Order Structure and Topological Methods in Nonlinear Partial Differential Equations: Maximum Principles and Applications, Volume 1

Yihong Du

Order Structure and Topological Methods in Nonlinear Partial Differential Equations voi 1 Maximum Principles and Applications

World Scientific

Order Structure and Topological Methods in Nonlinear Partial Differential Equations ^|i

Maximum Principles and Applications

Series on Partial Differential Equations and Applications Series Editor: Fang-Hua Lin (Courant Institute of Math. Sci., New York University)

Published Vol. 1

Semiclassical Analysis, Witten Laplacians, and Statistical Mechanics by B. Helffer (Univ. Paris-Sud, France)

Vol. 2

Order Structure and Topological Methods in Nonlinear Partial Differential Equations Vol. 1 - Maximum Principles and Applications by Yihong Du (Univ. of New England, Australia & Qufu Normal Univ, China)

Order Structure and Topological Methods in Nonlinear Partial Differential Equations #>i i

Maximum Principles and Applications

Yihong Du University of New England, Australia & Qufu Normal University, China

\fp

World Scientific

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Series on Partial Differential Equaitons and Applications — Vol. 2 ORDER STRUCTURE AND TOPOLOGICAL METHODS IN NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Vol. 1: Maximum Principles and Applications Copyright © 2006 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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Preface

This is volume one of a two volume series. The intention is to provide a reference book for researchers in nonlinear partial differential equations and nonlinear functional analysis, especially for postgraduate students who want to be led to some of the current research topics. It could be used as a textbook for postgraduate students, either in formal classes or in working seminars. In these two volumes, we attempt to use order structure as a thread to introduce the various versions of the maximum principles, the fixed point index theory, and the relevant part of critical point theory and Conley index theory. The emphasize is on their applications, and we try to demonstrate the usefulness of these tools by choosing applications to problems in partial differential equations that are of considerable concern of current research. An important work in this direction is H. Amann's classical review article (SIAM Rev. 18 (1976), 620-709), which discussed the combination of order structure and fixed point index theory and its applications to various problems of nonlinear partial differential equations. Much progress has been made since this article. The fixed point index theory has been further developed and found important new applications in partial differential equations. Moreover, the order structure has since been successfully combined with critical point theory and Conley index theory to study various nonlinear partial differential equation problems. Furthermore, the classical maximum principle in partial differential equations has found new applications in several important problems. All these are scattered in research articles published in various professional journals, and most of them are still active topics of current research. It is our hope that through these two volumes, we can present the reader

v

VI

Maximum

Principles

and

Applications

in a somewhat systematic way some of the new progresses in these topics. As the title suggests, volume 1 mainly considers the maximum principles and their various applications in some of the current research topics. The topological methods will be discussed in volume 2. There are 7 chapters and an appendix in this volume 1. In chapter 1, we use the Krein-Rutman theorem to derive several well-known properties of the principal eigenvalues, we then use these in chapter 2 to characterize the maximum principle. We briefly discuss the moving plane method in chapter 3. Existence results are not discussed until chapter 4, where we consider the methods of upper and lower solutions, also known as super and sub-solution methods. The weak theory here is based on the theory of monotone operators, whose basic result is recalled without proof. With these preparations, existence results can be considered in the later chapters. In chapter 5, the basic logistic model is discussed, where various comparison arguments, the upper and lower solution methods, together with a variety of elliptic estimates are used. Chapter 6 gives an introduction of some basic boundary blow-up problems. The last chapter considers again various symmetry properties of elliptic problems, where apart from the moving plane methods, other techniques are also used. In the appendix, we include a brief review of the classical elliptic theory for second order partial differential equations. Since this basic theory may take a long time for the beginners to master, we feel it might be practical for those readers such as postgraduate students to initially accept the relevant basic results in this theory and continue with their study of some current research topics. Some of the material is chosen to be included here for its usefulness, such as the various versions of the maximum principles, and the upper and lower solution methods. In such a case, we have tried to make the results as general as possible, provided that not too much complication of the presentation is caused. Some of the material is included here in order to introduce useful techniques and to lead the reader to some of the current research problems. In this situation, we usually put clarity in front of generality. The material presented and the references quoted here are mainly based on the author's taste and familiarity, which inevitably are biased with many important topics and references not included here. I apologize if these omissions inadvertently offend anyone. In volume 2, we will discuss some developments of the fixed point index theory (mainly due to E.N. Dancer), and their applications to various problems, in particular to several population models. We will also discuss the

Preface

VII

part of critical point theory and Conley index theory that can be combined with order structure to provide better applications. Some of the material in volume 1 here provides necessary preparation for volume 2. It is my great pleasure to thank all those who helped in one way or another in the writing of this first volume. In particular, I would like to express my deep thanks to Professor Norman Dancer for guiding me into nonlinear analysis, and for the constant help and encouragements. My sincere thanks to Professor Dajun Guo for taking me into nonlinear functional analysis, and to Professor Xingbin Pan for encouraging me to write this book. I'm grateful to my colleagues at the University of New England who freed me from teaching duties in the first half of 2005; that helped immensely in getting this belated volume ready before the end of the year. Part of the material here was presented at Qufu Normal University at a workshop in 2004, and my thanks go to the colleagues there for the help and support. Over the years, I have benefitted greatly from working with my collaborators. My sincere thanks to all of them, in particular, Florica Cirstea, Zongming Guo, Shujie Li, Lishan Liu, Li Ma, Tiancheng Ouyang, Shusen Yan, Feng Zhou, and my former PhD students and friends Qingguang Huang and Wei Dong, whose joint papers with me are used in this volume. Several friends helped me proof reading various parts of this volume, and I would like to thank in particular Florica Cirsta, Xing Liang, Rui Peng and Shusen Yan for their efforts in helping me reducing the mistakes. It is my responsibility for any remaining mistakes, and corrections from the readers are very much appreciated. My sincere thanks also go to the editors of World Scientific Publishing, especially Ms Zhang Ji, for all the help and advices. Finally I thank my family for the understanding and support during the writing of this book.

Yihong Du September, 2005, Armidale

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Contents

Preface

v

1.

Krein-Rutman Theorem and the Principal Eigenvalue

1

2.

Maximum Principles Revisited

9

3.

4.

5.

6.

2.1 Equivalent forms of the maximum principle 2.2 Maximum principle in W2'N(Cl)

9 11

The Moving Plane Method

17

3.1 Symmetry over bounded domains 3.2 Symmetry over the entire space 3.3 Positivity of nonnegative solutions

17 23 28

The Method of Upper and Lower Solutions

33

4.1 Classical upper and lower solutions 4.2 Weak upper and lower solutions

33 39

The Logistic Equation

61

5.1 The classical case 5.2 The degenerate logistic equation 5.3 Perturbation and profile of solutions

61 64 75

Boundary Blow-Up Problems

83

6.1 The Keller-Osserman result and its generalizations 6.2 Blow-up rate and uniqueness 6.3 Logistic type equations with weights ix

84 95 102

X

7.

Maximum

Principles

and

Applications

Symmetry and Liouville Type Results over Half and Entire Spaces

117

7.1 7.2 7.3 7.4

117 128 139 145

Symmetry in a half space without strong maximum principle Uniqueness results of logistic type equations over RN . . . . Partial symmetry in the entire space Some Liouville type results

Appendix A A.l A.2 A.3 A.4 A.5

Basic Theory of Elliptic Equations

Schauder theory for elliptic equations Sobolev spaces Weak solutions of elliptic equations LP theory of elliptic equations Maximum principles for elliptic equations A.5.1 The classical maximum principles A.5.2 Maximum principles and Harnack inequality for weak solutions A.5.3 Maximum principles and Harnack inequality for strong solutions

163 163 166 170 174 177 177 178 179

Bibliography

181

Index

189

Chapter 1

Krein-Rutman Theorem and the Principal Eigenvalue The Krein-Rutman theorem plays a very important role in nonlinear partial differential equations, as it provides the abstract basis for the proof of the existence of various principal eigenvalues, which in turn are crucial in bifurcation theory, in topological degree calculations, and in stability analysis of solutions to elliptic equations as steady-state of the corresponding parabolic equations. In this chapter, we first recall the well-known KreinRutman theorem and then combine it with the classical maximum principle of elliptic operators to prove the existence of principle eigenvalues for such operators. Let X be a Banach space. By a cone K C X we mean a closed convex set such that XK C K for all A > 0 and K n (-K) = {0}. A cone K in X induces a partial ordering < by the rule: u < v if and only if v — u G K. A Banach space with such an ordering is usually called a partially ordered Banach space and the cone generating the partial ordering is called the positive cone of the space. If K — K = X, i.e., the set {u — v : u, v £ K} is dense in X, then K is called a total cone. If K — K = X, K is called a reproducing cone. If a cone has nonempty interior K°, then it is called a solid cone. Any solid cone has the property that K — K = X; in particular, it is total. Indeed, choose XQ G K° and r > 0 such that the closed ball Br(uo) := {u G X : \\u — UQ\\ < r} is contained in K. Then for any u G X\{0}, v0 := uo+ru/||u]| e K and hence u = (||u||/r)(v 0 —«o) G K—K. We write u>viiu~v£K\ {0}, and u > w i f w - « € K°. Let X* denote the dual space of X. The set K* := {I G X* : l{x) > 0 Vx G K] is called the dual cone of K. It is easily seen that K* is closed and convex, and XK* C K* for any A > 0. However it is not generally true that K* n (-K*) = {0}. But if K is total, this last condition is satisfied and hence K* is a cone in X*. Indeed, if / G K* n (-K*), then for every

1

2

Maximum

Principles

and

Applications

x £ K, l(x) > 0, — l(x) > 0, and therefore l{x) = 0 for all x £ K. Since K — K = X, this implies that l(x) = 0 for all i £ l , i.e., / = 0. Let Q be a bounded domain in RN. It is easily seen that the set of nonnegative functions K in X = LP(Q.) is a cone satisfying K — K = X. However, it has empty interior. Similarly the set of nonnegative functions in W1'P{Q) gives a reproducing cone, and generally the nonnegative functions in Wk^{D) (k > 2 ,p > 1) form a total cone. On the other hand, the nonnegative functions form a solid cone in C(fl) but only form a reproducing cone in Co(O) := {u e C(£l) : u = 0 on 9Q}. If fi has C 1 boundary dQ, then it is easy to see that the nonnegative functions in Co(fi) := {u 6 C 1 (fl) : u = 0 on dQ} form a solid cone; for example, any function satisfying u(x) > 0 in Q and Dvu{x) < 0 on dQ is in the interior of the cone, where v denotes the outward unit normal of dfi. T h e o r e m 1.1 (The Krein-Rutman Theorem, [Deimling(1985)] Theorem 19.2 and Ex.12) Let X be a Banach space, K C X a total cone and T : X —* X a compact linear operator that is positive (i.e., T{K) C K) with positive spectral radius r(T). Then r(T) is an eigenvalue with an eigenvector u £ K \ {0}; Tu = r(T)u. Moreover, r(T*) = r(T) is an eigenvalue ofT* with an eigenvector u* G K*. Let us now use Theorem 1.1 to derive the following useful result. Theorem 1.2 Let X be a Banach space, K C X a solid cone, T : X —* X a compact linear operator which is strongly positive, i.e., Tu ^> 0 if u > 0. Then (a) r(T) > 0, and r(T) is a simple eigenvalue with an eigenvector v £ K°; there is no other eigenvalue with a positive eigenvector. (b) |A| < r(T) for all eigenvalues A ^ r(T). Let us recall that r is a simple eigenvalue of T if there exists v ^ 0 such that Tv = rv and (rl — T)nw — 0 for some n > 1 implies w e span{v}. Proof. Step 1: There exists v0 > 0 such that Tv0 = r(T)v0. Fix u £ K°. Then aTu > u for some a > 0, and we can find a > 0 such that Btj{u) c K. It follows that w < (cr) _1 ||ui||u for any w £ X. Let S = aT. Then u < Snu < a-^l^uWu

< 1.

Krein-Rutman

Theorem and the Principal

Eigenvalue

3

Hence \\Sn\\ > a/\\u\\ and r{S) = lim \\Sn\\^n > 0. By Theorem 1.1, r(S) is an eigenvalue of S corresponding to a positive eigenvector v0 £ K \ {0}. Clearly r(T) = r(S)/a > 0 and Tv0 = r(T)v0. Step 2: To prove that r(T) is simple, we show a more general conclusion: lfr>0 and Tv = rv for some v > 0, then r is a simple eigenvalue

ofT. Let us first show that (rl — T)w = 0 implies w £ span{v}. Suppose Tw = rw with w ^ 0. Then T(v ± tw) = r(v ± tw) for all t > 0. Since T is strongly positive, v £ K° and the above identity implies v ± tw $ dK unless v ± tw = 0. But v±tw £ K° for small t and this cannot hold for all large t for otherwise w £ K D (—-ft") = {0}. Therefore there exists t0 ^ 0 such that u -f t0w £ dK and hence v + tow — 0. This proves w £ span{v}. Let (rl — T)2w = 0. By what has just been proved, rw — Tw = tov for some t0 £ R1. If t 0 ^ 0, then we may assume to > 0 (otherwise change w to — iu). Since T(t; + sw) = r(v + sw) — sj^of 0, and v + sw £ K° for all small s > 0, we easily deduce v + sw £ K° for all s > 0. This implies that w £ K, and hence w = r _ 1 ( i o ^ + Tw) £ K°. We now have w -tv

£ K° for all small i > 0,

but not for all large t > 0 as this would imply v — 0. Therefore there exists t\ > 0 such that w — t\v £ dK. But then rw — tov — £irt> = T(w — *iv) > 0, w — t\v > r~ltoV 3> 0, contradicting w — t\v £ dK. Therefore we must have to = 0 and hence rw — Tw = 0, w £ span{v}. This proves that r is a simple eigenvalue. Step 3: Next we show that T cannot have two positive eigenvalues r\ > T2 corresponding to positive eigenvectors: Tv\ —r\V\, Tv2 — r2V2Let v(t) = V2 — tv\, t > 0. Since T is strongly positive, we have v\, v2 £ K°. As before we have v(t) £ K° for small t but not for all large t.

4

Maximum

Principles

and

Applications

Therefore there exists to > 0 such that v(to) £ K but v(t) g K for t > to. We now have v2 - to(n/r2)vi

= r^Tfa

- t0Vi) £ K,

which implies r\ < r2 due to the maximality of to- This contradiction proves step 3. S t e p 4: IfTw = Aw with w =£ 0 and A ^ r{T), then |A| < r(T). If A > 0, then by Step 3, w £ K. It follows that v0 + tw £ K for all small i > 0 but not for all large t. Therefore there exists to > 0 such that vo + tow £ K and vo + tw ^ K for t > to- It then follows that v0 + t0(X/r(T))w = r ( r ) ~ x T ( v 0 + t0w) £ K. The maximality of t0 implies that A < r(T) and hence A < r(T). If A < 0, then from T2w = \2w and T2v0 = r(T)2v0 and the above argument (applied to T 2 ) we deduce A2 < t ( T ) 2 and hence |A| < r(T). Consider now the case that A = o + ir with T / 0 . Then necessarily w = u + iv and Tu = ou — rv, Tv =

TU +

av.

(1-1)

We observe that u and v are linearly independent for otherwise we necessarily have r = 0. Let X\ := span{u,v}. Then (1.1) implies that Xi is an invariant subspace of T. We claim that K\ :— X\C\K — {0}. Otherwise K\ is a positive cone in X\ with nonempty interior, as for any w £ K\\ {0}, Tw £ Xi n K° - K^. We can now apply Step 1 above to T on Xi to conclude that there exists r > 0 and WQ £ K® such that Two = rwo- By Steps 2 and 3, we necessarily have r = r(T) and Wo € span{vo). In other words, t;o £ K\ and «o = &u + flv for some real numbers a and /3. But then one can use (1.1) and TVQ = r(T)vo to easily derive a = j3 = 0, a contradiction. Therefore K\ = {0}. From span{u, v}C\K = {0} we find that the set S := {(£, ri)£R2:v0+tu

+ riv£

K}

is bounded and closed. Since v0 £ K°, M := sup{£ 2 + rj2 : (£,r?) £ £ } > 0 and is achieved at some (£o,?7o) G S. Let 2 0 = VQ -\- ^Qu + r]ov. Then 2 0 £ K \ {0} and Tz0 G X°. Therefore we can find a £ (0,r(T)) such that T^o > oivo, i.e., ( r ( T ) - a ) v 0 + ( 6 « + »?iu)>0,

(1.2)

Krein-Rutman

Theorem and the Principal

Eigenvalue

5

where £1 = £o 0 if v € K. The strong maximum principle Theorem A.36 then implies that u = Tv > 0 in ft if v € K \ {0}, and the Hopf boundary lemma (Lemma A.35) gives further Dvu < 0 on 9ft. This implies that Tv € K°. Therefore T is strongly positive. It now follows from Theorem 1.2 that r(T) > 0 is a simple eigenvalue of T with an eigenfunetion v € K°: Tv = r(T)v. Thus u = Tv satisfies —Lu + £u = r(T)~1u

in ft, u = 0 on 9ft,

Maximum

6

Principles and

Applications

i.e., Lu + \\u

= 0 in ft, u = 0 on dCl,

with Ai = r ( T ) - 1 - £ . Generally, it is easily checked t h a t [i is an eigenvalue of T if and only if A = n~l — £ is an eigenvalue of Lu + AM = 0 in ft, u = 0 on 5ft.

(1.3)

Theorem 1.2 now implies the following result. T h e o r e m 1.3 Under the conditions of Theorem A.4 for L and ft, the eigenvalue problem (1.3) has a simple eigenvalue Ai £ R1 which corresponds to a positive eigenfunction; none of the other eigenvalues corresponds to a positive eigenfunction. If the boundary operator is of Neumann or Robin type, Bu = Dvu + a(x)u,

a > 0, a £ C 1 , a ( d f i ) ,

then we let X = C 1 , Q (ft) and let K be the cone of nonnegative functions in this space. We define the operator T analogously as in the Dirichlet case and again find t h a t it is compact on X and maps K t o itself, due to the weak maximum principle. Suppose now v £ K \ {0}. T h e n by t h e strong m a x i m u m principle, u = Tv > 0 in ft. Moreover, by t h e Hopf b o u n d a r y lemma, if U(XQ) = 0 for some XQ € dCl, then D^u^o) < 0 and hence BU(XQ) < 0, contradicting the boundary condition. Therefore u > 0 on dCl. Therefore Tv > 0 on ft, which implies t h a t Tv £ K°, i.e., T is strongly positive. Therefore we can apply Theorem 1.2 t o conclude t h a t Theorem 1.3 holds also for the Neumann and Robin b o u n d a r y conditions. T h e eigenvalue Ai in Theorem 1.3 is usually called the principle eigenvalue. T h e o r e m 1.4 If X ^ Ai is an eigenvalue of (1.3) but the boundary condition is either Dirichlet, or Neumann, or Robin type, then Re(X) > X\. Proof. Suppose w > 0 is an eigenvector corresponding to Ai and u is an eigenvector corresponding to A. Set v := u/w. Then —Xv = w~1L(vw)

= Lv — cv + 2w~lali

DjwDiV

— Xiv.

Writing Kv := aijDijV

+ VDiV, V := 6 i +

2vj-1aijDjw,

Krein-Rutman

Theorem and the Principal

Eigenvalue

7

we obtain Kv + (X - \i)v = 0. Take complex conjugates to yield Kv + (X-Xi)v

= 0.

Next we compute K(\v\2) = K(vv) = vKv + vKv + 2aijDivDjV

> vKv + vKv,

since aij^j

= a^(Re{^)Re(^)

+ Im^Imfa))

>0

for any complex vector £ £ CN. We now easily obtain i f ( H 2 ) > 2(i?e(A) - Ai)M 2 in ft. Suppose now the boundary operator B is either Neumann or Robin type. Then w > 0 over ft and a direct computation shows Dvv = 0 and D „ H 2 = 0. If Re(X) < Xi, then 0 := \v\2 > 0 satisfies K4> > 0 in ft, Dv(j> = 0 on 9ft. We now apply the strong maximum principle and Hopf boundary lemma and conclude that / = constant, that is u = cw and hence A = X\, a contradiction. Therefore we must have Re(X) > Ai. To prove the Dirichlet case, we replace w by ive := w1_e, 0 < e < 1, in the above discussion and obtain K{\v\2) > -2(Re(X) + — )\v\2 in ft. we Since Lwe = (1 - e)w~eLw - e(l - e)u> - 1 - V J DiivDjW + ecu)1-* < (1 - e)w~eLw + ecw 1 - 6 < (eC - (1 - e)Ai)iue, where C = max^c, we deduce K(\v\2) > 2((1 - e)A! - eC - Re(X))\v\2 in ft.

8

Maximum Principles and Applications

By the Hopf boundary lemma, we know Duw < 0 on d£l. Therefore, since u\gn = 0, u(x) _ z-txoGdfi w(x)

Duu(xp) DVW{XQ)

It follows that ii(x\

lim v — lim w(x)e—r-r x-*dQ

x-tdn

= 0.

W(X)

If Re(X) < (1 - e)Ai - eC, then, K(\v\2)>0ma. Let Qn be a sequence of smooth domains enlarging to Q, e.g., Qn := {a; £ fi : d(a;, 3Q) > So/n} with 0 small. We apply the maximum principle on Cln and deduce max^ |t>|2 < maxan^ \v\2. Letting n —• oo, it results \v\2 = 0 and hence u = 0, a contradiction. Therefore Re(\) > (1 — e)Ai — eC. Letting e - > 0 w e obtain Re(X) > \i• Remark 1.5 The above proof shows that in the Neumann and Robin boundary conditions case, Re(\) > \\. This is also true in the Dirichlet case; see Theorem 2.7 in the next chapter. Remark 1.6 Instead of (1.3), sometimes one also needs to consider the weighted eigenvalue problem Lu + \h(x)u = 0 in Q, Bu = 0 on d£l, where h(x) is a weight function, B is the Dirichlet, Neumann or Robin boundary operator. If h(x) is positive and suitably smooth, similar results to those in Theorems 1.3 and 1.4 can be analogously proved. If h(x) changes sign, similar results can still be proved by considerably different techniques, see [Hess-Kato(1980)].

Chapter 2

Maximum Principles Revisited

In this chapter we make use of the principal eigenvalues to formulate a useful characterization of the so called maximum principle property. The first section considers the classical case for C 2 solutions and the second section discusses solutions in W2'N.

2.1

Equivalent forms of the maximum principle

The principal eigenvalue can be used to formulate a necessary and sufficient condition for the validity of the maximum principle. In order to state the results in a concise way, we use a well-known framework that incorporates all three boundary conditions considered in the last chapter and also the mixed boundary conditions. We assume that To and Ti are two disjoint open and closed subsets of dVt with TQ U T\ = dfl, and define Bu:=l"

on To, [ Uvu + au on 11,

where a 6 C 1 , a (Q) is nonnegative. It is possible that either To or I \ is the empty set. As in the previous chapter, we assume that Q is bounded with C 2 , Q boundary and L is as in Theorem A.4. It is easy to modify the proofs in the previous chapter to obtain the following result. Theorem 2.1

The eigenvalue problem Lu + Xu — 0 in n, Bu = 0 on dfl

has a principal eigenvalue Ai = X\(L,B, ft), and any other eigenvalue A satisfies Re(X) > Ai. 9

10

Maximum

Principles and

Applications

Definition 2.2 We say that (L, B, Cl) has the strong maximum principle property if u G C2(Cl) n C1 (Cl U Ti) n C(Cl U To) and - L u > 0 in Cl, Bu > 0 on dCl imply u > 0 in Cl unless u = 0. Definition 2.3 of (L, B, Cl) if

A function u £ C2(Cl) n C 1 (fi) is called a supersolution - L u > 0 in Cl, Bu > 0 on df2.

It is called a strict supersolution if it is a supersolution but not a solution. Theorem 2.4 Under the above assumptions on (L,B,Cl), statements are equivalent:

the following

(i) (L,B,Cl) has the strong maximum principle property; (ii) (L,B,Cl) has a strict supersolution which is positive in CI; (Hi) Xi(L,B,Cl) > 0 . Proof. (i)=^(iii). Arguing indirectly, we assume Ai < 0. We may choose the corresponding eigenfunction of Ai negative: 4> < 0 in Cl. Therefore -L(p = Ai 0 in CI, B<j> = 0 on dCl.

By the strong maximum principle this implies 4> > 0, a contradiction. (iii)=^(ii). Let tp > 0 be the corresponding eigenfunction of Ai. Then - L V = AiV> > 0 in CI, Bij> = 0 on dCl. Hence I/J is a strict supersolution that is positive in CI. (ii)=^(i). Let v be a strict supersolution, positive in Cl. We first claim that v > 0 on Ti. Otherwise, that exists XQ € Ti such that v(x0) = 0. By the Hopf lemma, we deduce Bv(x0) = Duv(x0) < 0, contradicting the assumption that Bv > 0 on I V Next we show that v(x) > Sd(x,To) for some 5 > 0 and all x £ Cl. Otherwise we can find a sequence xn 6 Cl such that v(xn) < (l/n)d(xn, To). In particular, v(xn) —-> 0 as n —» oo. As v is positive in fiuri, we necessarily have, by passing to a subsequence if needed, xn —» XQ £ To. It follows by the continuity of v that V(XQ) — 0, and hence DVV(XQ) < 0 by the Hopf lemma. But this last inequality implies v(x) > Sd(x,T0) for some 6 > 0 and all x E Cl n BT(XQ), where BT(XQ) denotes a small ball centered at x$; this is in contradiction to the inequalities for xn with large n.

Maximum

Principles

Revisited

11

We are now ready to show that (L, B, ft) has the strong maximum principle property. Suppose that u ^ 0 and satisfies -Lu > 0 in ft, Bu > 0 on 5ft. If u > 0 in ft then there exists XQ £ ft such that U(XQ) > 0, as we have assumed u ^ 0. If there exists x\ £ ft such that u(x{) = 0, then we can find an open ball B C ft such that u{x) > 0 in 5 but u(x2) = 0 for some X2 € 0 in ft. The only other possibility is that min^u = u(xo) < 0. Since u > 0 on To and u € C 1 (ft), we easily see that for some M > 0 large, u > —Md(x, To). Therefore, recalling v > Sd(x,ro), we have u + £v > 0 in ft, V£ > M/8. This does not hold for small £ > 0 as U{XQ) < 0. Therefore, there exists a smallest positive £, say £o, such that u + £QV > 0 in ft. Now v := u + £0v is a strict supersolution that is nonnegative in ft. It is not identically zero because this would force v to be a solution due to the inequality satisfied by u. Therefore the argument in the previous paragraph can be repeated to show v > 0 in ft. Then, as for v before, we can find 8 > 0 such that v > 8d(x, To) and hence u + £o^ > —{8/M)u in ft, that is u + £o(l + 8/My1v

> 0 in ft,

contradicting the minimality of £o- Therefore the case min^u < 0 does not occur. This finishes the proof. • Remark 2.5 The careful reader might have observed that in the above proof we have not used a > 0 in the boundary condition. Indeed, this restriction can be removed; it is also not needed for Theorem 2.1 to hold. We refer to [Amann(1983)] and [Amann-L6pez(1998)] for more details.

2.2

Maximum principle in

W2'N(fl)

Let L be an elliptic operator in a bounded domain ft C RN, of the form Lu = alj(x)Dij

+ bl{x)DiU + c(x)u,

Maximum

12

Principles and

Applications

where a lJ € C(ft), bl,c £ L°°(ft), and for some positive constants CQ and Co, colCI2 < aij(x)Uj

< Col^l2, V£ £ fl", Vx 6 ft.

We say that the maximum principle holds for L and ft if for any v £

w?0f(n), Lv > 0 in ft, Irmx_+,9nv(z) < 0

(2.1)

implies i> < 0 in ft. A useful sufficient condition for the maximum principle of L in ft to hold is the following. Theorem 2.6 Let L be as above. Assume diam(ft) < d. Then there exists 6 > 0 depending on L and d such that the maximum principle for L in ft holds if the measure of ft satisfies |ft| < 5. Proof.

Write c = c+ — c~ and rewrite (2.1) as aijDijV + VDiV - c~v >

-c+v+.

We then apply the Aleksandrov weak maximum principle Theorem A.40 to obtain supv < C\\c+v+\\LN(n)

< Ci maxv+|ft| 1 / J v .

It follows that v < 0 in ft provided that C i j f t l 1 ^ < 1. Therefore we can take 6 = C^N. • In [Berestycki-Nirenberg-Varadhan(1994)], the results in Theorem 2.1 and Theorem 2.4 with Dirichlet boundary conditions are established for solutions in Wl(Jc (ft) with L as above; they do not require any smoothness condition on i G W^(i}) satisfying

13

D C(fi), Vp > N,

(L + Ai)(/>i = 0, / > 0 m Q, i = 0 on dfi. (MJ Suppose ip G W,20*(n) n C(H) satisfies (L + X)ip = 0, I/J > 0 in fi, ip — 0 on dCl. Then A = Ai and ^ e's a constant multiple of 4>\. (Hi) Suppose ip G Wj0£(fl) n C(fi) (possibly complex valued) satisfies (L + A)V> = 0 m f], V = 0 on A1; and i / ^ is real then it changes sign in fi. Proof, (i) We will use the Lp theory for elliptic operators, the strong maximum principle and the Krein-Rutman theorem. Choose £ > 0 so that c — ^ < 0 in fi, and denote L^u = Lu — £u. Let K be the positive cone in X :— Co(fl) consisting of nonnegative functions. For any v G C(Q) c Lp{Vt), Theorem A.33 guarantees that the problem —L$u = v in fi, w = 0 on dQ, has a unique solution u G W ^ ( f i ) n C(ft), Vp > 1. Clearly T : X -> X defined by Tv = u is a linear operator. Let us now show that it is bounded and compact. Let u\ = T(||u||oo), where |)u||oo = sup n v. We find that —Lj(u\ — u) > 0 in fi, («i — u) = 0 on 9 0 . Applying the Alksandrov weak maximum principle (Theorem A.40) we deduce u\ — u > 0 in fi, i.e., Tv < T(||u||oo) = IMIooT'fo, uo = 1 in &• Similarly, Tv > - H ^ T u o - Therefore, ||Tu||oo < M||t>||oo, M = | | T M 0 | U . So T is bounded. To see that T is compact, let vn be a bounded sequence in X, say ||wn||oo < C- Then from VQ := TUQ —> 0 uniformly as a; —> o < r u n < CVQ in Q, we see that Tvn —> 0 uniformly in n and —> dfi. For any fi' CC fi, by the interior estimate (Theorem A.26) and the Sobolev imbedding theorem, we find that {Tvn} is a bounded set in C 1 (fi'). Thus {Tvn} is bounded and equicontinuous in C(Cl). Hence it is precompact in X, by the well-known Arzela-Ascoli theorem. This proves that T is compact. Moreover, by the strong maximum principle (Theorem A.41), if v G K \ {0}, then Tv > 0 in ft. This fact implies that T is irreducible (see [Sweers(1992)] page 254 for a proof) and hence r(T) > 0 by [de Pagter(1986)]. As clearly K is reproducing, we can apply the Krein-Rutman

14

Maximum

Principles and

Applications

theorem to conclude that there exists v G K \ {0} such that Tv = r(T)v. Therefore (pi :=Tv satisfies (pi > 0 in ft and Lcfii + Ai^i = 0 in Q,, (pi = 0 on ! G Wfo'cp(0) n C(ft), Vp > N. This proves conclusion (i). (ii) We make use of Theorem 2.6 above. Firstly we choose a smooth subdomain K CC f2 such that jfi \ K\ is small enough such that the maximum principle for £(-A) in Q' :— f2 \ K holds. We can now find t > s > 0 such that sip < (pi < tip in if. For definiteness, we assume that A < Ai; the case A > Ai can be treated similarly. Therefore L{4>\ - sip) = Xstp - Ai^i < -\( 0 in fi, contradicting the maximality of so- Therefore, ip must be a constant multiple of (p\ and A = Ai. This proves conclusion (ii). (iii) If ip is real (and hence A must be real), then by (ii) ip must change sign in Q. If A < Ai then by considering the maximal so > 0 such that 0 in Q as in the proof of (ii) above, we deduce a contradiction in a similar way. Therefore A > Ai in this case. Suppose now ip is a complex function. Consider the product domain ft = fl x fl, with points (a;, y), x G Q, y G Q. Let Lx denote the action of L in the x variables, and Ly its action in the y variable and set L — Lx + Lv. It is easily checked that Lfa + 2Ai^i = 0 in Q,, (pi = 0 on d(l,

Maximum

Principles

Revisited

15

where <j>i(x,y) := <j>i(x)4>i(y) > 0. Therefore by conclusions (i) and (ii) applied to (L, ft) we find that 2X\ is its principal eigenvalue. (We note that ft has Lipschitz boundary.) By a direct computation, tp(x,y) := ip(x)tp(y) + / ip(x)tp(y) is a real valued function satisfying Lip + (A + X)tp = 0 in ft, -0 = 0 on dft. We now apply our argument in the last paragraph and conclude that A+A > 2Ai, i.e., Re(X) > Ai. This finishes the proof of conclusion (hi). • Using the technique in the proof of (ii) above, we can now prove the following result by arguing similarly as in the proof of Theorem 2.4. The details are left to the reader as an exercise. Theorem 2.8 Let the above assumptions on L hold and suppose that ft has Lipschitz boundary. Then the following are equivalent: (i) The maximum principle holds for L in fl. (ii) There exists (/> £ Wf^ (fl) n C(Tl) such that L(f> < 0 in ft, 4> > 0 on 9ft, and at least one inequality cannot be replaced by equality. (Hi) The principal eigenvalue \\ > 0. Remark 2.9 If 5ft is smooth enough, say C 2 , then we can apply the global Lp theory and prove similar results to Theorems 2.7 and 2.8 for Neumann and Robin (or mixed) boundary value problems. Similar results can also be proved for systems of equations that are cooperative in nature; see [Sweers(1992)] for more details. If u G C 2 (ft) has a local maximum at XQ € ft, then clearly DU(XQ) = 0 and [Diju(xo)] is a seminegative definite matrix. Let u e W^0'c (ft). Then by Theorem A.6, the classical partial derivatives Diu(x) and Dijii(x) exist for a.e. x in any ft' CC ft. The following theorem extends the above mentioned result for C 2 functions to W^c (ft) functions to some extent, and is known as Bony's maximum principle. (Bony first proved the result for W^'P(ft) functions with p > N, see [Bony(1967)]; P.L. Lions then extended it to include the case p = TV.) Theorem 2.10 ([Lions(1983)] Corrolary 2) Suppose u £ W^(fl) has a local maximum at XQ £ ft. Then there exists a subset A of fl such that the

16

Maximum

Principles

and

Applications

classical derivatives Dtu and DijU exist on A and \A n S r (a;o)| > 0 for any r > 0; \imx^XO:XeA^o,13Diju(x)

< 0,

lim

|Du(x)| = 0.

x—*xo,x£A

Using Theorem 2.10, it is easy to see that the Hopf boundary lemma (Lemma A.35) still holds for the case where the solution u 6 W2'N(Q). The proof is the same as the standard one. Theorem 2.11 The conclusions in Lemma A.35 still hold when u £ C 2 (n) is replaced by u £ W2>N({1).

Chapter 3

The Moving Plane Method

The moving plane method is a clever way of using the maximum principle to obtain qualitative properties of positive solutions of some elliptic equations, notably the symmetry of such solutions. It was introduced by A.D. Alexandroff in his study of surfaces of constant mean curvature, and successfully used by J. Serrin [Serrin(1971)] in proving symmetry properties for some over-determined elliptic problems. It has become well-known through the works of Gidas-Ni-Nirenberg [Gidas-Ni-Nirenberg(1979)] and [Gidas-Ni-Nirenberg(1981)], where it was used to prove symmetry results for positive solutions of rather general nonlinear elliptic problems. Since then, this method has been further developed and used in a variety of problems by many people. In this chapter, we will look at several symmetry problems and related questions.

3.1

Symmetry over bounded domains

Consider the following elliptic boundary value problem - A u = f(u) in fl, u = 0 on dSl,

(3.1)

where fi is a bounded domain in RN and / is a Lipschitz continuous function. The Laplacian operator Au(x) = T,?=1uXiXi(x) is invariant under many operations on the variable x € RN. For example, it is invariant under translations and rotations. Therefore, if u is a solution to (3.1) and if £1 is invariant under such an operation, say T, then UT{X) :— 17

18

Maximum

Principles

and

Applications

u(Tx) is also a solution. If there is a unique solution, then necessarily UT = u. An important problem it to understand when we always have UT — u even if uniqueness does not hold. In [Gidas-Ni-Nirenberg(1979)], Gidas, Ni and Nirenberg proved, by making use of the maximum principle, that any positive solution of (3.1) is radially symmetric when ft is a ball. The technique they employ is called the "moving plane method". Their original proof was later considerably simplified (see [Berestycki-Nirenberg(1991)]) by making use of Theorem 2.6 in the previous chapter. In this section, we will prove this symmetry result and look at some related problems. Definition 3.1 A domain ft C RN is called Steiner-symmetric with respect to the plane x\ = 0, if it is convex and symmetric in the xi-direction, i.e., for every (xi,x') := (x\,X2, ...,xpj) S ft, {{t,x') : \t\ < |xi|} C ft. Clearly an arbitrary ball in RN is Steiner-symmetric with respect to any plane passing through its center. However, the so called "Star of David" (see Fig. 1) in R2 is Steiner-symmetric with respect to the plane X\ = 0, but not with respect to x-i = 0. Theorem 3.2 Let ft be an arbitrary bounded domain in RN which is Steiner symmetric with respect to the plane x\ = 0. Let u be a positive solution of (3.1) belonging to C 2 (ft)nC(ft), and assume that f is Lipschitz continuous. Then u is symmetric with respect to x\, and uXl{x) < 0 for i e O with x\ > 0. Proof. For convenience of notation, we write x = {x\,y). We will show that uXl(x) > 0 if x = (xi,y) € ft satisfies xx < 0, and for (x\,y), {x[,y) G ft, u(xi,y)

—x\, and we deduce u{xi, y) < u(-xi,

y) if xi < 0.

Since we may prove the same for u(—x\,y), the above inequality implies u(xi,y) — u(—xi,y), i.e., u is symmetric in x\. As will become clear later, the proof of uXl(x) < 0 for xi > 0 also follows from (3.2). We now use the moving plane method to prove (3.2). Let a > 0 be given by -a — iaixenxiF ° r - a < A < 0, let T\ denote the plane x\ = A

The Moving Plane Method

19

and define S(A) = { i e f i : i i < A } . For x = {x\,y) £ £(A), set ux(x\,y)

= u(2X — x\,y),

w(x,A) = ux{x) — u(x).

We easily see that -Aux

= f(ux)

in E(A).

Since / is Lipschitz continuous, we can write f(ux(x))

- f(u(x)) = c(x,X)[ux(x)

-

u(x)\,

where c(x, A) is some bounded function for x € E(A), A e i J 1 . We can find some constant 6 > 0 such that \c(x, A)| < b for all such x and A. If A > —a is close to —a, then S(A) is narrow in the xi-direction, and by Theorem 2.6, the maximum principle can be applied to —Aw = c(x, \)w in S(A). On 0 in S(A). Let (j, := sup{A < 0 : w > 0 on S(A') for all - a < A' < A}. We want to show that fi = 0. We suppose \i < 0 and argue by contradiction. By continuity, w(x,fj,) > 0 in £(//). Since w is not identically 0 on 3S(,u), by the usual maximum principle or Harnack inequality (see Theorem A.42), w > 0 in S(/u). We will show that for all positive small e, w(x, fJ. + e) > 0 in T,(fi + e). Due to the definition of /z, this would give us the desired contradiction. By Theorem 2.6, we can find 6 > 0 small so that the maximum principle holds for Lu := An + c(x, X)u over any D C D, provided that \D\ < S. We now choose a closed set K in S(/x) such that |S(/i) \K\ < 5/2. Clearly, by compactness, w(x, ju) > 0 for x e K. Hence, by continuity we can find eo > 0 small so that w(x, fi + e) > 0, \E((j, + e)\K] 0 in E. Therefore w > 0 in E(/x + e), as we wanted. This proves that /j, = 0, and (3.2) thus follows. Finally we show that uXl(x) > 0 if x = (xi,y) G Q is such that xi < 0. Since for any fixed A e (—a,0), w(x, A) > 0 in E(A) and w;(a;,A) = 0 on T\, we can apply the Hopf boundary lemma (see Lemma A.35) to conclude that wXl(x, A) < 0 when x € fi and xi = A. Since u>Xl(:r, A) = — 2uXl(x) when £i = A, we obtain uXl > 0. • If fl is a ball, then by Theorem 3.2 we know that a positive solution of (3.1) is symmetric about any plane passing through the center of fi. Therefore, we have the following result. Corollary 3.3 If H is a ball and f is Lipschitz continuous, then any positive solution of (3.1) is radially symmetric. Remark 3.4 (i) Corollary 3.4 is not true for solutions of (3.1) which are not positive. For example, when Q is a ball, many eigenfunctions of —AM

= \u in fi, u = 0 on dVi

are not radially symmetric (though the one corresponding to the smallest eigenvalue does not change sign and hence must be radially symmetric). (ii) This corollary is also untrue if the Dirichlet boundary condition is replaced by other boundary conditions. A well-kown result of Ni and Takagi [Ni-Takagi(1993)] shows that the problem —Au = A(MP — u) in fi, Dvu = 0 on 3 denotes the dimension of the bounded smooth domain fi. (iii) By using the moving plane method and also a "rotating plane" method, it was proved in [Lin-Tagagi(2001)] that, if fi is a ball, then the positive solution in (ii) above (called the "least energy solution") is symmetric in any plane passing through the center of fl and XQ. Theorem 3.2 can be applied to domains with "corners". For example, if Q is a cube, then we can use Theorem 3.2 to conclude that any positive

The Moving Plane Method

21

solution of (3.1) is as symmetric as Q. So if 0 = {{x\,X2) Q R2 : \xi\ < 1, \x2\ < 1}, then any positive solution u of (3.1) is symmetric with respect to x\ = 0, to X2 = 0, to X\ + X2 — 0 and to x\ — X2 = 0, which in turn implies that u is invariant under rotations around the origin of degree 7r/2 and integer multiples of n/2. An interesting example is when ft. is the "star of David" in R2 (see Fig. 1 (a)). x2

O W m(a)

(b)

(c)

Fig.l Star of David

Let us use D to denote this particular f2. We easily see that D is Steinersymmetric with respect to any straight line passing through the center and an outward-pointing corner point. But it is not Steiner-symmetric with respect to any straight line passing through the center and an inwardpointing corner point. As a result, we can use Theorem 3.2 to see that any positive solution of (3.1) with fi = D is invariant under rotations around the center of D of degree 27r/3 and integer multiples of 27r/3 (see Fig.l (b)). But this does not reflect the full symmetry of D, as clearly D is invariant under rotations around its center of degree 7r/3. This problem was considered recently by Kawohl and Sweers [Kawohl-Sweers (2002)], who used a "sliding method" (another well-known technique of using the maximum principle) to show the following result. Proposition 3.5 Any positive solution u G C 2 (Q) n C(f2) of (3.1) with Q, — D is invariant under rotations around the center of D of degree 7r/3.

Maximum

22

Principles

and

Applications

Proof. Let us choose the coordinates as in Fig. 1 (c). Moreover, we use /i, I2 and I3 to denote the straight lines X2 = 0, \f3x\ — X2 = 0 and \/3x\ + X2 — 0. As indicated in Fig. 1 (c), we denote by D\ the part of D that is between l\ and l2 whose point has positive coordinates. Suppose that u G C 2 (fi) nC(fi) is a positive solution of (3.1) on D, and v is obtained by a rotation of M around the origin of degree 7r/3: v{x) = u(Rw/3x),

Re(xi,X2) = {xi cos6 + 2:2 sin #,2:1 sin# + 2:2 cos^).

Since D is Steiner-symmetric in the directions of l\, I2 and Z3, by Theorem 3.2 we know that u is symmetric with respect the the lines perpendicular to, respectively, h, h and ^3. Moreover, uXl(x\,X2) < 0 for (2:1,2:2) G D\. Similarly, vxl(xi,X2) < 0 for (xi,x2) G D\. Let us also note that the above symmetries of u imply that u{x) = v(x) for x G l\ UI2 U /3. We want to show that u = u on D\. This would imply u = v in .D. We use a sliding method. Let us denote by a the length of the side of D\ which lies on l\. For t G [0,a], set D\ := {(2:1,0:2) : (2:1 +t,X2) G I?i} and v*(2:1,12) = v(xi +t,x2),

(xi,x2)

G D\.

Clearly -Aw* = / V ) in D\. We set to show that v* < u on Di n Z>£ for all £ G [0,a]. For t < a but close to a, D\(t) := Di D £)j is a narrow set, and if/(2:) := u(x) — ^'(2:) satisfies - A w ' = c(2:,t)w/ in D(t), w* > 0 on dD(t), where c(x, t) is a bounded function and we have used the facts that u = v on Z2 and both w and v are decreasing in x\ in D\. Since w* is not identically 0 on dD(t), we can use Theorem 2.6 to conclude that w* > 0 in .D(i). Let to = inf{* G (0,a] : io*(a;) > 0 in D(s)Vs G [t,a]}. The above argument shows that to < a. If to > 0, then by continuity we obtain wto(x) >0 in D(to)- Moreover, - A w ' ° = c(x,t0)wto

in D{t0),

and wto is nonnegative and not identically 0 on dD(to). Therefore by the usual maximum principle or Harnack inequality, wto > 0 in D(t0). We now show that we can further "slide" v* under u to the right slightly, that is, there exists eo > 0 small such that wto~€ > 0 in D(to — e) for all e G (0,eo]. This would contradict the definition of to a n d hence prove

The Moving Plane Method

23

our claim that to = 0. Again we will use Theorem 2.6. Let 6 > 0 be small enough so that this theorem applies to Lu := Au + c(x, t)u when the underlying domain has measure less than 5. Then choose a closed set K in D(t0) such that |£>(t0) \K\ < 5/2. By compactness, wto~e(x) > 0 for x G K and all small e > 0, and \D(to — e) \ K\ < 5 for all such e. In D := D(to -e)\K, w(x) = wto~€(x) satisfies - A u ; = cw, and on dD, w is nonnegative and not identically 0. Therefore w > 0 in D by Theorem 2.6. Hence w > 0 in D(to — e), as we wanted. This proves to = 0 and thus v < u inDi. We can interchange the position of u and v in the above argument and hence deduce u < v in D\. Therefore u = v in D\. This finishes the proof. • Note that in the above proof, for the sliding method to work, it is important that we can first use Theorem 3.2 to guarantee that u = v on dD\ and both u(x) and v{x) are decreasing in x\ for x € Di. The symmetric domains in Fig. 2 below do not have any Steiner-symmetries and hence the above method does not work for them. Do positive solutions of (3.1) over these domains possess the symmetries of the domain? Kowohl and Sweers in [Kawohl-Sweers(2002)] conjecture that they do, but no proof (or counter example) is known yet.

Fig. 2 Symmetric domains without Steiner-symmetry

3.2

Symmetry over the entire space

The moving plane method was used in [Gidas-Ni-Nirenberg(1981)j to study the radial symmetry of positive solutions of the following entire space prob-

24

Maximum

Principles

and

Applications

lem: -AM

= f(u) in RN {N > 2),

lim u(x) = 0.

(3.3)

|x|—>oo

The following result improves some results in [Gidas-Ni-Nirenberg(1981)] and is proved by Li and Ni [Li-Ni(1993)]. T h e o r e m 3.6 Suppose that f(s) is Lipschitz continuous and nonincreasing for sufficiently small s > 0. Then any positive solution u € C2(RN) of (3.3) is radially symmetric about some XQ G RN and ur < 0 for r = \x — XQ\ > 0. Proof. Let x = (xx,y) be an arbitrary point in RN. Its reflection with respect to the hyperplane T\ := {x G RN : Xi = A} will be denoted by x*, namely, xx = (2A — X\,y). Denote Y,\ := {x € RN : x\ < A}. We observe that, for A > max{xi,0}, -A,

,„, _ 4A(A-a; 1 )

l»1-N= \x\ ,J.+ \x* J Let u € C2(RN)

>0-

(3-4)

be a positive solution of (3.3) and define

A := {A G R1 : u(x) > u{xx) Vx G E A , uXl(a:) < 0 Mx G T A }. By our assumption on f(s) we can find ro > 0 such that f(s) is nonincreasing for s G (0,ro]. Since u(:r) —> 0 as |rr| —> oo, there exist Ro < R\ such that Ro > 1/ro, max u(x) < ro, max u(x) < mo := min u(x). |x|>fi 0

\x\>Ri

(3.5)

\x\ R\, w{x) := u(x) — u(xx) satisfies, for xeZx\BRo, where BR = {x£RN : \x\ < R], —Aw = c(x)w in

SA

\

BR0,

w = 0 o n T\,

lim w(x) = 0, |a;|—*oo

where c{x) is a bounded non-positive function. Moreover, since A > R\ and x\ < A, we have |x A | > R\ and hence by (3.5) we find w(x) > 0 on ~BRa n S A .

(3.6)

The Moving Plane Method

25

Therefore we have —Aw = c(x)w in

SA

\

BR0,

W

> 0 on d(Y,\ \ BRo),

lim w(x) = 0. |x|—»oo

By the strong maximum principle (see Theorem A.36) we deduce w > 0 in SA \ BR0 . Then the Hopf boundary lemma and the Harnack inequality imply w > 0 in EA \ BRo and wxi < 0 on Tx. In view of (3.6), we have proved w(x) > 0 in SA for all A > R\. Hence [i?i,oo) C A. S t e p 2. We prove that A is an open set in (0, oo). Let A0 £ A n (0, oo). We want to show that (A0 - e, A0 + e) C A n (0, oo) for all small e > 0. Since we already knew that [Ri,oo) C A, we may assume that Ao G (0, i?i]. It follows from the assumption Ao € A that u(x) - u(xx°) > 0 in SA 0 , uXl < 0 on TXo.

(3.7)

By continuity, we can find e\ > 0 small such that uXl (x) < 0 if |a;| < i?i + 1 and A0 - 4ei < x: < A0 + 4ei.

(3.8)

It follows that, for any A £ (Ao — ei, Ao + ei), u(x) > u{xx) uXl < 0

in {x G S f i l + i : A0 - 2ei < xi < A}, on TxnBRl+1.

. >

Denote 6 = min{u(a;) - u(xx°) : -(Rr

+ 1)
0, and hence u(x) - u{xx) > 0 in {x G 5 R l + 1 : -(Ri

+ 1) < xx < A0 - 2ei}

(3.10)

for any A G (Ao — e, A + e) if we take e G (0, e\) small enough. Combining (3.9) and (3.10), we obtain, for A € (A0 - e, A0 + e), u{x) > u(xx) in B~Rl+i n E A , uXl < 0 on TxnBRl+i.

(3.11)

Now for A G (Ao — e, Ao + e), define w{x) = u(x) — u(xx) and we find that w jk 0 in SA \ -B_Rj+i and similar to before —Aw = cw in SA \ BRl+i,

w > 0 on C>(SA \ Bfl 1 + i),

lim w(a;) = 0. \x\—*oo

Maximum

26

Principles

and

Applications

By the choice of i?i, we know c(x) < 0 in SA \-Bfli+i- Hence by the strong maximum principle, for A G (Ao — e, Ao + e), w > 0 in

EA

\BRl+1,

wXl < 0 on T\

\-BR1+I,

that is, u(:r) - u(a;A) > 0 in EA \ BRl+1,

uXl < 0 on T\ \

BRl+i.

Together with (3.11), this proves (Ao — e, Ao + e) C A, as we wanted. Step 3. Either A n (0, oo) = (0,oo) or u{x) = u(xXl) for some Aj > 0. Let (Ai, oo) be the component of the open set A n (0, oo) containing (i?i,oo). By the continuity of u we have w{x) = u(x) - u(xXl) > 0 Vx 6 E A l . Moreover, —Aw = cw in

SAJ

, w = 0 on T\1,

lim w{x) = 0. \x\—+oo

Since w > 0 in S A l , though we do not know whether c is non-positive, we can apply the Harnack inequality to conclude that either w = 0 in T,\1 or w > 0 in S A l , and in the latter case, by the Hopf boundary lemma, wXl < 0 on T\ x . But then we find Ai G A in the later case. By Step 2, this is possible only if Ai = 0 . This finishes our proof of Step 3. Step 4. Completion of the proof. If u{x) = u(xXl) then u is symmetric with respect to the plane T\lt and since (Ai,oo) c A, we know that uXl < 0 for x\ > \\. If the other alternative occurs in Step 3, then other u(x) = u(x°) or u(x) > u(x°). In the former case u is symmetric with respect to To and as before uXl < 0 for xi > 0; in the latter case, we must have uXl < 0 on To and we can apply the argument in Step 3 to w(—x\, y) to deduce that for some A2 < 0, u(x) = u(xX2) and uXl > 0 for xi < A2. Hence u is always symmetric with respect to some T\, and is strictly decreasing away from T\. Since (3.3) is invariant under rotations, we may take any direction as the xi-direction and conclude that for any given direction, u is symmetric with respect to some hyperplane T perpendicular to that direction, and is strictly decreasing away from T. This implies that u is radially symmetric about some point XQ in RN and ur < 0 for r — \x — x0\ > 0. The proof is complete. • An important example covered by Theorem 3.6 is f(s) = sp — s, p > 1. However, Theorem 3.6 excludes important functions like f(s) = sp for s > 0

The Moving Plane Method

27

and p > 1. The following result, whose proof can be found in [Gidas-NiNirenberg(1981)] (see also [Fraenkel(2000)]), covers these cases. Theorem 3.7 Suppose that u £ C2(RN) is a positive solution of (3.3), N > 3 and u(x) = 0(|a;|~ m ) at infinity for some m > 0. Suppose further that (i) for s £ [0,uo], where UQ — maxflw u(x), f(s) = /i(s) + /2(s) with fx Lipschitz continuous and fi continuous and non-decreasing, (ii) for some a > max{(iV + l)/ro, (2/m) + 1}, f(s) = 0(sa) near s = 0. Then u(x) is radially symmetric about some point XQ G -R^, and ur < 0 for r = |x — XQ\ > 0. Moreover, there exists some k > 0 such that

lim l a r l " - 2 ^ ) = A;. |x|—»oo

A well-known example covered by Theorem 3.7 is / ( s ) = s(^+2)/(iV-2) In this case, we can take m = N - 2 and a = (N + 2)/(N — 2). Therefore, any positive solution of - A « = u(^+a)/(Ar-2) in RN 22 - ^ „ „ l „ l

u{x) = 0(\x\ ~

(N

>

3)

. _

) as \x\ —* co

(3-

1 2

)

must be radially symmetric about some XQ € RN • Let us note that all such solutions are explicitly known (see [Cerverno-Jacobs-Nohl(1977)]), and they are given by i(3!)=

(/+lx-xo|0

'A>M0€"

In [Chen-Li(1991)], by making use of the Kelvin transformation, the condition u(x) = 0(\x\2~N) as |x| —> oo in (3.12) was removed. We will discuss this in detail in Section 7.4 later. Further related results can be found in [Chen-Li-Ou(2005)], [Chen-Li-Ou(2003)], [Busca-Manasevich(2002)] and the references therein. R e m a r k 3.8 The moving plane method can also be applied to certain so called cooperative systems of elliptic equations and to deduce symmetry of the positive solutions; see [Troy(1981)], [de Figueiredo(1994)], [BuscaSirakov(2000)] and [Busca-Manasevich(2002)].

Maximum

28

3.3

Principles

and

Applications

Positivity of nonnegative solutions

The symmetry results in Sections 3.1 and 3.2 are valid for positive solutions. In many applications, the natural solution is nonnegative. Then one is interested to know whether such a solution is strictly positive; if it can vanish in a set of positive measure yet not identically zero, such a vanishing set is often called the dead core of the solution. Consider now the problem - A M = f(u) in

ft,

(3.13)

where ft is a domain in RN (not necessarily bounded). If f{u) is Lipschitz continuous and /(0) > 0, then f(u) > f{u) — /(0) > — Cu for some positive constant C, and we obtain, for any nonnegative solution u of the above problem, Au - Cu < 0 in ft. By the strong maximum principle (see Theorem A.38), we deduce either u = 0 in ft or u > 0 in ft. The strong maximum principle has been generalized to various nonlinear equations. The following well-known result is due to J.L. Vazquez [Vazquez(1984)] (with the converse part from [Benilan-BrezisCrandall(1975)]); see [Pucci-Serrin(2004)] and the references therein for further discussions. Theorem 3.9

Ljocity

Let u £ Lj0C(Q) be such that u > 0 a.e. in ft, Au €

and

Au < (3{u) a.e. in {x e ft : 0 < u(x) < a}, where a is a positive constant and /3 : [0, a] —> R1 is a continuous nondecreasing function with /?(0) = 0. / / either /3(d) = 0 for some 6 > 0 (and hence (5{t) = 0 on [0,6] due to the monotonicity), or (5{t) > 0 in (0, S) C (0, a) and 1-1/2 Jo

L

Jo

0{t)dt

ds = oo,

then u is either identically 0 in ft or strictly positive in ft.

(3.14)

The Moving Plane Method

29

Conversely, i//3(0) = 0, (3(t) > 0 in (0,6) for some 6 > 0 and (3.14) not satisfied, i.e.,

is

T-l/2

! \ f'mdt

as < oo, Jo l Jo then for every XQ G R N and every R > 0, there exists a function u € Cl{RN) with the properties that u > 0 , ^ 0, Au G L°°(RN), Aw = /3(u) ami w = 0 on RN

\

BR(XO).

From Theorem 3.9, we see immediately that if /(0) > 0, or /(0) = 0 and f(u) > —/3(u) for w > 0 small, where /? is a continuous non-decreasing function satisfying the conditions in the first part of Theorem 3.9, then any nonnegative solution of (3.13) is either identically 0 or strictly positive in When /(0) < 0, it turns out that, even if f(u) is very smooth, (3.13) may have nonnegative solutions which is not identically 0 but vanishes at certain interior points in fi. For example, the one dimensional problem —u" = u — 1 has a nonnegative solution u = 1 — cos a; that vanishes at x = 2kir, k = 0, ± 1 , ±2,.... More generally, consider -u" = f(u) in (0, a), «(0) - u(a) = 0,

(3.15)

where / : [0, oo) —* R1 is a continuous function such that for some constant

P>0, (i) f(s) 0 V s G ( / ? , o o ) ; (iii) f™ f(s)ds = +oo. Denote F(s) = f* f(t)dt. From the above assumptions on / we easily see that there exists a unique 6 > (3 such that F(0) = F(9) = 0, F(s) < 0 Vs G (0,0). Moreover, [-2F(U)]-1/2^M l-f(9)/2}(8

> [-/(0)/2]u for all small u > 0 and -F(u) - u) for all u < 9 with (9 - u) small.

=

Maximum

30

Principles

and

Applications

Let x = x(u) = / [-2F(s)}~1/2ds, Jo

0 0 in (0, 2L) U (2L, 4L). This gives an example where a nonnegative solution of (3.15) (with a — 4L) vanishes at certain point inside the domain but is not identically 0. Surprisingly, it has been shown by A. Castro and R. Shivaji [CastroShivaji(1988)] that examples like the one above can only be found in dimension 1! More precisely, if the interval (0, a) is replaced by a high dimension ball B, the corresponding problem of (3.15) in B cannot have a nontrivial nonnegative solution that vanishes somewhere in B. Theorem 3.10 If £1 is a ball of dimension no less than 2, f(u) is locally Lipschitz continuous on [0, oo) and /(0) < 0, then any nonnegative solution of (3.13) must be positive. (Note that 0 is no longer a solution due to

/(o) < o.; Proof. The theorem will be proved by making use of the moving plane method. For each given direction v in RN, i.e., a vector u £ RN with \v\ = 1, and each A G R1, we define a{u) = inf x • v\ Tx = {x G RN : x • v = A}; fi^ = {x G fi : x • u < A}.

The Moving Plane Method

31

Therefore Tx is a hyperplane in RN perpendicular to v, Qx is a part of fi lying on one side of Tx, and when A = a(i/), then Tx touches Ct at exactly one point on its boundary. Let R1^ be the reflection map in RN in the hyperplane Tx: R\{x) = x + 2(A - x • v)v, x e RN. We will use the notation

x\ = Rl(x), (niY = RWJ. Under the above assumptions, if A — a(v) is positive and small, (fi^)' fl. We now can define

c

\*(u) = sup{/i > a{v) : (Clx)' C Q for every A < /i}. We easily see that A*(v) can be characterized as the unique value of A such that Tx passes through the center of Q. Now as in the proof of Theorem 3.2, by comparing u(x) with u\(x) := u(xvx) over Qux for a(v) < A < \*(v), and making use of Theorem 2.6 and the usual weak maximum principle, we deduce that uvx(x) > u(x) on Q^. It follows that u(x) is non-decreasing in the direction v in the half ball fiw„NSuppose by contradiction that u(x§) = 0 for some XQ £ Cl. Then the above proved mono tonicity property of u(x) implies that u(x) = 0 on the line segment connecting XQ to a boundary point y 6 dCl such that the vector yxo is in the same direction as some v satisfying xo G fiw„\- All these line segments form a cone K mil with vertex XQ. (In fact, if we choose the coordinates such that ft = -B^(O) and XQ = (a, 0, ...,0) with 0 < a < R, then K = {x = (x\, ...,XN) £ BR(0) : X\ > a}.) Therefore u is identically 0 in K, and hence Aw = 0 in K, which is a contradiction to the assumption that /(0) < 0. • The above proof is modelled on the one given in [Damascelli-PacellaRamaswamy(2003)], and can be extended to bounded domains more general than a ball; see [Castro-Shivaji(1988)] and [Damascelli-PacellaRamaswamy(2003)] for more general results. However, it is unknown so far whether Theorem 3.9 holds for any bounded smooth domain in RN with N > 2. R e m a r k 3.11 The moving plan method has been extended to some quasilinear elliptic problems; see [Damascelli-Pacella(1998)], [DamascelliPacella-Ramaswamy(2003)], [Damascelli-Sciunzi(2004)] and the references therein.

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Chapter 4

The Method of Upper and Lower Solutions The "upper and lower solution method", also known as the super and subsolution method, is a basic tool in nonlinear partial differential equations. In this chapter, we discuss some classical and weak versions of this method. The weak version is often more convenient to use, though it has some structural requirements for the elliptic operator which do not appear in the classical version.

4.1

Classical upper and lower solutions

Let L, ft and B be as in section 2.1, namely, Lu = alj(x)DijU + bl(x)DiU + c(x)u has Ca(£l) coefficients and is strictly uniformly elliptic in the bounded domain CI which has C2'a boundary 0 . We now consider the elliptic boundary value problem —Lu = f(x, u) in Q, Bu = 4>(x) on dfl, 33

(4-1)

Maximum

34

Principles and

Applications

where / G Ca(fl x / ) , i" is some finite interval in R1, (f>\r0 G C 2 ' a (ro) and Definition 4.1 A function u 6 C 2 ' a (f2) is called an upper solution to (4.1) if it satisfies —Lu > f(x, u) in ft, ^ u > ^ on G C2'a(fl) are lower and upper solutions to (4-1), respectively, and v < w in fl. Moreover, suppose that f G Ca(fl x / ) satisfies, for some m > 0, / ( x , s) - / ( x , t) > - m ( s - J ) V i e n , s , t 6 [v(x), w(x)], s > t,

(4.2)

where I = [ram.^v{x),m.&x^iui{x)\. Then (4-1) has at least one solution u satisfying v < u < w in fl. Moreover, if we define {un} by —Lun + m*un = f(x, u„_i) + m*w„_i in fl, Bun = on dfl,

(4.3)

where m* > max{m, maxjy |c(x)|}, then, with UQ = v, un converges from, below to a solution u* of (4-1) and with UQ = w, un converges from above to a solution u* of (4-1)- Furthermore, the convergence of {un} holds in C2(fl), and any solution u of (4-1) with v < u < w satisfies w* < u < u*. In other words, M* and u* are the minimal and maximal solutions of (4-1) in the order interval \v,w\. Proof. Define —L*u = —Lu + m*u. Since — c{x) + m* > 0, the strong maximum principle holds for (L*,B, fl). With u0 — v, we find, —L*(u\ — M0) > 0 in fl, B{u\ — u0) > 0 on dfl, and -L*(ui

- w) < 0 in fl, B(ui - w) < 0 on dfl.

Therefore it follows from the maximum principle that w > u\ > UQ in fl. Now for i > 1, due to (4.2) we have, inductively, —L*(ui+i —Ui)>0

in fl, B(ui+i

- ut) > 0 on dfl,

—L*(uj+i - w) < 0 in fl, B(ui+i

- w) < 0 on dfl.

and

The Method of Upper and Lower

Solutions

35

Hence w > ui+i > m in 0 . Therefore {un} is an increasing sequence of functions bounded from above by w. We may define «» := limn^oo unSince {m*un + f(-, un)} is a bounded sequence in L°°(fl), by the Lp theory (see Theorems A.28 and A.29), {un} is a bounded sequence in W2,P(Q.) for any p > 1. By the Sobolev imbedding theorem (Theorem A.62 and Remark A.18), it follows that {un} is bounded in C 1 (fi), which in turn implies that {m*un + f(-,un)} is bounded in C a (ft). Therefore by the Schauder theory (Theorem 4.4), we find that {un} is bounded in C2'a(Q) and hence it has a convergent subsequence in C2(fi). As we already know that un converges to u , point-wisely in Q, we must have u„ —• u» in C 2 (fi). So we can now pass to the limit in (4.3) to obtain —Lu* + m*u* = f{x, u*) + TO*W* in fi, Bu» = on un > v, we find w > u» > v. If M is any solution of (4.1) satisfying w > u > v, then we can replace w in the above discussion by u and obtain u > un > v and hence u > u*. This shows that u* is the minimal solution of (4.1) in the order interval [v,w\. If we take UQ = w, we can analogously show that un decreases to a maximal solution u* in the order interval [v,w], and the convergence holds in C 2 (fi). We leave the details to the interested reader. D Condition (4.2) is always satisfied if f(x, s) is Lipschitz continuous in s. However, this condition is not necessary for the existence of a maximal and minimal solution in [v, w]. The following result is essentially due to H. Amann [Amann(1971)], where condition (4.2) is not needed. Theorem 4.3 Suppose that v, w G C2'a(Q) are lower and upper solutions to (4-1), respectively, and v <w in Q. Moreover, suppose that f £ Ca(fl X I) with I = [minQv(x),maXftW(x)]. Then (4-1) has a minimal solution M» and a maximal solution u* in the order interval \v,w\. If furthermore, c(x) < 0 and f(x, s) is non-increasing in s for s e / , then u , = u* and hence (4-1) has a unique solution in [v,w]. Proof. Firstly we may assume c > 0 in Q, for otherwise, we can replace L by L + rn and f(x, s) by f(x, s) + ms with some large positive m, and all the conditions in the theorem are retained (except for the uniqueness part, which will be treated separately).

36

Maximum

Principles

and

Applications

Since f(x, s) is Holder continuous, there exists some 7 > 0 such that \f(x, s) - f(x, t)\ < 7 k - t\a Vx e H, s, t G / .

(4.4)

We now divide our discussions below into several steps. Step 1. If f(x, s) is non-increasing in s for s £ I, and c(x) < 0 in fi, then (4.1) has at most one solution in [v,w] Let u\ and u2 be solutions of (4.1) in [v, w]. Set Qi := {x 6 fl : «i(a;) > ^ ( z ) } - We show that fii must be empty. Otherwise, it is an open set, and by the monotonicity assumption on / , we have —L(u\—u2) < 0 in ili, &(ui—U2) = 0 o n diliDdil,

ui—n?, — 0 on SfiiDfi.

Therefore, by the maximum principle (applied to every component of Qi), we obtain ui < u2 in fii, contradicting its definition. Hence Qi is empty and ui < 112 in Q,. We can similarly prove that 112 < ui- Therefore wi = ^2Step 2. If w, w S C 2 ' a (fi) are lower and upper solutions of (4.1) and v < w < w < w, then the problem -Lu — f(x, w(x)) — 7(14 — w(x))a in fl, Bu = <j> on dVL

(4.5)

has a unique solution w* in the order interval \w, w], and the problem -Lu = f(x, w(x)) + -y(w(x) - u)a in fl, Bu = on dQ,

(4.6)

has a unique solution w* in [w,w]. By Step 1 we know that each of (4.5) and (4.6) has at most one solution in [tD,w]. We will use Theorem 4.2 and a perturbation argument to show that (4.5) has a solution in [w, w]\ the proof for (4.6) is similar. Let e(x) be the unique solution of —Lu = 1 in il, Bu = 0 on Q. Now for every positive integer n, we define wn = w — (l/n)e and u-w(x)-\ where e > 0 is so small that 1/n > 7(—e(x)) .

e(x)\

,

The Method of Upper and Lower

Solutions

37

Then we easily checks that —Lwn < fn(x,wn)

in fi, Bwn < on dfi,

and by (4.4), —Lw > fn{x, w) in fi, Bw > / on Oil,. Let us observe the important fact that f„(x, s) is Lipschitz continuous in s when s€

[wn(x),w(x)].

Therefore we can use Theorem 4.2 to conclude that the problem —Lu = fn(x, u) in fl, Bu = on dfl

(4.7)

has at least one solution wn in [«;„,«)]. By Step 1, we know this solution is unique. Since / n + i(:r,u; n +i(a;)) > fn(x,wn+1(x)), and wn+1 > u)„ +1 > wn, we can apply Theorem 4.2 for the pair of lower and upper solutions {wn, wn+i) to (4.7) to conclude that it has at least one solution w' in [wn, w n +i]. Clearly w' is also in the order interval {w„,w]. Therefore by the uniqueness of wn we necesarily have w' = wn. It follows that wn < wn+i < w for all n. Let w* = lim n _ooW n . Then w < w* < w. Moreover, by the Lp theory, the Sobolev imbedding theorem and the Holder theory, we find that {wn} is bounded in C 2 , a (fi) and hence has a convergent subsequence in C2(Q). As we already know that wn —> u>* point-wisely, we must have wn —• u>» in C 2 (fi). Now we let n —> oo in (4.7) and find that w* is a solution to (4.5). The proof for the existence of a solution w* to (4.6) is analogous, where we replace wn by wn = w + (\/n)e and replace fn(x, s) by gn(x, s) = f(x, w(x)) + 7 f w(x) A

e(x) - sj .

The detailed argument is left to the interested reader. S t e p 3. Completion of the proof. In Step 2, we take, inductively (w,w) = (vn,w), where v0 — v and vn is the unique solution of (4.5) with w = v n - i - This is possible since by (4.4), we find that each vn is a lower solution of (4.1). Clearly v < vn-\ < vn < w. Let M» = limn-tooVn. Then as before, we can apply the Lp theory, the Sobolev imbedding theorem and the Holder theory to conclude that vn converges in C2(Q) to u„, and u* is a solution of (4.1) in the order interval [v, w]. We claim that w« is the minimal solution. Indeed, if u is any solution

38

Maximum

Principles and

Applications

of (4.1) in [v,w], then in the above argument, we replace w by u and we find that the vn is not changed and v < vn < u. Therefore u„ < u. We leave the proof for the existence of a maximal solution u* to the reader. • Remark 4.4 In [Amann(1971)], the case that 4>\ri depends on u was also discussed, where stronger smoothness conditions on the coefficients of L, etc. are needed. Consider next the problem -Au = f(x, u, Du) in ft, Bu = 0 on dfl,

(4.8)

N

where ft is a bounded domain in R as in Theorem 4.2 except that we only require dQ, to be C 2 , B is also as in Theorem 4.2 except that we only require a G C ^ T i ) , Au = ali (x)DijU, with aij e C(ft), oiJ' = a^ and a tf (a:)6& > 0 Vz € ft, £ e i? W \ {0}. Suppose that p > N. Then by the Sobolev imbedding theorem, W2'p(fl) is imbedded in C ^ Q ) . We say w G W2-P(Q) is a solution to (4.8) if -Au = f(x, u, Du) a.e. in Cl and Su = 0 on V* is called bounded if it maps bounded sets in V to bounded sets in V*. We have the following well-known result (see Theorem 2.3 in [Showalter(1997)]). Theorem 4.8 Let V be a separable reflexive Banach space, K a closed, convex non-empty subset of V, A : V —> V* a bounded pseudo-monotone operator, and f G V*. Assume that there is a VQ G K and p > 0 such that (Av, v-v0)

> (/, v - v0) \/v G K, \\v\\ > p.

(4.11)

Then there exists a solution of the following variational inequality: ueK

: (Au, v - u) > (/, v - u), Vv G K.

(4.12)

The Method of Upper and Lower

Solutions

41

Let us note that if we take K = V in (4.12), then u is a solution to (4.12) if and only if Au = f. Moreover, with K = V, (4.11) is satisfied if A is coercive, namely, (Av, v) MHoo \\v\\ hm

——-n— = ° ° -

Theorem 4.9 Suppose that v and w are weak lower and upper solutions of (4-10), respectively, and v < w a.e. in ft. Suppose further that there exist a constant c\ > 0 and a function k\ € Lq (ft) such that \p{x,t,0\Xidx+ Jn

[

p'(x,Tu,D(Tu))(j>dx

JQ

+P / ry{x,u)(j>dx, Jo. with u, 4> S W0 '*(fi) and some fixed constant /? > 0 to be determined below. For fixed u e Wo' 9 (fi), it is easy to check that A^(-,u,Du),

P'(;Tu,D(Tu)),

7(.,«)

e L*'(Q),

where q' = q/(q - 1). Therefore, £>(u, 0) is a bounded linear functional of / G W01,9(fi). Hence there exists a unique Fu G W~ 1 , 9 '(n) such that 6(u,0) = {Fu,4>)^ e ^ f i ) . S t e p 3. The operator F : W 0 1,9 ( fi ) ~» W~ 1,9 '(ft) is bounded, pseudomonotone and coercive. If u has a bound on its W 1,9 (fi) norm, say ||w|| < C, then it is easy to see that \b(u, 0 and all 0 G ^ ' ' ( f l ) . This implies that F is a bounded operator.

The Method of Upper and Lower

43

Solutions

To show that F is pseudo-monotone, we suppose that un converges weakly to u in WQ'9(Q) and lim„_ >00 (FM„,u n — u) < 0, i.e., lim n _ 0 0 6(u„,u n - u) < 0. Denote bi{u,4>) = Y$Ll / A-(:r,u,.Du)Xid:r, Jn b2{u,(fr)= / p'(x,Tu,T(Du))(/>dx

Jn

+ /3 / *y(x,u)(f>dx.

Jn

Since {un} is bounded in W0 ,q(0.), so is {Tu„}. Moreover, un —> w in L 9 (fi). Therefore, {p'(-,Tu n ,£>(Tu n ))} and {7(-,u n )} are bounded in L 9 '(fi). It follows that b2(un, un — u) —> 0 as n —» oo, which implies that lim„_oofri(wn, ti n - u) < 0.

(4-16)

Since A' satisfies (Al) — (A4), by a well-known result the operator F\ : W01,9(fi) -» W - 1 - ? ' ^ ) defined by b1(u,tf>) = (F1u,) \/cf>eW^(Q) is pseudo-monotone (see Theorem 6.1 in Chapter II of [Showalter(1997)]). Therefore (4.16) implies that h(u,u-v) / E ^ ! [ ^ ( a ; , w n , £ ) u ) - A'i(x,u,Du)](un Jn

-

u)Xidx u)Xidx.

Since u n —> u weakly in W0'9(Cl) and strongly in L9(f2), we easily see that E j l j [^(a;, un,Du)

- A[{x, u, Du)] -> 0 for a.e. a; e £1

Maximum

44

Principles

and

Applications

Therefore, for any given e > 0 there exists Qe C fi such that |fi £ | < e and HfLi [A'i(x, un, Du) - A'i(x, u, Du)] -> 0 uniformly on n \ Q€. By (A2), it follows that / E ^ [^-(z, u n , Du) - A'^x, u, Du)] (un Jn < f

u)Xidx

c0\un\q-l+c0\u\q-x+2c0\Du\q-l)\Dun-Du\dx

N{2k0 +

+ [ Jn\ne

\^1[A^(x,un,Du)-A!i(x,u,Du)]\\Dun-Du\dx

< Cdlfcoll^cn.) + llunlll;^.) + ll«lll; ( k) + \\Du\\l-lQc))\\Dun

-

Du\\LHa.)

+ | | E ^ ! [A'^x, un, Du) - Ai(x, u, Du)] || L ~(n\n e )\\Du n - £>«||z,i(n\ne)
( l / 2 ) H | * , ( n ) - C 4 ||U||.

By Young's inequality, for any e > 0, we have H ^ N I ^ n ) < (ei'/q')\\u\\i +

(e-yq)\\u\\lqm.

Therefore, if we choose e small enough and (3 > 0 large enough, we will have b{u,u) > (a/2)||u||« - C5\\u\\ -C0

Vu £

W^9(il).

This proves the coerciveness of F. We can now apply Theorem 4.8 with K = V = WQ'q(n) to conclude that the problem -A'u + p'{x,Tu,D{Tu))

+ /?7(x,u) = / in fi, w = 0 o n f f i

(4.19)

has a weak solution UQ € W0 'q(£l). Step 4. Let VQ = wo + g- Then w < vo < w and hence AVQ = Av0, Tvo = vo, 7(-, v0 — g) = 0, and vQ is a weak solution of (4.10). By definition, VQ satisfies E £ i / Ai{x,vo,Dv0)Xidx+ [ p(x,Tv0,D(Tv0))<j>dx Jn Jn +/? / 7 ( ^ ^ 0 - 5 ) ^ = / / ^ V^Wo1''^). in Vn

(4.20)

46

Maximum

Principles

and

Applications

Taking = (vo — w)+ we obtain / j(x,v0-g)4>dx=

/ [(v0 - w)+]qdx = \\(v0 ~

/ p(x,Tv0,D(Tv0))4> Jn

— / p(x, w,Dw)(v0 Jn

-

w)+\\lq{Q),

w)+dx,

and by (A3), T,^=1 / Jn

Ai(x,v0,Dv0)4>xidx Ai(x,w,Dv0)DXi(vQ-w)+dx

= EJIi / Jn

= S £ x / Ai{x,w,Dw)DXt(v0Jn +T,f=1 / LAi(x,W,DVQ) Jn

w)+dx — Ai(x,w,Dw)

> ££LX / Ai(x, w,Dw)DXi(v0 Jn

-

1

-

DXi(vo — w)+dx

+

w) dx.

Therefore from (4.20) we obtain / f(vo ~ w)+dx > Sf=! / ^4i(a;, w, Dw)DXi(vo — w)+dx Jn Jn + / p(x,w,Dw)(v0

- w)+dx + /3\\(v0 -

w)+\\"Lq{ny

Since w is a weak upper solution of (4.10) and (VQ — w)+ > 0, by definition, S^j

/ Aj(a;, to, Dw)DXi (v0 — w)+dx + / p(x,w,Dw)(v0 Jn Jn

— w)+dx

> / f(vo ~ w)+dx. Jn Thus we deduce ! f(vo - w) + dx > [ f(v0 - w)+dx + p\\(v0 - w)+\\L"(n)Jn Jn It follows that {3\\(v0 — w)+\\qLq,n) < 0 and hence (v0 — w)+ = 0, i.e., v0 < w. Similarly we can show that VQ > v. This proves Step 4 and hence finishes the proof of the Theorem. •

The Method of Upper and Lower

47

Solutions

Theorem 4.9 is due to Deuel and Hess [Deuel-Hess(1974)]. A natural question not addressed by this theorem is whether there exists a minimal and a maximal solution in the order interval [v,w] in W1,q(Q). Under a further condition on Ai, we can give a positive answer to this question by making use of the following result, which is essentially due to V.K. Le [Le(1998)] (see also [Dancer-Sweers(1989)] for a special case). L e m m a 4.10 that (A5)

Under conditions (Al) — (A4) for Ai, we suppose further Ai(x,t,£)

= Ai(x,£)

is independent

oft.

Let vi,V2 be weak lower solutions of (4-10); then m a x j v i , ^ } is a weak lower solution of (4-10). Similarly, if w\,W2 are weak upper solutions of (4-10), then so is min{u>i,u>2}Proof. We only prove the conclusion for weak lower solutions; the proof for weak upper solutions is analogous. Let v\,v2 S Wl'q(Q) be weak lower solutions of (4.10), and define v — max{ui,t>2}. We want to show that v is a weak lower solution of (4.10). We may write v = v\ + (u2 — ^ i ) + and by Theorem A.8, we find that

Dv1

in fti = { l £ [ l Vi > V2}

Dv = { Dvi Dvi = Dv Dv22

in ft<j ft0 •'= = {x {x £e ft •' vi Vi = — vV22}}

Dv2

in ft2 = { i e ( l V2 > V\ }

(4.21)

Therefore p(-,v,Dv)

=p{-,vi,Dvi)xsii

+p(-,v2,Dv2)xn\n1

€-L 9 '(ft).

Next we show that v < g on 5ft. To this end, we prove and use the following conclusion: If a,f3 e W^q{Q,), then max{a,/3} 6 WQ'"{Q.). Since (3 — a. £ WQ'q(fl), by definition, there exists rjn € Co°(fi) such that rjn —> j3 - a in the W 1,9 (ft) norm. It follows that, r)+ —> (/3 — a)+ in

Wx'q(ty.

It is clear that 77+ has compact support in ft, and therefore r)£ £ W£'9(n). Since w£'q{Sl) is a Banach space, we deduce (/3-a)+ e W£'q(Q) and hence max{a,(3} = a + ((3 — a)+ € W0,9(ft). Now we take a — (vi — g)+ and (3 = (V2 — g)+ and find that m a x ^ ! - g)+, (v2 - g)+} G

W^O).

Maximum

48

Principles

and

Applications

On the other hand, it is direct to check that max{(wi — g)+(x), (v2 — g)+(x)} = (v — g)+(x)

for a.e. i £ f i .

Hence (v - g)+ e W01,9(ft), i.e., v < g on dfl. It remains to show that a(v, )+/ p(x, v, Dv)dx < (/, ) V^ e W01,9(fi) with / > 0 a.e. in fi. By classical density results for Wl,q{0) (see Theorem A.11), there exists a sequence {wn} C C°°(f2) such that w„ ^ w := V2 — vi in W /1 ' 9 (fi). It follows that wn(x) —-> w(x) for a.e. x G fi. By passing to a subsequence, we may assume that

I K - H I

V

(4-22)

«-

Let 7 : R1 —> i? 1 be a function such that (i) 7 e C - ^ 1 ) , (ii) 7 is nondecreasing in R1, (hi) 0 < 7 ( s ) < 1, (iv) 7(s) = 0 for s < 0, 7(s) = 1 for s > 1. Define Jn{s) = ^(ns). Then clearly j

n

satisfies (i)-(iii) above and

7„(s) = 0 for 5 < 0; 7„(s) = 1 for s > l/n. Moreover, let M — max{7'(s) : s G [0,1]}; then 0 < 7 ^(s) = nf'{ns)

< Mn.

(4.23)

Now for any <j> G Co°(f2), we define Tpl = V"" = (1 - In ° Wn)([>, 1p2=^2

= (in ° Wn)4>.

Clearly ipi,^ S Co°(^) a n d both functions are nonnegative. Since Vi, i = 1,2, are weak lower solutions to (4.10), by definition we have a(vi, ipi) + / p(x, Vi, Dvi)4>idx < (/, ip,), i = 1, 2. Jo. It follows that / ^LiAi{x,Dv{) -in{wn)(wn)Xi(j)-\-[\-^n{wn)\4)Xi dx L Jn + / p(x,vi,Dvi)[l-'yn(wri)]dxdx

p(x,v2,Dv2)-p(x,vi,Dvi) ^n{wn)cj)dx.

By the dominated convergence theorem, we find, as n —> oo, / p(x,v2,Dv2)-p(x,vi,Dvi) Jn2 -> /

J

jn(wn)4>dx

[p{x, v2, Dv2) - p(x, vi, Dui)J 4>dx,

and / ./ni

p(x,v2,Dv2)

L

-p(x,vi,Dvi)

J

^n(wn)4)dx —> 0

Therefore, as n —> oo, / p(x,v2,Dv2) Jn L -> /

-p(x,vi,Dvi)

dx.

(4.27)

Maximum

50

Principles

and

Applications

Similarly, as n —> oo,

hi

Ai(x, Dv2) - Ai(x,Dvi)

"/n(wn)cf>Xidx

JQ, 'i=l

^ Jf2i Jn2 j T,f=l Ai{x,Dv2) JQ.1

Ai(x,Dv2)

- Ai(x,Dvi)

- Ai(x,Dvi)

L

J

^n(wn)^Xidx

(4.28)

4>Xidx.

Next we use (AS), 7^(u>„) > 0 and > 0 to obtain

hi Jn

v4i(a;,Dv2) -Ai(a;,Dui) 7^(w n )(u 2

-vx)Xi4>dx

JQ „

(4.29)

+ / E ^

r

^ ( x , Dv 2 ) - Ai(x,Dvi)

r

~f'n{wn)(wn - w)Xi4>dx

i

> - / E-li A(a;,Dv2) -Ai(x,Dvi) L Jn

J

^'n{wn)(wn

- w)Xi<j>dx

By (4.22) and (4.23) we deduce that \\(wn - w)Xil'n(wn)4>\\Lq{n)

-> 0.

It follows that Jn

Ai(x,Dv2)

- Ai(x,Dvi)

i'n(wn)(wn

- w)Xidx

0.

Therefore, from (4.29) we obtain Iim„_ 0 0 / E j l i LU ( a ; , Z M ) - ^ ( x ^ i J) ^ ^ ) ^ ) * , ^ > 0. (4.30) Jn We now let n -> oo in (4.26) and make use of (4.27), (4.28) and (4.30) to obtain (/,-\

Ef=1

Ai(x,Dv2)-Mx,Dvi)

f kl.Mx, Dv )cf> 1

Jn

+p(x,vi,Dvi)<j> dx

L

p(x,v2,Dv2) Jn2 — j T,i=1Ai(x,Dv)^)Xidx Jn

+/

Xi

,.dx

-p(x,vi,Dvi) + / p(x,v,Dv)dx. Jn

The Method of Upper and Lower

Solutions

51

Since Ai(-,Dv),p(-,v,Dv) G Lq (fi), we find that the above inequality also holds for arbitrary G W^'g(fl) with > 0 provided that any such is the limit in W 0 M (ft) of a sequence of nonnegative CQ° (fi) functions. Since CS°(Q) is dense in Wo'^fi), for any <j> G W01,9(fi) with 4> > 0, we can find 4>n G Co°(^) such that 0„ —» / in the W1) be as denned in Step 2 of the proof of Theorem 4.9. Then for any u G S, we have b(u-g,) = (f,tf>), V ^ e ^ ' ( n ) . By the proof of Step 3 there, b(u,u) > (a/2)||u||« - CslMI, Vu G W^q(tt). Therefore, for any u G S, (a/2)||u - S ||« - C 6 ||« - g\\ q(Sl). If {ui}i£i is a totally ordered set in S, we will show that u(x) := s\xpi€lUi(x) and u{x) = infjejUj(x) are elements in S. We only prove for u; the other case is similar. The conclusion is clear if I is finite. So we assume that / is an infinite set. Then we can find a sequence {un} in this family such that v < «i < U2 < ••• < w; lim un(x) = u(x). n—>oo 19

Since S is bounded in VF ' (r2), un must converge weakly to u in W1,q(Q). Since each un is a weak solution of (4.10), it then follows easily that u is a weak solution of (4.10). By Zorn's lemma, we conclude that (4.10) has a maximal solution u* and a minimal solution u* in [v, w] in the sense of

Maximum

52

Principles

and

Applications

partial ordering. We now use Lemma 4.10 to show that any weak solution of (4.10) satisfies u» < u < u*. Indeed, if u £ S does not satisfy u < u* , then u\ := max{«, «*} > u* and wi ^ u*. By Lemma 4.10, u\ is a weak lower solution of (4.10). Since u\ < w, by Theorem 4.9, there is a weak solution U2 of (4.10) such that ui < u u for all u £ S. Similarly, we can show that «» < u for all u € S. • The results in Theorem 4.9, Lemma 4.10 and Theorem 4.11 can be easily extended to Neumann boundary value problems. Let v = (V-\.,...,VN) denote the outward unit normal of dQ. and suppose that Ai satisfies (Al) — (A4). We say that u is a weak solution of -Au + p(x, u, Du) = f in ft, E£LjAi(x, u, Du)^ = 0 on dtt, if u e W 1 , 9 (n), p(x,u(x),Du(x)) a(u, v)+

belongs to L«'(ft) and

[ p{x, u, Du)vdx = (/, v) Vu (/, v) Vu G W^q(n),

W1,q(il),

v>0.

It is called a weak lower solution if the above inequality is reversed. If the imbedding W 1,9 (fi) t-» Lq(Q) is compact, one easily checks that the proofs for Theorem 4.9, Lemma 4.10 and Theorem 4.11 carry over with only minor natural changes when (4.10) is replaced by (4.31). Therefore we have the following result. T h e o r e m 4.12 The results in Theorem J^.9, Lemma 4-10 and Theorem 4-11 remain true if (4-10) is replaced by (4-31) and if the imbedding W 'q(Q.) ^-> Lq(Q) is compact (which is guaranteed if dQ, is Lipschitz continuous, see Remark A. 18). The above results can be extended to more general boundary value problems by making use of proper variational inequalities. Suppose that Ai satisfies (Al) — (AA). For convenience of notation, we write A(x,u,Du) = (AI(X,U,DU),...,AN(X,U,DU)). The other notations used above will be kept. Let K be a nonempty closed and convex subset of Wrl'