+0 and satisﬁes (3.3.36), i.e. VRN (x) +

lim

VBr+ (x)

x→0

= 1.

0

By Hopf boundary lemma ∂n φ

S

N −1 +

N −1 ≤ γ < 0 on ∂S+ , hence there exists

c2 > 0 such that ρ(x) c−1 2 |x|

≤φ

S

N −1 +

x |x|

≤ c2

ρ(x) . |x|

Then u(rk , σ) = u(rk , σk )

u(rk , σ) u(rk , σk ) −β

N −1 S ρ(rk , σ) rk + rk −1 −1 ≥ kc1 c2 VRN (rk , σ) +

≥ kc−1 1

−1 ≥ kc−1 1 c2 VBr+ (rk , σ)

N −1 ∀σ ∈ S+ .

0

−1 + + By the comparison principle u ≥ kc−1 1 c2 VBr+ in Br0 \ Brk . For j ﬁxed 0

−1 + + and any k ≥ j, u ≥ kc−1 1 c2 VBr+0 in Br0 \ Brj , which yields a contradiction when k → ∞.

Corollary 6.1.29 γ ≥ 0 such that

Under the assumptions of Theorem 6.1.28 there exists β S

lim r

N −1 +

r→0

u(r, σ) = γφ

S

1+β

lim r

S

N −1 +

r→0

N −1 +

(σ)

∇ u(r, σ) = γ∇ φ

S

(6.1.117) (σ) N −1

+

N −1 . uniformly in S+

Proof. By Theorem 6.1.28 and Lemma 3.3.2 there holds |∇u(x)| ≤ c |x|

−1−β

S

N −1 +

∀x ∈ Br+0 ,

and |∇u(x) − ∇u(y)| ≤ c |x|

−1−α−β

S

N −1 +

α

|x − y|

∀x, y ∈ Br+0 , |x| ≤ |y| .

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The remaining of the proof follows the one of Theorem 3.3.14 in the charaterization of weak singularities. Remark. In the case p = N , the assumption that Ω ∩ Br0 = Br+0 is not needed if we use conformal transformations as in Chapter 3. In that case the critical exponent q˜1 is 2N − 1. 6.2 6.2.1

Equations with measure data The case p = 2

Let Ω ⊂ RN be a bounded domain and g : R → R a continuous function satisfying rg(r) ≥ |r|q+1

∀r ∈ R,

(6.2.1)

for some q > 1. If u is a nonnegative weak solution of the semilinear problem −Δu = g(u) + μ

in Ω

u=0

on ∂Ω,

(6.2.2)

where μ is a positive measure in Ω, there holds, by deﬁnition of weak solutions, − uΔψdx = g(u)ψdx + ψdμ, Ω

Ω

Ω

for any ψ ∈ C01 (Ω) such that −Δψ ∈ L∞ (Ω). Thus, if we assume that ψ is the ﬁrst eigenfunction of −Δ in W01,2 (Ω) we have using (6.2.1),

−

uψdx ≤ λΩ

uΔψdx = λΩ Ω

Ω

q1 1 q g(u)ψdx ψdx ,

Ω

Ω

which implies ψdμ ≤ Ω

λqΩ

1 1 − q q q

ψdx.

(6.2.3)

Ω

This is an upper estimate of the norm of μ in Mρ (Ω). Notice that the following estimate is also satisﬁed g(u)ψdx ≤ λqΩ ψdx. (6.2.4) Ω

Ω

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If we take ψ = ϕq for some ϕ ∈ C0∞ (Ω), ϕ ≥ 0, there holds q −1 q Δϕdx ≥ g(u)ϕ dx + ϕq dμ. −q uϕ Ω

Ω

Ω

Because 1q 1 q q q uϕq −1 Δϕdx ≤ g(u)ϕ dx |Δϕ| dx , Ω

we infer

Ω

g(u)ϕq dx ≤ q q

Ω

Ω

q

|Δϕ| dx, Ω

from which inequality it follows 1 1 q − q ϕq dμ ≤ q q |Δϕ| dx. q q Ω Ω

(6.2.5)

¯ ⊂ Ω is a Borel set, there holds Hence, if E ⊂ E μ(E) ≤ c(q)capΩ 2,q (E).

(6.2.6)

This is an estimate of Lipschitz continuity of μ with respect to capΩ 2,q . (E) > τ for any E = ∅ Furthermore, if N > 2 and 1 < q < NN−2 , capΩ 2,q and some ﬁxed τ > 0. Hence condition (6.2.6) is fulﬁlled for any measure μ such that μ(Ω) ≤ c(q)τ . Estimates (6.2.3) and (6.2.6) show that in the case p = 2 there are two types of necessary conditions linking μ and q, for the existence of positive solutions of (6.2.2). Actually this situation holds for all p > 1. 6.2.2

The subcritical case

In this section Ω ⊂ RN is a C 2 bounded domain, A : Ω × R × RN → RN a Caratheodory vector ﬁeld satisfying (4.1.1) with p > 1 and g : Ω×R×RN → R a Caratheodory function satisfying (5.1.18). If μ is a measure in Ω, we consider the problem −div A(x, ∇u) = g(x, u, ∇u) + μ u=0

in Ω on ∂Ω.

(6.2.7)

Existence of a renormalized solution to (6.2.7) is based upon LeraySchauder’s theorem.

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405

Suﬃcient condition

The next result proved in [Bidaut-V´eron (2002)] extends to the quasilinear case the result presented in the previous paragraph for the case p = 2 and q , has has to be compared with Lemma 5.2.5. We recall that qp = q+1−p been introduced in (5.2.27). Proposition 6.2.1

Assume μ is a bounded positive measure. If g satisﬁes q+1

∀(x, r, ξ) ∈ Ω × R × RN ,

rg(x, r, ξ) ≥ |r|

(6.2.8)

for some q > p − 1, then for any R > pqp any nonnegative renormalized solution u of problem (6.2.7) veriﬁes

g(x, u, ∇u)ζdx ≤ c1

ζdμ + Ω

ζ

Ω

1−R

pqRp |∇ζ| dx , R

(6.2.9)

Ω

for any ζ ∈ W01,p (Ω) ∩ W 1,s (Ω) (s > N ) satisfying 0 ≤ ζ ≤ 1, for some c1 > 0 depending on N , p, q, R and Ω. Furthermore, for any α < 0, there exists c2 = c2 (N, p, q, R, α, Ω) > 0 such that Rq p R α 1−R (1 + u) |∇u| ζdx ≤ c2 1 + g(x, u, ∇u)ζdx ζ |∇ζ| dx . Ω

Ω

Ω

(6.2.10)

+ Proof. We decompose μ = μ0 + μ+ s , μ0 0 α and α ∈ (1 − p, 0) we set hα,k, (r+ ) = (Tk (r+ ) + ) , uk = Tk (u). We 1,p choose ζhα,k, (u) for test function where ζ ∈ W0 (Ω) ∩ W 1,s (Ω) (s > N ), takes values in [0, 1]. We obtain, using (4.1.1), (uk + )α ζdμ0 + (k + )α ζdμ+ (uk + )α ζg(x, u, ∇u)dx s + Ω

Ω

Ω

A(x, ∇u), ∇u(uk + )α−1 ζdx

+ |α| Ω

A(x, ∇u), ∇ζ(uk + )α dx

= Ω

p−1 |∇ζ| (uk + )α dx + λ(k + )α ≤ λ |∇uk | Ω

|α| ≤ 2

p

|∇u| (uk + ) Ω

α−1

{u>k}

Ω

|∇u|p−1 |∇ζ| dx,

p−1

|∇ζ| dx

(uk + )α+p−1 ζ 1−p |∇ζ|p dx

ζdx + c3

+ λ(k + )α

{u>k}

|∇u|

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Local and global aspects of quasilinear elliptic equations

406

where c3 = c3 (α, λ). From H¨older’s inequality with θ =

1 1 pθ 1−pθ (uk + ) ζdx θ ζ |∇ζ| dx θ .

α+p−1 1−p

(uk +)

ζ

q p−1+α ,

p

|∇ζ| dx ≤

q

Ω

Ω

Ω

If we take k ≥ 1, we get

|α| 2

p

|∇u| (uk + )α−1 ζdx ≤ c3 Ω

1 1 pθ (uk + )q ζdx θ ζ 1−pθ |∇ζ| dx θ

Ω

+

{u>k}

Ω

|∇u|

p−1

|∇ζ| dx, (6.2.11)

and, letting → 0, yields

|α| 2

p

Ω

|∇u| uα−1 ζdx ≤ c3 k

Ω

+

{u>k}

1θ 1 θ pθ uqk ζdx ζ 1−pθ |∇ζ| dx Ω

|∇u|

p−1

|∇ζ| dx.

(6.2.12) Since u is a weak solution, we can take ζ for test function and derive

Ω

Ω

ζg(x, u, ∇u)dx =

ζdμ+ s +

ζdμ0 +

Ω

A(x, ∇u), ∇ζdx Ω

α−1 1−α p−1 |∇ζ| uk p dx + λ ≤ λ uk p |∇uk | Ω

≤

|∇uk |

+λ

Ω

{u>k}

p

1

uα−1 ζdx k

|∇u|

p−1

p

Ω

{u>k}

(1−α)(p−1) uk

|∇u| p

p−1

|∇ζ| ζ

|∇ζ| dx

1−p

1 p dx

|∇ζ| dx. (6.2.13)

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Since q > p − 1, we ﬁx α ∈ (1 − p, 0) such that τ =

ζg(x, u, ∇u)dx ≤ λ

ζdμ + Ω

{u>k}

Ω

|∇uk |

+ ≤λ 0

p

Ω

{u>k}

+ c3 Ω

× Ω

|∇u|

p−1

|∇u|

1 p

uα−1 ζdx k

Ω

p−1

407

q (1−α)(p−1)

|∇ζ| dx

τ1p

uqk ζdx

ζ

1−τ p

|∇ζ|

τ p

1 τ p dx

Ω

|∇ζ| dx 1 p1

θ1 1 θ pθ q 1−pθ uk ζdx ζ |∇ζ| dx +

{u>k}

Ω

τ1p 1 τ p τ p uqk ζdx ζ 1−τ p |∇ζ| dx. ,

|∇u|

p−1

|∇ζ| dx

Ω

from (6.2.12) and (6.2.13). Because g(., u, ∇u) + |∇u| using (6.2.8), we can let k → ∞ and obtain

ζg(x, u, ∇u)dx ≤ c4

ζdμ + Ω

Ω

p−1

∈ L1 (Ω), and

1 + τ1p p θ uq ζdx

Ω

1 1 p θ τ p pθ τ p 1−pθ 1−τ p ζ |∇ζ| dx ζ |∇ζ| dx .

× Ω

(6.2.14)

Ω

We notice that p1θ + derive in particular

> 1. Hence,

1 τ p

= 1−

1 qp

=

1 p θ

+

1 τp

. Using again (6.2.8) we

1− p−1 q ζg(x, u, ∇u)dx

Ω

≤ c4

ζ

1−pθ

|∇ζ|

1 1 (6.2.15) p θ τ p τ p 1−τ p dx ζ |∇ζ| dx .

pθ

Ω

Because τ < ζ

p q−1

1−pθ

Ω

< θ, we have by H¨ older’s inequality

|∇ζ|

pθ

dx ≤

ζ

Ω

1−τ p

Ω

1− τθ

≤ |Ω|

|∇ζ|

θ 1− θ τ τ dx ζdx Ω

ζ Ω

τ p

1−τ p

|∇ζ|

τ p

θ τ dx ,

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since 0 ≤ ζ ≤ 1. Combined with (6.2.15) it yields

ζg(x, u, ∇u)dx ≤ c5

ζ

Ω

1−τ p

|∇ζ|

τ p

qp τ dx .

(6.2.16)

Ω

Jointly with (6.2.14) it implies

ζdμ ≤ c6 Ω

ζ 1−τ p |∇ζ|

τ p

qp τ dx .

(6.2.17)

Ω

We take now α < 0 so that pqp < pτ =

q ≤ R, q + 1 − p − |α|(p − 1)

and we infer (6.2.9) from (6.2.16) and (6.2.17) by H¨older’s inequality. If we let k → ∞ in (6.2.11), we also deduce from the assumptions on g, |α| p ( + u)α−1 |∇u| ζdx 2 Ω 1θ 1 θ pθ 1−pθ (6.2.18) ≤ c6 g(x, u + , ∇u)ζdx ζ |∇ζ| dx Ω

≤ c2

Ω

Rp R 1−R g(x, u + , ∇u)ζdx ζ |∇ζ| dx ,

Ω

Ω

for any 0 ≤ ≤ 1, where c2 depends on N , p, q, α, R and Ω.

It follows from this result a series of necessary conditions in order to solve (6.2.7) with a measure data. Corollary 6.2.2 Assume g satisﬁes the assumptions of Proposition 6.2.1 and μ is a bounded nonnegative measure. If there exists a renormalized solution u ≥ 0 to problem (6.2.7), then for any R > pqp and any subdomain ¯ ⊂ Ω there exist two positive constants c1 depending on N , p, q, Ω, G⊂G G and R and c2 on N , p, q, Ω, G and R such that where μ M

ρR

μ M

ρR

≤ c1 ,

(6.2.19)

ρR d|μ| and ρ(x) = dist (x, ∂Ω, and

= Ω

μ(E) ≤ c2 cΩ 1,R (E)

∀E ⊂ G, E Borel.

(6.2.20)

We will see in Section 5.2.3 that this condition is not a necessary and suﬃcient condition for solving (6.2.7).

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409

The case 1 < p < N

The following general existence result for problem with forcing has to be compared with Theorem 5.1.2. It extends [Grenon (2002), Th 1.1] and [Bidaut-V´eron, Hamid (2010), Th 2.18]. Theorem 6.2.3 Let 2 − N1 ≤ p < N , Ω ⊂ RN be a bounded domain, A satisﬁes the assumptions (4.1.1) and g : Ω × R × RN → R veriﬁes (6.0.3), (5.1.18), ∞ p(N −1) N −1 g˜ s, s N −p s− N −p ds < ∞, (6.2.21) 1

and

g˜(as, bt) ≤ αa(p−1)˜q + βb(p−1)˜r g˜(s, t),

∀(a, b, s, t) ∈ R4+ ,

(6.2.22)

for some q˜ > 1, r˜ > 1 and α, β ≥ 0 verifying α2 + β 2 > 0. Then there exists m > 0 such that for any positive measure μ in Ω with μ M ≤ m there exists a renormalized solution u to (6.2.7). Proof. We recall that (5.1.18) is |g(x, s, ξ)| ≤ g˜(|s|, |ξ|)

∀(x, s, ξ) ∈ Ω × R × RN ,

where g˜ is a continuous function nondecreasing in its two variables. Furthermore (6.2.8) combined with (6.2.21) implies q < q1 = NN(p−1) −p . Let a ∈ C ∞ (R∗ ) such that a(r) = 1 for 0 ≤ r ≤ 1, a(r) = 0 for r ≥ 2 and a (r) ≤ 0. For n ∈ N∗ we set an (r) = a(n−1 r) and gn (x, r, ξ) = g (x, Tn (r), an (|ξ|)ξ)

∀(x, r, ξ) ∈ Ω × R × RN .

(6.2.23)

Then |gn (x, r, ξ)| ≤ g˜(Tn (|r|), an (|ξ|)|ξ|) := g˜n (|r|, |ξ|) ≤ g˜(|r|, |ξ|), and we notice that |gn (x, r, ξ)| ≤ g˜(n, 2n). Step 1: The approximation. We set N (p−1) N (p−1) X = v ∈ L N −p ,∞ (Ω) s.t. ∇v ∈ L N −1 ,∞ (Ω) . Since 2 −

1 N

≤ p < N , X is a Banach space for the norm v X = v

L

N (p−1) ,∞ N −p

+ ∇v

L

N (p−1) ,∞ N −1

.

(6.2.24)

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We notice that W01,p ⊂ X with continuous imbedding by GagliardoNirenberg inequality. In the same way as in the proof of Theorem 4.3.8 we consider an approximation of μ by the converging decomposition μ = −1,p (Ω). f − div g + μ+ s, (there is no negative singular part). Thus μ ∈ W If v ∈ X, we denote by u = Tn (v) the unique solution of −div A(x, ∇u) = gn (x, v, ∇v) + μ u=0

in Ω on ∂Ω.

(6.2.25)

Because the right-hand side of equation (6.2.25) is bounded in W −1,p (Ω), Tn (v) is uniformly bounded in W01,p (Ω) and therefore in X. If {vj } ⊂ X is such that vj X ≤ R, then up to a subsequence, gn (., vj , ∇vj ) converges in the weak-star topology of L∞ to some f ∈ L∞ (Ω). By RellichKondrachov theorem the imbedding of W01,p (Ω) in L1 (Ω) is compact, and by Schauder duality theorem the imbedding of L∞ (Ω) into W −1,p (Ω) shares this property ([Brezis (2010), Th 6.4]). Thus gn (., vj , ∇vj ) converges to f in W −1,p (Ω) which implies that uj = Tn (vj ) converges to some u in W01,p (Ω) and therefore in X and there holds −div A(x, ∇u) = f + μ u=0

in Ω on ∂Ω,

(6.2.26)

by Theorem 4.3.8. If we assume moreover that vj → v in X, then f = gn (., v, ∇v), thus u = Tn (v) which proves that Tn is compact. Step 2: The a priori bound. We ﬁrst show that g(., v, ∇v) is integrable in Ω if v ∈ X. Avσ = {x ∈ Ω : |v(x)| > σ}

and av (σ) = meas (Avσ ),

N −1

Bσv = {x ∈ Ω : |∇v| > σ N −p } and bvσ = meas (Bσv ), and Cσv = Avσ ∩ Bσv with cv (σ) = meas (Cσv ). Then $ % N (p−1) N (p−1) , |Ω| , av (σ) ≤ min σ − N −p v NN−p (p−1) ,∞ L

N −p

$ % N (p−1) N (p−1) bv (σ) ≤ min σ − N −p ∇v NN−1 , |Ω| , (p−1) ,∞ L

N −1

and

% $ $ % N (p−1) N (p−1) N (p−1) − N −p N −1 σ , ∇v , |Ω| . cv (σ) ≤ min min v NN−p (p−1) N (p−1) ,∞ ,∞ L

N −p

L

N −1

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We set N −p

v σ1 =

∇v NN−1(p−1) ,∞

N (p−1) ,∞ L N −p

|Ω|

N −1

L

, σ2 =

N −p N (p−1)

|Ω|

,

N −p N (p−1)

and σ3 = min{σ1 , σ2 }. Then

|g(x, v, ∇v)| dx ≤

Ω

g˜(|v|, |∇v|)dx Ω

g˜(|v|, |∇v|)dx +

≤

v Ω∩Cσ 3

v ∩(Av )c Ω∩Bσ σ3 3

g˜(|v|, |∇v|)dx

+ c v c Ω∩(Av σ3 ) ∩(Bσ3 )

g˜(|v|, |∇v|)dx +

v c Ω∩Av σ3 ∩(Bσ3 )

g˜(|v|, |∇v|)dx.

Because g˜ is nondecreasing in its two variables and (6.2.21) and (6.2.22) hold, we obtain

N −1

c v c Ω∩(Av σ3 ) ∩(Bσ3 )

g˜(|v|, |∇v|)dx ≤ g˜(σ3 , σ3N −p ) |Ω| (p−1)(N −1) r˜ (p−1)˜ q N −p g˜(1, 1). ≤ ασ3 + βσ3

Since dcv (σ) = 0 on [0, σ3 ),

g˜(|v|, |∇v|)dx = −

v Ω∩Cσ 3

$

≤ c1 min v

∞

N −1 g σ, σ N −p dcv (σ)

σ3 N (p−1) N −p N (p−1) ,∞ L N −p

≤ c2

∞

, ∇v

N (p−1) N −1 N (p−1) ,∞ L N −1 N −1

%

g˜(σ3 t, (σ3 t) N −p )t−

∞

N −1

g˜(σ, σ N −p )σ −

p(N −1) N −p

dσ

σ3 p(N −1) N −p

dt

1

(p−1)(N −1)˜ r (p−1)˜ q N −p ≤ c2 ασ3 + βσ3

1

∞

N −1

g˜(t, t N −p )t−

p(N −1) N −p

dt,

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where c2 = c2 (N, p, |Ω|) > 0. Similarly dav (σ) = 0 on [0, σ1 ), hence ∞ N −1 g˜(|v|, |∇v|)dx ≤ − g˜(σ, σ3N −p )dav (σ) v c Ω∩Av σ3 ∩(Bσ3 )

σ1

≤ c1 v

N (p−1) N −p N (p−1) ,∞ L N −p

∞

σ1

N −1

g˜(σ, σ3N −p )σ −

(p−1)(N −1)˜ r (p−1)˜ q N −p ≤ c2 ασ1 + βσ3

∞

g˜(t, 1)t−

p(N −1) N −p

dσ

p(N −1) N −p

dt,

p(N −1) N −p

dt.

1

and, mutatis mutandis, g˜(|v|, |∇v|)dx v ∩(Av )c Ω∩Bσ σ3 3

(p−1)(N −1)˜ r (p−1)˜ q N −p ≤ c2 ασ3 + βσ2

∞

N −1

g˜(1, t N −p )t−

1

Combining these inequalities, we infer (p−1)(N −1)˜ r (p−1)˜ q |g(x, v, ∇v)| dx ≤ c3 ασ1 + βσ2 N −p Ω

(p−1)˜ q (p−1)˜ r ≤ c4 α v N (p−1) ,∞ + β ∇v N (p−1) ,∞ , L

where

c3 = 4c2

∞

N −p

L

(6.2.27)

N −1

p(N −1) (p−N )q˜ (1−N )˜ r N −1 . g˜ s, s N −p s− N −p ds and c4 = c3 min |Ω| N , |Ω| N

1

Notice that these inequalities hold uniformly with respect to n if g is replaced by gn and g˜ by g˜n . If v ∈ X and u = Tn (v), then, by (4.3.26) and (4.3.27) in Proposition 4.3.9, p−1

u

L

N (p−1) ,∞ N −p

p−1

+ ∇u

L

N (p−1) ,∞ N −1

≤ c5

|gn (x, v, ∇v)| dx +

Ω

dμ , Ω

(6.2.28) with c5 = c5 (N, p, |Ω|) > 0. Combining (6.2.27) and (6.2.28) we derive q˜ r˜ u N (p−1) ,∞ + ∇u N (p−1) ,∞ ≤ c6 α v N (p−1) ,∞ + β ∇v N (p−1) ,∞ L

N −p

L

N −1

L

N −p

L

N −1

1 p−1

dμ

+ c7 Ω

(6.2.29)

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413

1

where α = α p−1 and β = β p−1 . For R > 0 we deﬁne a bounded closed convex subset KR by KR = {v ∈ X : v

L

N (p−1) ,∞ N −p

+ ∇v

L

N (p−1) ,∞ N −1

≤ R}.

(6.2.30)

If v ∈ KR and u = Tn (v), there holds by (6.2.29) u

+ ∇u

N (p−1) ,∞ L N −p

N (p−1) ,∞ L N −1

1 p−1

where A = c7

dμ

≤ c6 Rq˜ + Rr˜ + A

(6.2.31)

. Put Φ(R) = c6 Rq˜ + Rr˜ + A − R for R ≥ 0.

Ω

Then Φ is convex, Φ(0) = A and Φ(R) → ∞ when R → ∞. Furthermore Φ (R) = c6 q˜Rq˜−1 + r˜Rr˜−1 − 1, hence there exists a unique R0 > 0 such that Φ (R0 ) = 0. Consequently there exists a unique A0 such that Φ(R0 ) = 0. Therefore u

L

N (p−1) ,∞ N −p

+ ∇u

L

N (p−1) ,∞ N −1

− R0 ≤ Φ(R0 ) = 0,

(6.2.32)

which implies that u ∈ KR0 . By the Leray-Schauder theorem, Tn admits a ﬁxed point un in KR0 . Note that the value of R0 depends in N and p, but not on n, and it is the same with A0 . Step 3: Convergence. Let {un } ⊂ KR0 be a sequence of ﬁxed points of Tn . By (6.2.28) the set of functions {gn (., un ∇un )} is bounded in L1 (Ω). Let E ⊂ Ω be a Borel set and σ > 0. Following the method used in the proof of Theorem 5.1.2, we put Auσn = {x ∈ Ω : |un (x)| > σ}

Bσun = {x ∈ Ω : |∇un (x)| > σ

and aun (σ) = meas (Auσn ),

N −1+δ N −p

}

and bun (σ) = meas (Bσn ),

and Cσun = Auσn ∩ Bσun with cun (σ)(σ) = meas (Cσn ). Then N (p−1) max{aun (σ), bun (σ), cun (σ)} ≤ θun (σ) ≤ min cR s− N −p , |Ω| .

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Hence we get |g(x, un , ∇un )| dx ≤ E

un E∩Cσ

g˜(|un | , |∇un |)dx

+ un n c E∩Bσ ∩(Au σ )

g˜(|un | , |∇un |)dx

+

un c n E∩Au σ ∩(Bσ )

g˜(|un | , |∇un |)dx

+

un c n c E∩(Au σ ) ∩(Bσ ) ∞

g˜(|un | , |∇un |)dx

N −1

g˜(t, t N −p )t−

≤ c8

p(N −1) N −p

N −1

+ g˜(σ, σ N −p ) |E| .

σ

This implies that the set of functions {g(., un , ∇un )} is uniformly integrable in Ω. Since it is bounded in L1 (Ω), it is relatively weakly compact by the Dunford-Pettis theorem. Furthermore Tk (un ) converges to Tk (u ) in W01,p (Ω) for any k > 0 and ∇un → ∇u in (Lr (Ω))N ) for all r < NN(p−1) −1 (see Step 6 in the proof of Theorem 4.3.8). Therefore, up to a subsequence, gn (., un , ∇un ) converges weakly to some f ∈ L1 (Ω). By Theorem 4.3.8 {Tn (un )} converges a.e. in Ω to a renormalized solution u of −div A(x, ∇u ) = f + μ u = 0

in Ω on ∂Ω.

(6.2.33)

Furthermore, Tk (un ) converges to Tk (u ) in W01,p (Ω) for any k > 0 and ∇un → ∇u in (Lr (Ω))N ) for all r < NN(p−1) −1 (see Step 6 in the proof of Theorem 4.3.8). Up to a subsequence ∇un → ∇u a.e. in Ω. By the Vitali convergence theorem we infer that gn (., un , ∇un ) → g(., u , ∇u ) = f in L1 (Ω) which proves that u is a renormalized solution of −div A(x, ∇u ) = g(., u , ∇u ) + μ u = 0

in Ω on ∂Ω.

(6.2.34)

Step 4: End of the proof. Since the set {g(., u , ∇u )} is uniformly bounded in L1 Ω, it is uniformly integrable as in Step 3 thanks to Theorem 4.3.8. We ﬁrst obtain that g(., u , ∇u ) converges weakly (up to a subsequence) in L1 Ω to some f , then u → u a.e. in Ω. Then u is a renormalized solution of −div A(x, ∇u) = f + μ u=0

in Ω on ∂Ω.

(6.2.35)

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Next Tk (u ) to Tk (u) in W01,p (Ω) and ∇u → ∇u in (Lr (Ω))N ) for all r < NN(p−1) −1 . We use again Vitali’s theorem and derive that f = g(., u, ∇u) and u is a renormalized solution of (6.2.7). Applications - If g(x, u, ∇u) = |u|q then (6.2.21) and (6.2.22) are satisﬁed provided p − 1 < q < q1 . r

- If g(x, u, ∇u) = |∇u| then (6.2.21) and (6.2.22) are satisﬁed provided p − 1 < r < qc = NN(p−1) −1 . r

- If g(x, u, ∇u) = |u|q + |∇u| , (6.2.21) is satisﬁed if p − 1 < q < q1 and p − 1 < r < qc . Since (a|u|)q + (b |∇u|)r ≤ (aq + br )(|u|q + |∇u|r )

∀a, b ≥ 0,

(6.2.22) is veriﬁed. - If g(x, u, ∇u) = |u|q |∇u|r with qr > 0, we write brκ aqκ r r q r q r q + |u|q |∇u| . (a|u|) (b |∇u|) = a b |u| |∇u| ≤ κ κ The condition p − 1 < min{κq, κ r} is satisﬁed for some κ ∈ (1, ∞) if q + r > p − 1, and (6.2.21) holds if 0 < q(N − p) + r(N − 1) < N (p − 1). 6.2.2.3

The case p = N

When p = N the Gagliardo-Nirenberg inequality does not apply since W01,N (Ω) does not imbed in L∞ (Ω). The corresponding inequality is due to [Pohozaev (1965)], [Moser (1967)] and [Trudinger (1980)]. The extension of Moser-Pohozaev-Trudinger inequality to Sobolev-Lorentz spaces is essentially obtained in [Brezis, Wainger (1980)] (under a slightly diﬀerent form) and in [Alvino, Ferone, Trombetti (1996)]. Proposition 6.2.4 Let Ω be a bounded domain in RN . For any 1 < q ≤ ∞ there exists cN,q > 0 such that q q−1 sup eθN,q |u(x)| dx ≤ cN,q |Ω| , (6.2.36) ∇v LN,q ≤1 1 N

where θN,q = (N ωN )

q q−1

Ω

and ωN is the volume of the unit ball in RN . 1

In the sequel we set θN,∞ = θN = N ωNN and cN,∞ = cN . A consequence of this estimates is the following result.

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Lemma 6.2.5 Let Ω be a bounded domain in RN . Then for any v ∈ C01 (Ω) there holds $ % θN σ − av (σ) ≤ min cN e ∇v LN,∞ , 1 |Ω| , (6.2.37) where Avσ = {x ∈ Ω : |v(x)| > σ} any 1 < m < ∞,

and av (σ) = meas (Avσ ). Moreover, for

! m av (σ) ≤ min dN,m ∇v LN,∞ σ −m , 1 |Ω| ,

(6.2.38)

−m Γ(m + 1). where dN,m = cN θN

Proof. Put v˜ =

v ∇v LN,∞

a (σ ∇v LN,∞ )e v

. Since inequality

θN σ

v ˜

= a (σ )e

θN σ

eθN |˜v (x)| dx ≤ cN |Ω| ,

≤ ˜ Av σ

holds by (6.2.36) with q = ∞, (6.2.37) follows if we set σ = σ ∇v LN,∞ . Furthermore, for m > 1, m θN |v(x)|m dx ≤ cN |Ω| ∇v m LN,∞ , Γ(m + 1) Ω (Γ(m + 1) = r! if m ∈ N∗ ) which yields 1

−1 v Lm ≤ θN (cN Γ(m + 1) |Ω|) m ∇v LN,∞ ,

(6.2.39)

and (6.2.38) follows.

The next result is the mere extension of Theorem 6.2.3 to the case p = N. Theorem 6.2.6 Let Ω ⊂ RN be a bounded domain, A satisﬁes the assumptions (4.1.1) with p = N and g : Ω × R × RN → R veriﬁes (6.0.3), (5.1.18), ∞ q g˜ s, s N s−q−1 ds < ∞, (6.2.40) 1

for q ≥ 1 and

g˜(as, bt) ≤ αa(N −1)˜q + βb(N −1)˜r g˜(s, t),

∀(a, b, s, t) ∈ R4+ ,

(6.2.41)

for some q˜ > 1, r˜ > 1 and α, β ≥ 0 verifying α2 + β 2 > 0. Then there exists m > 0 such that for any positive measure μ in Ω with μ M ≤ m there exists a renormalized solution u to (6.2.7).

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Proof. Up to some modiﬁcations that we indicate the proof follows the one of Theorem 6.2.3 with the same notations. We deﬁne Avσ and av (σ) as in Theorem 6.2.3 and we have ! q av (σ) ≤ min dN,q ∇v LN,∞ σ −q , 1 |Ω| . We set

q Bσv = x ∈ Ω : |∇v| > σ N and bv (σ) = meas(Bσv ),

then

N −1 N bv (σ) ≤ min ∇v LN,∞ σ −q , |Ω| = min |Ω| ∇v LN,∞ σ −q , 1 |Ω| ,

and cv (σ) = meas(Bσv ∩ Avσ ), then q −1 N cv (σ) ≤ min min dN,q ∇v LN,∞ , |Ω| ∇v LN,∞ σ −q , 1 |Ω| . We put 1

− q1

q ∇v LN,∞ , σ2 = |Ω| σ1 = dN,q

N

∇v LqN,∞ and σ3 = min{σ1 , σ2 }.

After some computation (6.2.27) is transformed into (N −1)˜ r (N −1)˜ q N , |g(x, v, ∇v)| dx ≤ c1 ασ1 + βσ2 Ω

where c1 depends on N , q, |Ω| and holds u BMO + ∇u LN,∞ ≤ c2

6∞ 1

(6.2.42)

q

g(t, t N t−q−1 dt. If u = Tn (v), there

∇v qL˜ N,∞

+

˜ ∇v rL N,∞

N1−1

+

dμ

,

Ω

(6.2.43) where c2 depends on c1 . When the integration is not performed on whole Ω but on a Borel set E, the condition (6.2.40) yields the uniform integrability of the sequence {gn (., un , ∇un } and {g(., u, ∇u } as in Theorem 6.2.3. The conclusion of the proof follows. Applications ∗

∗

If g(x, u, ∇u) = 1 |u|q + 2 |u|q |∇u|r + 3 |∇u|r , then the assumptions of Theorem 6.2.6 are satisﬁed if N − 1 < q and N − 1 < r < N . Theorem 6.2.7 Let Ω ⊂ RN be a bounded domain, A satisfy the assumptions (4.1.1) with p = N and g : Ω × R × RN → R verify (6.0.3),

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(5.1.18), ∀(s, t) ∈ R2+ ,

g˜(s, t) ≤ α sinh(λs) + βtν

(6.2.44)

for 0 < ν < N , some α, β, λ ≥ 0 such that α2 + β 2 > 0. Then for any λ0 > 0 there exists m0 = m0 (Ω, ν, N, λ0 ) > 0 such that for any (α, β, λ) ∈ R+ × R+ × [0, λ0 ] and any positive measure μ in Ω, the condition αλ + β + μ(Ω) ≤ m0 ,

(6.2.45)

implies that there exists a renormalized solution u to (6.2.7). Proof. We proceed as in Theorem 6.2.3 using the Moser-Trudinger type estimates as in the case p = N in Theorem 5.1.2. We deﬁne gn and g˜n by (6.2.23) and denote by XN the space of functions v in BM O(Ω) such that ∇v LN,∞ It is known that BM O(Ω) ⊂ XN [Tartar (1998)] and (4.3.43) holds, therefore XN is a Banach space for the norm v XN = ∇v LN,∞ ∼ v BMO + ∇v LN,∞ .

Step 1: Estimate of g(., v, ∇v) L1 . We assume that v ∈ XN and deﬁne for σ > 0, Avσ = {x ∈ Ω : |v(x)| > σ} and av (σ) = meas (Avσ ). From (6.2.37), we get for σ > 0, θN σ , 1 |Ω| . av (σ) ≤ min 2−N cN cschN N ∇v N,∞

(6.2.46)

L

Next, for some constant c > 0 which will be made precise later on, Bσv = {x ∈ Ω : |∇v(x)| > sinh(cσ)} and bv (σ) = meas (Bσv ), then N csch (cσ), |Ω| . bv (σ) ≤ min ∇v N LN,∞ If we take c =

θN N ∇v LN,∞

, we obtain

N θN σ , |Ω| . bv (σ) ≤ min ∇v LN,∞ cschN N ∇v N,∞ L

(6.2.47)

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Finally Cσv = Avσ ∩ Bσv , cv (σ) = meas (Cσv ), hence N θN σ , |Ω| . cv (σ) ≤ min min ∇v LN,∞ , 2−N cN |Ω| cschN N ∇v LN,∞ (6.2.48) Set 1 N N ∇v LN,∞ N ∇v LN,∞ cN ∇v LN,∞ , σ2 = , arsinh arsinh σ1 = 1 θN 2 θN |Ω| N and σ3 = min{σ1 , σ2 }. As in the proof of Theorem 6.2.3 we write |g(x, v, ∇v)| dx ≤ g˜(|v|, |∇v|)dx Ω

Ω

g˜(|v|, |∇v|)dx +

≤ v Ω∩Cσ 3

g˜(|v|, |∇v|)dx

v ∩(Av )c Ω∩Bσ σ3 3

g˜(|v|, |∇v|)dx +

+

c v c Ω∩(Av σ3 ) ∩(Bσ3 )

g˜(|v|, |∇v|)dx.

v c Ω∩Av σ3 ∩(Bσ3 )

Firstly

θN σ3 g˜(|v|, |∇v|)dx ≤ g˜ σ3 , sinh( N ∇v ) |Ω| N,∞

c v c Ω∩(Av σ3 ) ∩(Bσ3 )

where c1 = arsinh

1

N cN 2

L

≤ g˜ σ3 , sinh( c1σσ1 3 ) |Ω| ,

. Then

∞

g˜(|v|, |∇v|)dx ≤ −

v c Ω∩Av σ3 ∩(Bσ3 )

≤− ≤

−N

2 c N θN ∇v LN,∞

≤ 2−N N cN c1

σ3 ∞

σ1

θN σ3 g˜ σ, sinh( N ∇v ) dav (σ) N,∞ L

g˜ σ, sinh( c1σσ1 3 ) dav (σ)

N θN σ θN σ csch dσ g˜ σ, sinh( c1σσ1 3 ) coth N ∇v N ∇v N,∞ N,∞

∞

σ1

1

L

L

g˜ σ1 t, sinh( c1σσ1 3 ) coth (c1 t) cschN (c1 t) dt

∞

≤ c2 1

g˜ σ1 t, sinh( c1σσ1 3 ) cschN (c1 t) dt,

∞

where c2 = 2−N N cN c1 coth c1 . Next

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g˜(|v|, |∇v|)dx ≤ −

v ∩(Av )c Ω∩Bσ σ3 3

≤ θN

∞

g˜ (σ3 , sinh(cσ)) dbv (σ) σ3

g˜(σ3 , sinh(cσ)) coth(cσ) cschN (cσ)dσ

σ2

≤

∞

N c1 ∇v N LN,∞

∞ σ2 σ1

g˜ (σ3 , sinh(c1 t)) coth(c1 t) cschN (c1 t)dt.

Finally

g˜(|v|, |∇v|)dx ≤ −

v Ω∩Cσ 3

∞

g˜(σ, sinh(cσ))dcv (σ) σ3

N ≤ N c1 min ∇v LN,∞ , 2−N cN |Ω| ×

∞ σ3 σ1

g˜(σ1 t, sinh(c1 t)) coth(c1 t) cschN (c1 t) dt.

Using (6.2.44) we obtain

g˜(|v|, |∇v|)dx ≤ α sinh(λσ3 ) + β sinhν ( c1σσ1 3 ) |Ω| ,

c v c Ω∩(Av σ3 ) ∩(Bσ3 )

g˜(|v|, |∇v|)dx

v c Ω∩Av σ3 ∩(Bσ3 )

≤ c2

∞ 1

≤ c3

α sinh(λσ1 t) + β sinhν

c1 σ3 σ1

cschN (c1 t) dt,

αλσ1 β ν c1 σ3 , + sinh σ1 N 2 c21 − λ2 σ12 N c1

where c3 depends only on N , g˜(|v|, |∇v|)dx v ∩(Av )c Ω∩Bσ σ3 3

≤ N c1 ∇v N LN,∞

∞ σ2 σ1

(α sinh(λσ3 ) + β sinhν (c1 t)) coth(c1 t) cschN (c1 t) dt

Nβ N cschN −ν c1σσ1 2 , ≤ ∇v LN,∞ α sinh(λσ3 ) cschN c1σσ1 2 + N −ν

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421

N g˜(|v|, |∇v|)dx ≤ N c1 min ∇v LN,∞ , 2−N cN |Ω|

v Ω∩Cσ 3

×

∞ σ3 σ1

(α sinh(λσ1 t) + β sinhν (c1 t)) coth(c1 t) cschN (c1 t) dt,

which is estimated in two steps: Firstly

1 σ3 σ1

α sinh(λσ1 t) coth(c1 t) cschN (c1 t)dt = +

αλc1 N

1 σ3 σ1

αc1 sinh(λ σσ31 ) cschN c1 σσ31 N

cosh(λσ1 t) cschN (c1 t)dt

1−N −N αλ cosh(λ σσ31 ) αc1−N σ3 σ3 1 + sinh(λ σσ31 ) −2 N σ1 σ1 N (N − 1)cN 1 1−N σ3 2αλ cosh λ ≤ . −2 σ1 (N − 1)cN 1 ≤

Hence ∞

α sinh(λσ1 t) coth(c1 t) cschN (c1 t)dt ≤ c4 α

1

∞

sinh(λσ1 t)e−c1 N t dt

1

c4 αλσ1 , ≤ 2 2 N c1 − λ2 σ12 where c4 = c4 (N ) > 0. From these inequalities it follows N g˜(|v|, |∇v|)dx ≤ N c1 min ∇v LN,∞ , 2−N cN |Ω| v Ω∩Cσ 3

×

Nβ 2αλ cosh λ cschN −ν c1σσ1 3 + −2 N −ν (N − 1)cN 1

σ3 σ1

1−N

c4 αλσ1 + 2 2 N c1 − λ2 σ12

.

Step 2: The a priori estimates. For R > 0 we set KR = {v ∈ XN : v XN ≤ R}, with R ≤ Rκ , where

# Rκ = min

1

(cN |Ω|) N κθN , 2 λ

) ,

(6.2.49)

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R , for κ ∈ (0, 1). If v ∈ XN there holds σ1 ≤ Ncc21 R , σ2 ≤ Nc2R arsinh 1 |Ω| N ∇v LN,∞ N σ3 = σ2 , c1 σσ21 = arsinh , min ∇v LN,∞ , 2−N cN |Ω| = 1 |Ω| N

N ∇v LN,∞

and ≤ (1−καλ Summarizing the 2 )θ c ∇v LN,∞ . N 1 estimates from step 1, we infer +1 2 g˜(|v|, |∇v|)dx ≤ c6 αλ ∇v N LN,∞ + ∇v LN,∞ + cosh λ ∇v LN,∞ αλσ1 N 2 c21 −λ2 σ12

Ω

ν

+ c7 β ∇v LN,∞ , (6.2.50) where c6 , depends on N , κ and |Ω|, and c7 on N , ν and |Ω|. In particular, since v ∈ KR , g˜(|v|, |∇v|)dx ≤ c5 αλ RN +1 + R2 + cosh λ R + c6 βRν . (6.2.51) Ω

Step 3: End of the proof. We proceed as in the proof of Theorem 6.2.3Steps 2,3 in deﬁning the same operator Tn in X by (6.2.25). For the same reason as here above, Tn is a compact operator and the only problem is to prove that Tn (KR ) ⊂ KR . If u = Tn (v) there holds if v ∈ KR , −1 ∇u N ≤ c |g (x, v, ∇v)| dx + dμ N,∞ 5 n L Ω

≤ c8 βR + αλ R ν

≤ c8 αλR

N +1

Ω

N +1

2

+ R + cosh λ R

+ c5 μ(Ω)

(6.2.52)

+ c5 μ(Ω) + c8 (β + αλ(1 + cosh λ))

+ c8 (βM (ν) + αλ(cosh λ M (1) + M (2))) RN −1 , since, by Young’s inequality, τ R ≤1+ N −1 τ

1

there holds, for any 0 < τ ≤ N − 1,

N −1 N −1−τ

1− Nτ−1 RN −1 := 1 + M (τ )RN −1 .

We put . N +1 Ψ(S) = c8 αλS N −1 − 1 − c8 (βM (ν) + αλ(cosh λ M (1) + M (2))) S + A, 1 Here we assume that N > 3 in order M (2) be defined. When N = 3 only M (ν) and M (1) need to be introduced.

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where A = c8 (β + αλ(1 + cosh λ)) + c5 μ(Ω). Then Ψ is convex on [0, ∞) and . 2 N +1 αλS N −1 − 1 − c8 (βM (ν) + αλ(cosh λ M (1) + M (2))) . Ψ (S) = c8 N −1 If we take λ ≤ λ0 , arbitrarily and then αλ and β small enough so that 1−c8 (βM (ν) + αλ(cosh λ0 M (1) + M (2))) > 0, there exists a unique R0 > 0 such that Ψ (R0N −1 ) = 0. Furthermore R0 can be assumed smaller than R∗ by increasing αλ or β. Hence there exists m0 > 0 such that the identity c8 (β + αλ(1 + cosh λ)) + c5 μ(Ω) = m0 implies that . c8 αλR0N +1 − 1 − c8 (βM (ν) + αλ(cosh λ M (1) + M (2))) R0N −1 + m0 = 0. This yields Tn (KR ) ⊂ KR . Since Tn is compact, there exists un ∈ KR such that Tn (un ) = un . We write the equation under the form −divA(x, ∇un ) = gn (x, un , ∇un ) + β := M,n un = 0

in Ω on Ω.

(6.2.53)

The measure M,n is positive and bounded by Step 1. Since β → μ in the sense of measures, there exists m > 0, independent of and n such that m−1 ≤ M,n M ≤ m.

(6.2.54)

By Proposition 4.3.11 there exists a constant c9 > 0 such that 1

1

(i)

N −1 ≤ c9 m N −1 un BMO(Ω) ≤ c9 M,n M

(ii)

N −1 M,n M m N −1 meas {|∇un | > σ} ≤ c9 ≤ c9 N . N σ σ

N

N

(6.2.55)

We put −1 A˜uσn = x ∈ Ω : |un (x)| ≥ N θN ∇un LN,∞ σ

!

and a ˜un (σ) = meas(A˜uσn ),

˜σun ), ˜σun = {x ∈ Ω : |∇un (x)| ≥ sinh σ} and ˜bun (σ) = meas(B B and ˜ un and c˜un (σ) = meas(C˜ un ). C˜σun = A˜uσn ∩ B σ σ By (6.2.46) and (6.2.55)-(ii) there holds a ˜un (σ) ≤ min 2−N cN cschN (σ), 1 |Ω| ,

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and ˜bun (σ) ≤ min c9 m NN−1 cschN (σ), |Ω| . The set of functions {gn (., un , ∇un )} (with the notations of the proof of Theorem 6.2.3) is uniformly bounded in L1 (Ω) since un ∈ KR . In order to prove its uniform integrability, we consider σ > 0 and a Borel set E ⊂ Ω that we decompose into ˜ un ∩(A˜un )c )∪(E∩A˜un ∩(B ˜ un )c )∪(E∩(B ˜ un )c ∩(A˜un )c ). E = (E∩C˜σun )∪(E∩B σ σ σ σ σ σ Then, as in Step 1, and since ∇un LN,∞ ≤ R,

∞ −1 −1 u gn (., un , ∇un )dx ≤ g˜ N θN Rσ, sinh σ |E| − g˜ N θN Rt, sinh σ d˜ a n (t)

E

−

σ

∞

σ

−1 g˜ N θN Rσ, sinh t d˜bun (t) −

∞

σ

−1 u g˜ N θN Rt, sinh t d˜ c n (t). N

Up to changing m we can assume that 2−N cN |Ω| ≤ c9 m N −1 , therefore N max{˜ aun (t), ˜bun (t), c˜un (t)} ≤ min c9 m N −1 cschN (t), |Ω| . Using the previous inequalities and some integrations by parts, we derive E

−1 gn (., un , ∇un )dx ≤ g˜ N θN Rσ, sinh σ |E| N

∞

+ 3N c9 m N −1 σ

−1 g˜ N θN Rt, sinh t coth t cschN (t)dt.

−1 −1 −1 Since g˜ N θN Rt, sinh t ≤ α sinh(λN θN Rt) + β sinhν t, λN θN R ≤ N N −N t (t) ∼ 2 e as t → ∞, the function t → Nκ and coth t csch −1 Rt, sinh t coth t cschN (t) is integrable on (0, ∞). The same arg˜ N θN gument as in the proof of Theorem 5.1.2 yields the uniform integrability of {gn (., un , ∇un )}. Letting n → ∞ we obtain the existence of a solution of (6.2.34). Since {g(., u , ∇ug e)} satisﬁes the same estimates as {gn (., un , ∇un )}, we end the proof as in Theorem 6.2.3-Steps 4.

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425

The supercritical case Necessary and suﬃcient conditions

In this section we consider the problem −divA(x, ∇u) = uq + μ u=0

in Ω (6.2.56) on ∂Ω,

where Ω ⊂ RN is a domain, A : Ω × RN → RN satisﬁes (4.1.1) and μ is a positive measure in Ω. The critical exponent q for this equation is again qc . As for the similar problem (5.4.2) with absorption treated in Theorem 5.4.6, the treatment of the supercritical case q ≥ q1 with a source reaction necessitates potential estimates and Bessel or Riesz capacities. The following results have been published in [Phuc, Verbitsky (2006)], [Phuc, Verbitsky (2008)]. The next local estimate is a sharpening of Proposition 6.2.1. Theorem 6.2.8 Assume 1 < p < N , p − 1 < q, μ is a positive measure in Ω and u is a nonnegative locally renormalized solution of −divA(x, ∇u) = uq + μ

in Ω.

(6.2.57)

Then for any closed cube Q ⊂ Ω and any compact set E ⊂ Q there holds N (i) uq dx + μ(E) ≤ c∗1 c˙R if q > q1 , p,qp (E) E (6.2.58) N uq dx + μ(E) ≤ c∗2 cR (E) if p − 1 < q ≤ q , (ii) 1 p,qp E

q R R , q1 = (p−1)N with qp = q+1−p N −p , where c˙ p,qp and cp,qp are respectively the Riesz and the Bessel capacities and where the constant c∗1 depends only on N , p and q, while c∗2 depends on N , p, q and the length of the side of Q. Moreover, if Ω is bounded and q > q1 , Ω uq dx + μ(E) ≤ c∗3 Cp,q (E) ∀E ⊂ Ω, E compact, (6.2.59) p N

N

E

Ω denotes the fractional Sobolev where c∗3 depends on N , p, q and Ω and Cp,q p p,qp capacity associated to the space W0 (Ω).

The proof uses the local lower estimate of [Kilpel¨ainen, Mal` y (1994)] in terms of Wolﬀ potential that every p-superharmonic satisﬁes. One of its main originality lies in the construction of a discrete model of this equation using dyadic decomposition of Ω [Phuc, Verbitsky (2006)]. The next metric

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estimate follows easily (see e.g. [Adams, Hedberg (1999), Prop 5.1.2 and 5.1.3]) Corollary 6.2.9 Under the assumptions of Theorem 6.2.8 there holds uq dx + μ(Br ) ≤ c∗4 rN −pqp if q > q1 , (i) Br

u dx + μ(Br ) ≤ q1

(ii) Br

c∗5

ln

(6.2.60)

2R 1− Np

if q = q1 ,

r

for all r > 0 such that B r ⊂ Ω in case (i) and B r ⊂ BR ⊂ B2R ⊂ Ω in case (ii). The following inequality is a suﬃcient condition for the existence of a renormalized solution to (6.2.56). Theorem 6.2.10 Assume Ω ⊂ RN is a bounded domain, 1 < p < N and p − 1 < q and μ is a positive bounded measure in Ω. If there holds 2diam(Ω)

W1,p

2diam(Ω)

[W1,p

[μ]q ] ≤ C ∗ W1,p

2diam(Ω)

[μ]

a.e. in Ω,

(6.2.61)

where ∗

C =

q+1−p qc max{1, 2p −2 }

q(p −1)

p−1 q+1−p

,

(6.2.62)

and where c = c(N, p) > 0 is the constant appearing in estimate (5.4.3) in Theorem 5.4.1, then there exists a nonnegative renormalized solution u to (6.2.56). Furthermore there exists m > 0, depending on N , p, q and the constant λ appearing in (4.1.1) such that 2diam(Ω)

u ≤ mW1,p

[μ]

a.e. in Ω.

(6.2.63)

Proof. We consider a sequence {uk } of nonnegative functions deﬁned by the following scheme: u0 is a renormalized solution of −divA(x, ∇u0 ) = μ u0 = 0

in Ω (6.2.64) on ∂Ω,

and, for k ∈ N∗ , uk is a renormalized solution of −divA(x, ∇uk ) = uqk−1 + μ uk = 0

on ∂Ω.

in Ω (6.2.65)

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By regularizing μ as we did in the proof of Theorem 5.4.5 we can assume that the sequence uk is nondecreasing. By (5.4.3) there holds 2diam(Ω)

u0 ≤ cW1,p

2diam(Ω)

[μ] and uk ≤ cW1,p

[uqk−1 + μ]

∀k ≥ 1. (6.2.66)

The second identity implies 2diam(Ω) q 2diam(Ω) [uk−1 ] + W1,p [μ] . uk ≤ c max 1, 2p −2 W1,p By induction we construct a sequence {ck } with the property that uk ≤ 2diam(Ω) [μ] such that c0 = c and ck W1,p q(p −1) ck = c max 1, 2p −2 ck−1 C ∗ + 1 ∀k ≥ 1. Since ck ≤ cqp max 1, 2p −2 , we conclude that the sequence {uk } is 2diam(Ω)

[μ] for some m > 0 and that uqk is bounded from above by mW1,p dominated by a ﬁxed integrable function by (6.2.61). We end the proof by Theorem 4.3.8. The next result proved in [Phuc, Verbitsky (2006)] exhibits a necessary and suﬃcient condition for the solvability of (6.2.56). Theorem 6.2.11 Assume Ω ⊂ RN is a bounded domain, 1 < p < N and p − 1 < q and μ is a positive measure with compact support in Ω and set R = diam(Ω). Then the following assertions are equivalent. (i) There exists some 0 > 0 such that for any 0 ≤ ≤ 0 there exists a nonnegative renormalized solution u to −divA(x, ∇u) = uq + μ u=0

in Ω (6.2.67) on ∂Ω.

(ii) There exists c1 > 0 such that for all compact set K ⊂ Ω there holds μ(K) ≤ c1 cR p,qp (K). N

(6.2.68)

(iii) There exists c2 > 0 such that for any ball B ⊂ Ω, q 2diam(Ω) W1,p [χB μ](x) dx ≤ c2 μ(B).

(6.2.69)

B

(iv) There exists c3 > 0 such that 2R 2R 2R [W1,p [μ]q ] ≤ c3 W1,p [μ] W1,p

a.e. in Ω.

(6.2.70)

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6.2.3.2

Removable singularities

The previous results allow us to characterize sets which are removable for any positive solution of (6.2.57). Theorem 6.2.12 Assume Ω ⊂ RN is any domain, 1 < p < N and p − 1 < q and E ⊂ Ω is compact. Any nonnegative locally renormalized solution u ∈ Lqloc (Ω \ E) of −divA(x, ∇u) = uq

(6.2.71)

in Ω \ E is a locally renormalized solution in Ω if and only if cR p,qp (E) = 0. N

R Proof. Assume that cR p,qp (E) = 0. Then c1,p (E) = 0. Since u is psuperharmonic in Ω \ K, it follows from Theorem 4.3.6 that it is psuperharmonic in Ω and thus there exists a nonnegative Radon measure μ in Ω such that (6.2.57) holds. By Corollary 6.2.9, uq ∈ L1loc (Ω), and by N Theorem 6.2.8, μ vanishes on sets with zero cR p,qp -capacity, in particular on E. Since μ(K c ) = 0 it follows that μ = 0 and (6.2.71) holds in Ω. The N reverse assertion is true since, if cR p,qp (E) > 0 there exists a capacitary measure with support in E and this measure satisﬁes (6.2.68) since it belongs q to the dual space L−p, p−1 (RN ). N

6.3

N

Quasilinear Hamilton-Jacobi type equations

Let Ω ⊂ RN be a domain and p > q > p − 1 > 0. This section is concerned with properties of the solutions of −Δp u = |∇u|

q

(6.3.1)

in Ω or Ω \ {0}. By opposition to the Lane-Emden equation, the a priori estimates are easy to obtain and have already been proved in Section 3.4.3. 6.3.1

Separable solutions of quasilinear Hamilton-Jacobi equations

We denote by (r, σ) ∈ R+ × S N −1 the spherical coordinates in RN . Any separable solution of (6.3.1) has the form u(x) = u(r, σ) = r−γp,q ω(σ),

(6.3.2)

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p−q where γp,q = q+1−p (see Section 5.5) and ω is a solution of 2 2 p−2 q 2 2 2 2 γp,q ω + |∇ ω| ∇ ω − γp,q ω + |∇ ω|2 2 −div p−2 2 2 − γp,q Λ(γp,q ) γp,q ω + |∇ ω|2 2 ω = 0 in S N −1 .

(6.3.3)

We recall that Λ(γp,q ) = γp,q ((p − 1)γp,q + p − N ). A necessary and suﬃcient condition for the existence of a positive solution is Λ(γp,q ) < 0, that is γp,q

qc := . p−1 N −1

(6.3.4)

If this is satisﬁed there exists a singular radial solution of (6.3.1) in RN \{0}, ˜s (r) = μ ˜N,p,q r−γp,q , U

(6.3.5)

where 1

−1 (N − p − (p − 1)γp,q ) p+q−1 . μ ˜N,p,q = γp,q

(6.3.6)

It is easy to check that there exists no branch of solutions bifurcating from the value (q, μ ˜N,p,q ), contrary to the Lane-Emden equation (see Proposition 6.1.1). N −1 Singular solutions in RN } vanishing on + = {(r, σ) : r > 0 , σ ∈ S+ \ {0}, under the form (6.3.2) are solutions of 2 2 p−2 q 2 2 2 2 γp,q ω + |∇ ω| −div ∇ ω − γp,q ω + |∇ ω|2 2 p−2 2 2 N −1 − γp,q Λ(γp,q ) γp,q ω + |∇ ω|2 2 ω = 0 in S+

∂RN +

N −1 on ∂S+ . (6.3.7)

ω=0 The critical exponent for existence is deﬁned by (3.5.3): βS N −1 = γp,q ⇐⇒ q := q˜c = p − +

βS N −1 +

βS N −1 + 1

.

+

It is the same as in the absorption case. Theorem 6.3.1

Assume p > 1, then the following holds.

(i) If p − 1 < q ≤ q˜c there exists no nontrivial nonnegative solution of (6.3.7).

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(ii) If q˜c < q < p there exists at least one positive solution ω := ωh of (6.3.7). The next lemma will be used in the proof. Lemma 6.3.2 Assume q˜c < q < p, then the set of positive solutions of (6.3.7) is bounded. Proof. We recall that any solution of (6.2.56) in RN + satisﬁes (3.4.32), i.e. |∇u(x)| ≤ cN,p,q (ρ(x))

1 − q+1−p

∀x ∈ RN +,

N −1 , where ρ(x) = dist (x, ∂RN + ) = xN . If σN denotes the north pole of S+ this implies

|u(r, σN ) − u(1, σN )| ≤

q+1−p cN,p,q r−γp,q −1 p−q

∀r ∈ (0, 1]. (6.3.8)

If u(r, σ) = r−γp,q ω(σ) it yields in particular |ω(σN )| ≤

q+1−p cN,p,q . p−q

(6.3.9)

N −1 we In order to extend this estimate from the point σN to whole S+ use [Pucci, Serrin (2007), Th 8.2.2] which asserts that a positive solution of a nondegenerate quasilinear equation in a ball B which vanishes on ∂B is radially symmetric. Their proof is performed by using the moving plane method which can be easily adapted to the geometric setting on the sphere provided the symmetries with respect to hyperplanes be replaced by symmetries with respect to great circles, i.e. intersection of S N −1 by hyperplanes. As a consequence ω depends only on the azimuthal angle ˜ . FurtherθN −1 := arccos xrN ∈ (0, π2 ) (see (2.2.50)) and is denoted by ω more θN −1 → ω(θN −1 ) is decreasing. As a consequence

ω ˜ (θN −1 ) ≤ ω ˜ (0) = ω(σN ) ≤ which ends the proof.

q+1−p cN,p,q , p−q

Proof of Theorem 6.3.1. The non-existence is a consequence of Theoq 2 2 N −1 rem 6.1.18 with S = S+ , β = γp,q and g(., ω, ∇ω) = γp,q ω + |∇ ω|2 2 . The existence part follows the method of the proof of Theorem 6.1.19. We

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N −1 N −1 deﬁne the homeomorphism A from W01,p (S+ ) onto W −1,p (S+ ) by (6.1.86) and put p−2 F˜ (η, t) = A−1 β(β + Λ(β)) β 2 η 2 + ∇ η|2 2 η (6.3.10) q +(1 + t)| β 2 η 2 + |∇ η|2 2 .

For the same reason as in Theorem 6.1.19, F is a bounded mapping N −1 N −1 from C01 (S + ) into C01,γ (S + ) for some γ > 0, thus compact in X := N −1

C01 (S + ). We check now condition (i) in Proposition 6.1.20 by contradiction in supN −1 posing that there exist sequence {sn } ⊂ [0, 1] and {ηn } ⊂ C01,γ (S + ) N −1

such that ηn → 0 in C01 (S + ) and sn F (ηn , 0) = ηn and sn → s ∈ [0, 1]. −1 This implies that ω := limn→∞ ωn = ηn X ηn is nonnegative and satisﬁes ω X = 1 2 2 p−2 2 2 β ω + |∇ ω| ∇ω −div p−2 2 2 = βΛ(β) β ω + |∇ ω|2 2 ω, N −1 N −1 in S+ and vanishes on ∂S+ , which is impossible since β < βS N −1 . + Next, the argument developed in Step 2 of the proof of Theorem 6.1.19 applies similarly: there exists T > 1 such that

p−2 β(β + Λ(β)) β 2 ω 2 + |∇ ω|2 2 ω q + (1 + t) β 2 ω 2 + |∇ ω|2 2 ≥ (λ1,β + δ)ω p−1

∀t ≥ T , ∀ω ≥ 0.

Thus, if t ≥ T and ω ≥ 0, ω = 0 we infer that it satisﬁes (6.1.91). The contradiction follows as in the previous proof. Thus condition (ii) in Proposition 6.1.20 holds and it is independent of ω X . In order to prove that condition (iii) in Proposition 6.1.20, we have to show that for ω X = R2 large enough the equation F˜ (ω, t) = ω cannot be satisﬁed. Assuming the contrary, there exists a function ωt such that ωt X = Rt . Then ωt satisﬁes 2 2 2 2 p−2 q ωt + |∇ ωt |2 2 ∇ ωt − (1 + t) γp,q ωt + |∇ ωt |2 2 −div γp,q p−2 2 2 − γp,q Λ(γp,q ) γp,q ωt + |∇ ωt |2 2 ω = 0,

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N −1 N −1 in S+ and vanishes on ∂S+ . Thus (1 + t) q+1−p ωt is a positive solution of (6.3.10). It is bounded by a constant depending only on N, p, q N −1 , conby Lemma 6.3.2. Therefore ωt , is also uniformly bounded on S+ tradiction. Thus the condition holds. We conclude by Proposition 6.1.20. 1

Quasilinear Hamilton-Jacobi equations with measure data

6.3.2

We give below a result of removability of singularity Theorem 6.3.3 Assume Ω ⊂ RN is a domain and p − 1 < q < p. Let K ⊂ Ω be a compact set such that cΩ 1,qp (K) = 0 and μ ∈ M+ (Ω) vanishes on K. If u is a nonnegative locally renormalized solution of (6.0.2) in Ω\K where A satisﬁes (4.1.1) and g : Ω×R×RN → R is a Caratheodory function such that g(x, r, ξ)r ≥ |ξ|q

∀(x, r, ξ) ∈ Ω × R × RN ,

(6.3.11)

then u can be extended to whole Ω into a locally renormalized solution of (6.0.2) in Ω. Proof. The function u is p-superharmonic in Ω \ K. Since p ≥ q, qp ≥ p, thus cΩ 1,p (K) = 0. From Theorem 4.3.6, u can be extended into a psuperharmonic function in Ω by formula (4.3.18). The p-harmonic measure μu coincides with g(., u, ∇u) + μ on Ω \ K. In particular g(., u, ∇u) ∈ L1loc (Ω), thus ∇u ∈ Lqloc (Ω). Furthermore u satisﬁes the regularity estimates given in Proposition 4.3.9. If ζ ∈ C01 (Ω) and {ηn } ⊂ C01 (Ω) is a sequence of nonnegative functions with value 1 in a neighborhood of K, 0 ≤ ηn ≤ 1 and ηn W 1,qp → 0, we put η˜n = 1 − ηn and we have (A(x, ∇u), ∇ζ˜ ηn − A(x, ∇u), ∇ηn ζ) dx Ω

g(x, u, ∇u)ζ η˜n dx +

= Ω

(6.3.12) ζ η˜n dμ.

Ω

By the dominated convergence theorem

g(x, u, ∇u)ζ η˜n dx +

lim

n→∞

Ω

Ω

g(x, u, ∇u)ζdx +

ζ η˜n dμ = Ω

ζdμ, Ω

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and

n→∞

A(x, ∇u), ∇ζ˜ ηn dx =

lim

433

A(x, ∇u), ∇ζdx.

Ω

Ω

Hence, if the support of ζ is the compact set Θ ⊂ Ω, A(x, ∇u), ∇ηn ζdx ≤ λ χ ∇u p−1 Θ Lq ∇ηn Lqp , Ω

and the right-hand side term tends to 0 when n → ∞. Consequently A(x, ∇u), ∇ζdx = g(x, u, ∇u)ζdx + ζdμ. (6.3.13) Ω

Ω

Ω

Therefore u is a weak solution of (6.0.2) in Ω. Finally let uk = min{u, k}, k > 0, and assume ζ is nonnegative with value in [0, 1] and support in Θ, then (A(x, ∇u), ∇ζuk η˜n − A(x, ∇u), ∇ηn uk ζ + A(x, ∇u), ∇uk ζ η˜n ) dx Ω

g(x, u, ∇u)dx + kμ(Θ).

≤k Θ

Since

(6.3.14)

|∇uk |p ζ η˜n dx,

A(x, ∇u), ∇uk ζ η˜n dx ≥ Ω

Ω

and the two other terms on the left-hand side of (6.3.14) admit a limit when n → ∞, we derive by Fatou’s lemma, p |∇uk | ζdx ≤ A(x, ∇u), ∇ζuk dx + k g(x, u, ∇u)dx + kμ(Θ). Ω

Ω

Θ

We conclude by Proposition 4.2.2 that u is a p-superharmonic and therefore it is a renormalized solution of (6.0.2) in Ω. The method used in the previous theorem allows to give a necessary condition for solving (6.0.2) [Phuc (2014)]. Theorem 6.3.4 Let A and g satisfy the assumptions of Theorem 6.3.3. If μ is a positive measure in Ω and there exists a locally renormalized solution of (6.0.2) in Ω, then for any compact subset Q ⊂ Ω there exists a positive constant c1 = c1 (N, p, q, Q) such that g(x, u, ∇u)dx + μ(E) ≤ c1 cΩ (6.3.15) 1,qp (E), E

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for any compact subset E of Q. Proof. Let {ηn } ⊂ C0∞ (Ω) where supp (ηn ) ⊂ Q and 0 ≤ ηn ≤ 1, ηn = 1 q on E and ∇ηn Lpqp → cΩ 1,qp (E) as n → ∞. Since qp qp A(x, ∇u), ∇ηn dx = g(x, u, ∇u)ηn dx + ηnqp dμ, (6.3.16) Ω

taking

q ηnp

as test function, we derive by H¨ older’s inequality, using (4.1.1),

|∇u|

λqp

Ω

q

p−1 q

ηnqp dx

Ω

|∇ηn |

qp

q1 p q dx ≥ |∇u| ηnqp dx + dμ.

Ω

Ω

E

This implies

|∇u|

q

ηnqp dx

≤ (λqp )

Ω

and

|∇ηn |

qp

dx,

0≤

qp

Ω

A(x, ∇u), ∇η qp dx ≤ (λqp )qp

|∇ηn |

qp

dx.

(6.3.17)

Ω

Combined with (6.3.16), this yields (6.3.15).

In order to state existence results we introduce the following deﬁnitions (the ﬁrst one has already been used in Chapter V). Deﬁnition 6.1

Let Ω ⊂ RN be a domain.

I- The R-truncated fractional maximal operator of order α ∈ [0, N ) is deﬁned for a bounded measure in Ω by Mα,R [μ](x) = sup tα−N |μ|(Bt (x)), 0 1 and 1 ≤ m ≤ ∞,

MR [μ] q,m is a norm on the set Mα,q,m (Ω) of bounded measures such α L q,m (Ω). that MR α [μ] ∈ L II- For γ > 0 and s > 1 we denote by Mγ,s (Ω) the set of bounded measures in Ω satisfying # ) |μ| (K ∩ Ω) RN : K compact s.t. cγ,s (K ∩ Ω) > 0 < ∞. μ Mγ,s = sup N cR γ,s (K ∩ Ω) (6.3.19) γ,s Clearly μ → μ Mγ,s is a norm on M (Ω).

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These deﬁnitions are justiﬁed by the following regularity results dealing with solutions of (4.2.3) proved in [Phuc (2014), Th 1.4, Th 5.6]. We will not go into their proofs which are delicate and based upon the use of sharp estimates of the measure of the level sets of the diﬀerents functions involved in their expression. Furthermore, the Lebesgue measure dx can be replaced by a weighted measure wdx where w belongs to the class of Muckenhoupt-Wheeden weights [Muckenhoupt, Wheeden (1974)], and the C 1 condition on the boundary can be replaced by a more general one called the Reifenberg ﬂatness condition [Toro (1997)]. Proposition 6.3.5 Assume Ω ⊂ RN is a bounded domain with a C 1 boundary with R =diam (Ω), 2 − N1 < p ≤ N and A satisﬁes (4.1.1). I- If 1 ≤ q < ∞, 1 ≤ m ≤ ∞ and μ ∈ Mq,m (Ω), any renormalized solution u of problem (4.2.3) satisﬁes

1 1

≤ c1 M1,R [μ] p−1q ,m , (6.3.20) |∇u| Lq,m ≤ c1 (M1,R [μ]) p−1 Lq,m

L p−1

for some c1 > 0 depending on N , m, q, Ω and λ. II- If m > 1 and μ ∈ M1,m (Ω), any renormalized solution u of problem (4.2.3) satisﬁes

m m(p−1)

m−1 (6.3.21)

|∇u| m−1 1,m ≤ c2 (M1 [μ]) M 1,m , M

for some c2 > 0 depending on N , m, Ω and λ. The next theorems provide suﬃcient conditions for solving the supercritical quasilinear Hamilton-Jacobi equation (6.0.2). Theorem 6.3.6 Let Ω ⊂ RN be a bounded C 1 domain, R = diam(Ω) and 2 − N1 < p ≤ N . Assume A satisﬁes (4.1.1), g : Ω × R × RN → R+ is a Caratheodory function such that g(x, r, ξ) ≤ g˜(|ξ|) ≤ a |ξ|

q p−1+ N

∀(x, r, ξ) ∈ Ω × R+ × RN , (6.3.22)

+b

where a > 0, b ≥ 0, p − 1 < q ≤ p and g˜ : R+ → R+ is a continuous nondecreasing function such that ∞ g˜(s)s−1−q ds < ∞. (6.3.23) 1

There exists 0 > 0, depending on p, q, a, N and Ω such that if μ ∈ Mb+ (Ω), the inequality M1,R [μ]

q

L p−1

,∞

+ b ≤ 0 ,

(6.3.24)

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436

implies that the problem −div A(x, ∇u) = g(x, u, ∇u) + μ u=0

in Ω on ∂Ω,

(6.3.25)

admits a renormalized solution. Proof. We extend μ by 0 outside Ω, then, by Theorem 5.3.2, we have c−1 3 G1 [μ]

q

L p−1

,∞

≤ M1,R [μ]

q

L p−1

,∞

≤ c3 G1 [μ]

q

L p−1

,∞

.

(6.3.26)

If μ = (μ0 +μs )∗η −div g is an approximation of μ by the convolution with q the smooth regularizing sequence {η }, then G1 [μ ] → G1 [μ] in L p−1 ,∞ (Ω). q Therefore M1,R [μ ] → M1,R [μ] in L p−1 ,∞ (Ω). We follow the ideas introduced in Theorem 6.2.3. We set X = {v ∈ L with norm v X = v

N (p−1) ,∞ N −p

L

N (p−1) N −p ,∞

(Ω) : ∇v ∈ Lq,∞ (Ω)}

+ ∇v Lq,∞ ,

and, for k > 0, put Kk = {v ∈ X s.t. v X ≤ k} .

(6.3.27)

Note that we have assumed that p < N in the deﬁnition of X, the case p = N needing only a minor adaptation. Then Kk is a closed convex set of the Banach space X. If v in Kk , we have ∞ g(x, v, ∇v)dx ≤ g˜(|∇v|)dx ≤ − g˜(t)dbv (t) Ω

Ω

0

where Btv = {x ∈ Ω : |∇v(x)| > t} and bv (t) = meas (Btv ). Since ∇v ∈ Lq,∞ (Ω), bv (t) ≤ max { ∇v Lq,∞ t−q , |Ω|}, hence ∞ g(x, v, ∇v)dx ≤ q ∇v Lr,∞ g˜(t)t−1−q dt Ω

≤ kq

|Ω|

∞

(6.3.28)

g˜(t)t−1−q dt.

|Ω|

Step 1: Approximation and estimates. We introduce also the approximation gn of g by truncation of the variables r and |ξ| used in the proof of

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Theorem 6.2.3. If v ∈ Kk let u = Tn (v) be the renormalized solution of −divA(x, ∇u) = gn (x, v, ∇v) + μ

in Ω (6.3.29)

u=0

on ∂Ω.

By (6.3.22), 0 ≤ gn (x, v, ∇v) ≤ a |∇v|

q p−1+ N

+b

a.e. in Ω.

Hence M1,R [g(., v, ∇v)] ≤ aM1,R [|∇v| By Theorem 5.3.2

q

p−1+ N ]

M1,R [|∇v|

q ,∞ L p−1

q p−1+ N

] + c4 b.

q

p−1+ N ≤ c4 G1 [|∇v| ]

q

L p−1

,∞

.

The Riesz kernel satisﬁes the following Gagliardo-Nirenberg inequality in Lorentz spaces which extends the classical ones valid in Lp spaces: I1 [Θ] Ls∗ ,∞ ≤ c5 Θ Ls,∞ ,

(6.3.30)

[ where s∗ = NsN −s and c5 depends on N and s. Its proof follows from Adams, Hedberg (1999), Prop 3.1.4] which asserts that I1 is a continuous linear ∗ ∗ map from L1 (RN ) into L1 ,∞ (RN ) and from Lp (RN ) into Lp (RN ) (see e.g. [Adams, Hedberg (1999), Prop 3.1.4]). By a classical real interpolation theorem [Lions, Peetre (1964), Th 3.1], I1 extends as a continuous linear maps between all the real interpolations spaces as they are deﬁned in [Lions, Peetre (1964)]. In particular, for any θ ∈ (0, 1) . - ∗ 8 7 ∗ . I1 ∈ L L1 , Lp θ,∞ , L1 ,∞ , Lp θ,∞

It is a classical property of interpolation of Lorentz spaces that 7

L1 , Lp

hence - ∗ . ∗ L1 ,∞ , Lp

8 θ,∞

= Ls,∞

= Ls˜,∞

with

θ,∞

As a consequence, we obtain

q

p−1+ N ] q

I1 [|∇v|

L p−1

,∞

with

θ 1 =1−θ+ , s p

1−θ 1 1 1 θ 1 = = ∗. + ∗ = − s˜ 1∗ p s N s

q

≤ c5 |∇v|p−1+ N

˜ Lq,∞

,

(6.3.31)

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438

where p−1 1 1 Nq = − ⇐⇒ q˜ = , q q˜ N N (p − 1) + q and

q

p−1+ N

|∇v|

p−1+

˜ Lq,∞

with

q

N = ∇v LQ,∞ , ˜

˜ = q˜ p − 1 + q = q. Q N

Since G1 ≤ I1 we ﬁnally derive M1,R [g(., v, ∇v)]

q ,∞ L p−1

q p−1+ N

≤ ac6 ∇v Lq,∞

(6.3.32)

+ c4 b.

Therefore, if u = Tn (v) and v ∈ Kk there holds by Proposition 6.3.5-I q q ∇u p−1 Lq,∞ ≤ c7 M1,R [gn (., v, ∇v)] p−1 ,∞ + M1,R [μ] p−1 ,∞ L

≤

p−1+ q c8 a ∇v Lq,∞ N

L

+ c9 b + c7 M1,R [μ]

q ,∞ L p−1

.

(6.3.33) Combining this estimate with (6.3.28) and Proposition 4.3.9 we ﬁnally infer p−1

u X

p−1+

q

N q ≤ c9 v X + c10 b + c11 M1,R [μ] p−1 ,∞ L p−1 q 1+ ≤ 2c9 v X N (p−1) + B ,

(6.3.34)

where B = 2 c10 b + c11 M1,R [μ]

1 p−1 q ,∞ L p−1

.

In particular, since v ∈ Kk , q

u X ≤ 2c9 k 1+ N (p−1) + B. If we set − N (p−1) q N (p − 1) + q qk0 , and B0 = k0 = 2c9 N (p − 1) q + N (p − 1) q

then the function k → 2c9 k 1+ N (p−1) + B0 is positive and decreasing on [0, k0 ) and vanishes at k = k0 . Moreover c9 and B are independent of n

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and . If we suppose that v ∈ Kk0 there holds r 1+ N (p−1)

u X − k0 ≤ 2c9 k0

+ B0 − k0 = 0,

thus Tn (Kk0 ) ⊂ Kk0 . It is interesting to notice that the dependence of B0 r ,∞ can be expressed by with respect to b and M1,R [μ] L p−1 c10 b + c11 M1,R [μ]

q

L p−1

,∞

≤ 21−p B0p−1 .

(6.3.35)

This is condition (6.3.24). Step 2: Convergence. Because q ≤ p, there holds W01,p (Ω) ⊂ X. Consequently, the proof of the compactness of the operator Tn is the same as the one of Theorem 6.2.3, Step 1. Hence Tn admits a ﬁxed point u = un in Kk0 . The set of functions {gn (., un , ∇un )} is bounded in L1 (Ω) independently of n and . If E is a Borel subset of Ω and σ > 0, we put Bσun = {x ∈ Ω : |∇un | > σ}

and bun (σ) = meas (Buσn ).

Since un ∈ Kk0 , bun (σ) ≤ k0 σ −q , thus |gn (x, un , ∇un )| dx ≤ g˜(|∇un |)dx E

E

≤ g˜(σ) |E| −

∞

g˜(τ )dbun (τ ) σ

≤ g˜(σ) |E| + qk0

∞

g˜(τ )τ −q−1 dτ.

σ

The integrability condition (6.3.23) implies that the set {gn (., un , ∇un )} is uniformly integrable. We conclude as in Theorem 6.2.3-Step 3 that, up to a subsequence, un converges a.e. to u , Tk (un ) converges to Tk (u ) and u is a renormalized solution of −divA(x, ∇u) = g(x, v, ∇v) + μ u=0

in Ω (6.3.36) on ∂Ω.

Similarly as in the proof of Theorem 6.2.3-Step 4, u converges a.e. to a renormalized solution of (6.3.25). Remark. The condition that q ≤ p can be removed if μ has compact support. In this case we approximate it by μ = μ ∗ η for a sequence of molliﬁers {η } ⊂ C0∞ (RN ). If we write μ = (μ0 + μs − div g) ∗ η , then

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it is a converging decomposition in the sense of Theorem 4.3.8. By regularity the operator Tn maps Kk0 into a bounded subset of C 1,α (Ω). Thus compactness follows. Remark. Assumption (6.3.24), jointly with Theorem 5.3.2 and the links between the Bessel potential G1 and the Riesz potential I1 , implies that r r μ ∈ M+ (Ω) ∩ L−1, p−1 ,∞ . Since L−1, p−1 ,∞ (RN ) is the dual space of r N L1, r+1−p ,1 (RN ), μ vanishes on compact sets K ⊂ Ω with zero cR r 1, r+1−p ,1 [ ] capacity. The following result Phuc (2014), Th 1.6 is more precise with a more restrictive assumption on g. Theorem 6.3.7 Assume Ω ⊂ RN is a bounded C 1 domain, A satisﬁes (4.1.1) with 2 − N1 < p ≤ N and q > p − 1. Then there exists c0 depending on N , p, q, λ and diam (Ω) such that for all bounded measure μ in Ω, the q ≤ c0 , see (6.3.19), implies that the problem assumption μ 1, q+1−p M

q

−divA(x, ∇u) = |∇u| + μ u=0

in Ω (6.3.37) on ∂Ω,

admits a solution in W01,q (Ω). Furthermore there exists c1 depending on the same elements as c0 such that N q |∇u| dx ≤ c1 cR (K), (6.3.38) 1, q K∩Ω

q+1−p

for all compact set K ⊂ RN . 6.4

Notes and open problems

6.4.1. Concerning problems on the whole sphere it is not known if all positive solutions of (6.1.2) are constant when q1 < q < qs . The same question arises with positive solutions of (6.3.3) when qc < q < p. 6.4.2. It is an open problem to prove that there is only one positive soluN −1 tion to the problems (6.1.77) with S = S+ and (6.3.7) in the respective range of exponents q˜1 < q < qs and q˜c < q < p. Since it is easily proved that such solutions depend only on the azymuthal variable θN −1 the problem is reduced to prove uniqueness of positive solutions of some ODE. 6.4.3. It is an open problem to prove that there is only one positive N −1 and (6.3.7). Similarly, solution to the problems (6.1.77) with S = S+

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is it true that any positive solution of (6.3.1) in RN + which vanishes on −γp,q \ {0} is bounded by c |x| ? ∂RN + 6.4.4.

The quest of singular separable solutions of −Δp u + |u|q−1 u |∇u|s = 0,

(6.4.1)

in RN \ {0} or RN + , where = ±1 and (q + s + 1 − p)(p − s) > 0, leads to p−s . As for ω it satisﬁes u(r, σ) = r−τp,q,s ω(σ) where τ := τp,q,s = q+s+1−p −div

τ ω + |∇ω| 2

2

2

p−2 2

∇ω

2

p−2 2

= τ Λ(τ ) τ ω + |∇ω| ω 2s − |ω|q−1 ω τ 2 ω 2 + |∇ω|2 , 2

2

(6.4.2)

N −1 N −1 with zero boundary conditions on ∂S+ . A either on S N −1 or on S+ ﬁrst critical set of exponents (q, s) is

Λ(τ ) = 0 ⇐⇒ q + s = p − 1 +

1 , N

for the problem in RN and Λ(τ ) = Λ(βS N −1 ) ⇐⇒ (q + s)βS N −1 + s = (p − 1)βS N −1 + p, +

+

+

N −1 for the problem in S+ . Existence of nontrivial positive solutions of (6.4.2) N −1 N −1 vanising on ∂S+ is an interesting question. The proof of an a on S+ priori estimate for any solution in RN \ {0} is a deeper question.

6.4.5.

Not much is known for the supercritical problem −Δp u + |u|

q−1

s

u |∇u| = μ.

(6.4.3)

A natural critical threshold is expressed by q(N − p) + s(N − 1) = N (p − 1). 6.4.6.

The study of −Δp u + |u|

q−1

u + |∇u| = 0, s

(6.4.4)

where , = ±1 is still essentially open, except in the case = = 1 where one of the two reaction terms can be dominant, the critical case being s=

pq . q+1

II = = −1 no a priori estimate is known, even if one term dominates the other. Another interesting question is to ﬁnd condition (necessary,

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suﬃcient) to solve the problem with measure data, −Δp u + |u|q−1 u + |∇u|s = μ.

(6.4.5)

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List of symbols

W1,p , ix N c˙Rα,p (K), 21 βS N −1 , 121

(M d , g), 2 C k (K), C(K), C ∞ (K), 10 C k (Ω), C(Ω), C ∞ (Ω), 10 C0k (Ω), C0 (Ω), C0∞ (Ω), 10 Dα T (φ), 11 Dα φ, 11 Lα,p,s (RN ), 18 Lα,p , 18 Rp [v], 65 S N−1 (S N−1 , g0 ), 87 T M , Tx M , 2 RN + = {x := (x , xN ) : xN > 0}, 1, 61 B2,q , 150, 153 D1,p , 378 ˆ 12 F(φ), φ, H, 150, 153 Hk , 153 Hk , 150 Δp , vii Δ∞ φ, 73 Λ(β), 98, 103, 104, 128 Λp , viii Σ0 , 1 Σβ , 1 Ω0 , 1 Ωβ , 1 Ωλ = λ1 Ω, 176, 239 Ωλ = λ1 Ω, 180 Ωβ , 1 Ωλ := λ1 Ω, 157 Ωλ = λ1 Ω, 192

+

βp,q , 129, 142, 168 Λ(βp,q ), 130 ωc , 130, 142 γp,q , 201 λ1 λ1 (p, Ω), 70 μN,p,q , 201 σ[−Δp ; W01,p (Ω)], 67 σ(Δg ; W 1,2 (M )), σ(Δg ; W01,2 (M )), 5 σ(Δp ; W 1,p (M ), 78 ωc , 130, 142, 171 μ ˜N,p,q , 203 cΩ 1,p -capacity, 250 N cRα,p,s , 19 N cRα,p,s (K), 19 q qp = q+1−p , 425, 427, 428 Λp (β) = β(p − 1) + p − N , 87 critical exponents internal singularities q1 = N(p−1) , N−p 131 critical exponents, Emden-Fowler boundary singularities q˜1 = β Np −1 + p − 1, 144, S

+

145, 168, 178, 179, 196, 401, 440 , internal singularities q1 = N(p−1) N−p 453

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Local and global aspects of quasilinear degenerate elliptic equations

130, 131, 142, 320, 344, 347, 357, 368, 370, 372, 373, 409, 425, 440 critical exponents, Hamilton-Jacobi boundary singularities βS N −1 + q˜c = p − , 231, βS N −1 + 1 + 233, 234, 246, 429, 440 , internal singularities qc = N(p−1) N−1 201, 202, 208, 214, 216, 370, 372, 429, 440 curvature Gaussian curvature Kg , 4 Ricci curvature Riccg , 4 scalar curvature Scalg , 4 sectional curvature secg (π), 4 decreasing rearrangement f ∗ , 7 distance to boundary ρ(x), 1

divergence divg (X), 5 Euclidean unit ball volume ωN , 10, 66, 75, 217, 415 Fractional Sobolev spaces W s,p (Ω, 16 fundamental solution μp , 56 Laplacian Δg u, 5 measure ⊥ singular, 255, 274, 298, 310, 405 signed distance to boundary ρ(x), ˙ 1 Sobolev constant SN,p , 14, 76 Sobolev space W k,p (Ω), 13 Sobolev space W0k,p (Ω), 14 tempered distributions S (RN ), 12

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Glossary

(φ, p)-scaling-growth property, 175 p-Laplacian, 52 Hopf comparison principle, 54 strict comparison principle, 54 strict monotonicity inequality, 54 p-harmonic, 53 p-harmonic measure, 254–256 p-superharmonic, 249, 250, 336 weakly p-superharmonic, 252 p-harmonic analog, 181

coercive, 33, 35, 51 comparison hypothesis, 115 conformal invariance, 52 conformal transformation, 58 convex, 2 convexity radius, 3, 223 curvature Gaussian curvature, 4 Ricci curvature, 4 scalar curvature, 4 sectional curvature, 4 cut locus, 2

eventually uniformly continuous, 185 spherical p-harmonic eigenvalue problem, 87

degree, 34 homotopy, 34 domain smooth exhaustion, 154 smooth exterior approximation, 154

Bessel capacity, 256 Bessel-Lorentz capacity, 347 Bessel-Lorentz potential spaces, 18 boundary singular solution, 116 Cantor diagonal process, 214 capacitary measure, 20, 428 capacity Bessel capacity, 19 Riesz capacity, 21 Sobolev capacity, 19 Caratheodory, 28, 45 Caratheodory function, 249 Caratheodory functions, 22 Carleson estimate, 61, 114 Carleson inequality, 63 Co-area formula, 16

eigenfunction, 67 eigenvalue, 67 entropy solution, 289 ergodic constant, 88, 96 ergodic limit, 93 eventual convergence eventually C 1 convergence, 185 eventually uniformly convergent, 185 exponential map, 3 exterior domain, 158 455

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9850 - Local and Global Aspects of Quasilinear Degenerate Elliptic Equations 9789814730327

Local and global aspects of quasilinear degenerate elliptic equations

ﬁne topology, 260 ﬂow coordinates, 2 Fourier transform, 12 fractional maximal operator, 321 R-truncated fractional η-maximal operator, 321 fundamental solution, 56, 135

narrow topology, 268

geodesic, 2 geodesic distance, 2 geodesic distance, 88 signed, 89 geodesically complete, 3

quasiclosed, 21 quasicontinuous, 252 quasieverywhere, 20 quasiopen, 21

Harnack inequality, vii, 23, 114 boundary Harnack inequality, 63, 114, 241 Hausdorﬀ capacity, 347 Hausdorﬀ measure, 9 dimensional Hausdorﬀ capacity, 9 Hausdorﬀ dimension, 10 hemicontinuous, 35, 40 Hessian, 5 isotropy threshold, 144, 152 Kelvin transform, 363 kernel Bessel kernel, 17 Riesz kernel, 21 Krasnoselskii genus, 74 Kronecker’s symbol, 169 large solution, 154 Leray-Lions class, 33, 36, 38, 49 Liouville rigidity theorem, 58 local continuous graph property, 163 locally renormalized solution, 262, 265, 305, 311, 428 Lorentz-Bessel spaces, x manifold which possesses a pole, 3 molliﬁers, 132 Morse function, 84

p-Laplacian, vii polar set, 266 potential Bessel potential, 18 Riesz potential, 21

Rayleigh quotient, 65 regular for the p-Laplacian, 156 renormalized solution, 249, 260, 268, 276, 286, 289, 298, 302 Riemann curvature tensor, 3 Riesz capacity, 21, 425 Sobolev capacity, vii, 22, 156, 425 Sobolev spaces, 13 Extension operator, 15 Fractional Sobolev spaces, 16 Gagliardo-Nirenberg inequality, 14 Gagliardo-Nirenberg interpolation inequality, 14 Meyers-Serrin approximation, 14 Morrey inequality, 14 Poincar´e inequality, 14 Rellich-Kondrachov theorem, 15 Trace, 15 sphere condition exterior sphere condition, 158 sphere condition interior sphere condition, 158 spherical p harmonic equation, 98 spherical p harmonic function, 181 spherical inﬁnite harmonic eigenvalue problem, 106 tangent bundle tangent space, 2 thick, thin, 260 very weak gradient, 253

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Glossary

viscosity solution, 73 weakly p-superharmonic, 53, 63 weakly p-subharmonic, 53

457

Weitzenb¨ ock formula, 6, 83, 90 p-Bochner formula, 86 Wolﬀ potential, ix, 255, 256, 320, 333 R-truncated Wolﬀ potential, 320

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