Chipot m quittner p (eds ) handbook of differential equations stationary partial differential equations vol 2

CHAPTER 1The Dirichlet Problem for Superlinear Elliptic Equations Thomas Bartsch Mathematisches Institut, Universität Giessen, Arndtstrasse 2, 35392 Giessen, Germany E-mail: Thomas.Barts

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Department of Applied Mathematics and Statistics, Comenius University,

Bratislava, Slovak Republic

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This handbook is Volume II in a series devoted to stationary partial differential equations.Similarly as Volume I, it is a collection of self-contained, state-of-the-art surveys written

by well-known experts in the field

The topics covered by this handbook include existence and multiplicity of solutions ofsuperlinear elliptic equations, bifurcation phenomena, problems with nonlinear boundaryconditions, nonconvex problems of the calculus of variations and Schrödinger operatorswith singular potentials We hope that these surveys will be useful for both beginners andexperts and speed up the progress of corresponding (rapidly developing and fascinating)areas of mathematics

We thank all the contributors for their clearly written and elegant articles We also thankArjen Sevenster and Andy Deelen at Elsevier for efficient collaboration

M Chipot and P Quittner

v

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List of Contributors

Bartsch, T., Universität Giessen, 35392 Giessen, Germany (Ch 1)

Dacorogna, B., EPFL, 1015 Lausanne, Switzerland (Ch 2)

Du, Y., University of New England, Armidale, NSW 2351, Australia (Ch 3)

López-Gómez, J., Universidad Complutense de Madrid, 28040 Madrid, Spain (Ch 4) Melgaard, M., Uppsala University, S-751 06 Uppsala, Sweden (Ch 6)

Rossi, J.D., Universidad de Buenos Aires, 1428 Buenos Aires, Argentina (Ch 5)

Rozenblum, G., Chalmers University of Technology, and University of Gothenburg, S-412 96 Gothenburg, Sweden (Ch 6)

Solimini, S., Politecnico di Bari, 70125 Bari, Italy (Ch 7)

Wang, Z.-Q., Utah State University, Logan, UT 84322, USA (Ch 1)

Willem, M., Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium (Ch 1)

vii

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1 The Dirichlet Problem for Superlinear Elliptic Equations 1

T Bartsch, Z.-Q Wang and M Willem

2 Nonconvex Problems of the Calculus of Variations and Differential Inclusions 57

G Rozenblum and M Melgaard

7 Multiplicity Techniques for Problems without Compactness 519

S Solimini

ix

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Contents of Volume I

1 Solutions of Quasilinear Second-Order Elliptic Boundary Value Problems via

C Bandle and W Reichel

2 Stationary Navier–Stokes Problem in a Two-Dimensional Exterior Domain 71

G.P Galdi

3 Qualitative Properties of Solutions to Elliptic Problems 157

W.-M Ni

4 On Some Basic Aspects of the Relationship between the Calculus of Variations

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CHAPTER 1

The Dirichlet Problem for Superlinear

Elliptic Equations

Thomas Bartsch

Mathematisches Institut, Universität Giessen, Arndtstrasse 2, 35392 Giessen, Germany

E-mail: Thomas.Bartsch@math.uni-giessen.de

Zhi-Qiang Wang

Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA

E-mail: wang@math.usu.edu

Michel Willem

Institut de mathématique pure et appliquée, Université catholique de Louvain,

1348 Louvain-la-Neuve, Belgium E-mail: willem@amm.ucl.ac.be

Contents

0 Introduction 3

0.1 Conditions on the nonlinearity 4

1 Positive solutions 4

1.1 Existence of positive solutions 4

1.2 Uniqueness of positive solutions 10

1.3 The Nehari manifold 13

1.4 Existence of ground states 16

1.5 Symmetry of the ground state solution 17

1.6 Multiple positive solutions 17

1.7 The method of moving planes 18

1.8 A priori bounds for positive solutions 21

2 Nodal solutions on bounded domains 22

2.1 A natural constraint 22

2.2 Localizing critical points 24

2.3 Upper bounds on the number of nodal domains 27 HANDBOOK OF DIFFERENTIAL EQUATIONS

Stationary Partial Differential Equations, volume 2

Edited by M Chipot and P Quittner

1

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2 T Bartsch et al.

2.4 The existence of nodal solutions 28

2.5 Geometric properties of least energy nodal solutions on radial domains 31

2.6 Multiple nodal solutions on a bounded domain 33

2.7 Perturbed symmetric functionals 35

3 Problems on the entire space 35

3.1 The compact case 36

3.2 The radially symmetric case 39

3.3 The steep potential well case 40

3.4 Ground state solutions for bounded potentials 43

3.5 More on periodic potentials 45

3.6 Strongly indefinite potentials 51

Acknowledgments 51

References 52

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The Dirichlet problem for superlinear elliptic equations 3

0 Introduction

Boundary value problems for nonlinear elliptic partial differential equations have been

a major focus of research in nonlinear analysis for decades In this survey we discusssemilinear equations like

−u + a(x)u = f (x, u), x ∈ Ω,

where Ω is a domain in RN, N 2, and f : Ω × R → R is superlinear, that is,

f (x, t )/t→ ∞ as |t| → ∞ The model nonlinearity is the homogeneous function

f (x, t )= |t|p −2t with p > 2. (0.2)

The continuation method or other classical methods based on the Leray–Schauder gree do not apply easily to (0.1) because there are no a priori bounds for the solutions.This is inherent to superlinear nonlinearities In fact, for the model nonlinearity (0.2) with

de-p > 2, de-p < 2N/(N− 2) if N 3, a ∈ L∞(Ω), and bounded Ω , there exist infinitely

many solutions which are unbounded in the H1norm On the other hand, if Ω= RN isstar-shaped, a(x)≡ const 0, f is as in (0.2) with p 2N/(N − 2), N 3, then (0.1)

has no solution except the trivial one u≡ 0 Here we know a posteriori that the solutions

are bounded but the Leray–Schauder methods do not apply due to a lack of compactness.After some initial work during the 1960s and early 1970s, Ambrosetti and Rabinowitzestablished in the seminal paper [5] several variational methods to obtain solutions of (0.1)

on bounded domains, most notably the mountain pass theorem and variations thereof.These methods have been refined and extended to deal, for instance, with unbounded do-mains or the critical exponent case f (x, t )= b(x)|t|4/(N −2)t which is closely related to

the Yamabe problem from differential geometry With these methods more complicatedpartial differential equations with variational structure can now be investigated Moreover,qualitative properties of the solutions have been discovered in recent years, in particular,

on the nodal structure and the symmetry of the solutions

The goal of this chapter is to present some basic ideas in a simple setting and to surveyselected results on the Dirichlet problem (0.1) with superlinear nonlinearity The chap-ter consists of three sections In Section 1 we deal with positive solutions of (0.1), inSection 2 with sign-changing solutions on bounded domains and in Section 3 we treatthe unbounded domain Ω= RN No effort is being made to be as general as possible.Neither did we try to write a comprehensive survey on (0.1) For example, we do notpresent results on the bifurcation of solutions nor for the p-Laplace operator, nor do wetreat singularly perturbed equations in detail These topics require separate surveys For-tunately, there are a number of well written monographs about (0.1) where the reader canfind additional information and further references We would like to mention the books byRabinowitz [79], Struwe [87], Chang [41], Ghoussoub [52], Kavian [56], Schechter [81],Willem [95], Chabrowski [39,40], Kielhöfer [57] We concentrate on topics not being cov-ered in these monographs, although a certain overlap cannot be avoided for natural reasons

Of course, the choice of topics is also influenced by our own research interests

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0.1 Conditions on the nonlinearity

For the convenience of the reader we list here conditions on f which we use at variousplaces in the chapter The critical exponent is defined by

(f0) f : Ω× R → R is a Carathéodory function with f (x, t) = o(t) as t → 0 There

exist C > 0 and p 2∗such that|f (x, t)| C(|t| + |t|p −1) for all x∈ Ω, t ∈ R.(f0′) f : Ω× R → R is differentiable in t ∈ R, and the derivative ft is a Carathéodoryfunction with ft(x, 0)= 0 There exist C > 0 and p 2∗ such that |ft(x, t )| C(1+ |t|p−2) for all x∈ Ω, t ∈ R

(f1) For every x∈ Ω the function R \ {0} → R, t → f (x, t)/|t|, is strictly increasing.(f1′) ft(x, t ) > f (x, t )/t for every x∈ Ω and every t = 0

(f2) There exist R 0 and θ > 2 such that 0 < θ F (x, t ) f (x, t )t for all x∈ Ω,

1.1 Existence of positive solutions

We consider first the problem

where Ω is a smooth domain in RN, a∈ L∞(Ω) and 2 < p 2∗ In order to obtain a

solution of (1.1) we assume that− + a is positive, i.e., there exists c > 0 such that, for

Our main tool is the Rellich compactness theorem

THEOREM 1.1 Let Ω be bounded and let 1 p < 2∗ Then the injection H01(Ω)⊂

Lp(Ω) is compact.

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THEOREM1.2 Assume that Ω is bounded, 2 < p < 2∗and (1.2) is satisfied Then there

is achieved by some¯v After replacing ¯v by | ¯v|, we may assume that ¯v 0 It follows from

the Lagrange multiplier rule that

− ¯v + a(x) ¯v = µ ¯vp −1.

A solution of (1.1) is then given by ¯u = µ1/(p −2)¯v Indeed, ¯u > 0 on Ω by the strong

REMARK 1.3 There are other ways to prove Theorem 1.2 Instead of minimizing as

in (1.3) one can minimize the functional

It maps the minimizer ¯v of (1.3) to the minimizer ¯u of Φ on N The solution can also be

obtained via the mountain pass theorem from [5] In fact,

Φ(¯u) = inf

u =0maxt 0Φ(t u)= inf

γ ∈Γ tmax∈[0,1]Φ γ (t )

,

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where Γ consists of all continuous paths γ :[0, 1] → H1

0(Ω) with γ (0) = 0 andΦ(γ (1)) < 0 These three different approaches are equally valid for (1.1) They allow

different generalizations The mountain pass approach leads to the most general existenceresults for positive solutions of−u + a(x)u = f (x, u) with Dirichlet boundary condi-

tions This approach is most widely used in the literature Minimizing over the Neharimanifold requires more conditions on the nonlinearity f When these are satisfied one canfind nodal solutions on the Nehari manifold and obtain useful additional information, inparticular on the nodal structure and the symmetry Minimizing as in (1.3) only makessense for homogeneous nonlinearities

The critical case p= 2∗and the supercritical case p > 2∗are more delicate.

THEOREM1.4 Suppose that N 3, p 2∗, Ω= RNis star-shaped with smooth ary, and a(x) ≡ λ 0 Then there is no solution of (1.1).

bound-PROOF By the Pohozaev identity [78], if

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We assume now that Ω is bounded and a(x)≡ −λ, where 0 < λ < λ1(Ω) and

LEMMA1.7 Let N 4 and λ > 0 Then Sλ< S

PROOF The instanton

U (x)=[N(N − 2)](N−2)/4

[1 + |x|2](N −2)/2 (1.6)

is a minimizer for S Since U|Ω∈ H/ 1

0(Ω), we have to use a truncation ψ We can assume

that Bρ(0)⊂ Ω Let ψ ∈ D(Ω), ψ 0, be such that ψ ≡ 1 on Bρ(0) Using

Uε(x)= ψ(x)ε(2−N)/2U

xε

REMARK 1.8 When N= 3, the situation is more delicate Consider the unit ball Ω =

B1(0)⊂ R3 Then we have

0 < λ λ1(Ω)

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and

λ1(Ω)

4 < λ < λ1(Ω) ⇒ Sλ< S

The next result is due to Brezis and Nirenberg [32]

THEOREM1.9 Let N 4, p= 2∗, 0 < λ < λ1(Ω) and a(x) ≡ −λ Then Sλ is achieved and there exists a solution of (1.1).

THEOREM 1.10 Suppose Ω⊂ RN is a bounded domain with nontrivial topology in the sense that it has a nontrivial homology group Hk(Ω; Z2)= 0 for some k 1 Then

(1.1) with a(x) ≡ 0 and p = 2∗has a solution.

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We now consider the case Ω= RN, a(x)≡ 1, 2 < p < 2∗ and the boundary valueproblem

Let us recall that H1(RN)= H1

0(RN) We shall use the Schwarz symmetrization and radial

See [34] or [96] for a simple proof

We denote by Hr1(RN) the space of radial functions of H1(RN) Let us recall that a

function u is radial if u= u(|x|) The Schwarz symmetrization of a measurable function is

radial

The following results are due to Strauss [85]

LEMMA1.12 Let N 2 There exists c(N ) > 0 such that, for every u∈ H1

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PROOF Let (vn)⊂ H1(RN) be a minimizing sequence for Sp: vn L p= 1,

R N|∇vn|2+ v2ndx→ Sp

By Theorem 1.11, we can replace (vn) by (vn∗) By Theorem 1.1 and Lemma 1.12, we can

assume, going if necessary to a subsequence:

vn∗⇀ v in H1 RN

,

vn∗→ v in Lp RN

,

where v is a radial function Clearly, v is a minimizer for Spand (1.7) is solvable

The existence of nonradial entire solutions u∈ H1(RN) of (1.1) and more general

equa-tions will be discussed in Section 3

1.2 Uniqueness of positive solutions

The problem of uniqueness of positive solutions is mostly solved for symmetric domainsand is closely related to the symmetry of solutions Let us first recall a celebrated resultproved in 1979 by Gidas, Ni and Nirenberg [53]

THEOREM 1.14 Let Ω be the unit ball in RN Assume that f is C1 and u∈ C2(

Then u is a radial function and u′(r) is negative.

Consider the problem

where Ω= B1(0) is the unit ball inRN Uniqueness is proved when

λ= 0, 2 < p < 2∗, Gidas, Ni and Nirenberg [53], 1979,

λ > 0, 2 < p < 2∗, Kwong [59], 1989,

λ < 0, 2 < p 2∗, Srikanth [84], 1993

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Concerning problem (1.7), uniqueness for 2 < p < 2∗is proved in [59] after the pioneeringwork of Coffman [42] in 1972

The situation differs when Ω is an annulus, Ω= {x ∈ RN: r <x < R} for some

R > r > 0 Let us recall a particular case of the principle of symmetric criticality proved

Fix(G)= {u ∈ X: gu = u for all g ∈ G} =u∈ X: Gu = {u}

A function ϕ : X→ R is invariant if ϕ ◦ g = ϕ for every g ∈ G

EXAMPLE1.16 Assume that Ω is invariant by rotations: for every g∈ SO(N), gΩ = Ω.

The action of SO(N ) on H01(Ω) is defined by

gu(x)= u g−1x

The space Fix(SO(N )) is the space H0,r1 (Ω) of radial functions in H01(Ω) From

Theo-rem 1.1 and Lemma 1.12, it follows that the injection H0,r1 (Ω)⊂ Lp(Ω) is compact for

2 < p < 2∗ Moreover, if Ω is an annulus

Ω=x∈ RN: ρ <|x| < R

or an exterior domain

Ω=x∈ RN: ρ <|x| ,

the injection H0,r1 ⊂ Lp(Ω) is compact for 2 < p ∞

Let us recall that, for every open subset Ω ofRN,

S(Ω):= inf

u ∈H 10(Ω)

u2∗ =1∇u22= S

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and S(Ω) is never achieved except when Ω= RN

Since the “radial infimum” is achieved when Ω is an annulus, we have

For 2∗− ε < p < 2∗ inequality (1.9) yields a nonradial solution of problem (1.10) Thus

we have proved the following theorem

THEOREM 1.17 For 2∗− ε < p < 2∗, problem (1.10) has at least two solutions, one radial and one nonradial Moreover, the least energy solution is nonradial.

The above result is due to Brezis and Nirenberg [32] Following their work, there arerelated results for multiple positive nonradial solutions of semilinear elliptic equations onexpanding annular domains, by Coffman [43] for N= 2, by Li [62] for N 4, and by

Byeon [35] (and Catrina and Wang [38] independently) for N = 3 We set Ωa= {x ∈

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pos-The Dirichlet problem for superlinear elliptic equations 13

1.3 The Nehari manifold

Let Ω⊂ RNbe a (not necessarily bounded) smooth domain We consider the problem

−u + a(x)u = f (x, u), x ∈ Ω,

where a∈ L∞(Ω) Concerning the nonlinearity we assume:

(f0) f : Ω×R → R is a Carathéodory function with f (x, t) = o(t) as t → 0, uniformly

in x There exist C > 0 and p 2∗such that|f (x, t)| C(|t| + |t|p −1) for all

We begin with a geometric description of N We assume that f satisfies (f0) and (f1),

or the following differentiable versions:

(f0′) f : Ω× R → R is differentiable in t ∈ R with f (x, t) = o(t) as t → 0, uniformly

in x The derivative ft is a Carathéodory function There exist C > 0 and p 2∗such that|ft(x, t )| C(1 + |t|p −2) for all x∈ Ω, t ∈ R

(f1′) ft(x, t ) > f (x, t )/t for every x∈ Ω and every t = 0

Clearly, (f0′) and (f1′) imply (f0) and (f1) For u∈ SE := {u ∈ E: u = 1}, the map,

is strictly decreasing by (f1) The set

U:=u∈ SE: ψu(λ) < 0 for some λ > 0

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is a homeomorphism with inverse N → U , v → v/v

(b) If f satisfies (f0′) and (f1′) then N is a C1-manifold with tangent space TuN ={v ∈ E: Φ′′(u)[u, v] + Φ′(u)v= 0} The map h is a C1-diffeomorphism If u ∈ N is a

critical point of the constrained functional Φ|N, then u is a critical point of Φ.

PROOF (a) is a simple consequence of the fact that ψu is strictly decreasing and that

ψu(λ) is continuous in (u, λ)

(b) Here Φ∈ C2(E) and the implicit function theorem applied to the map

U× R+, (u, λ)→ ψu(λ)

yields that u→ λuis C1 The claims follow because TuN is transversal to R+u.

We can describe the set U if f is superlinear:

(f2) There exist R > 0 and θ > 2 such that 0 < θ F (x, t ) f (x, t )t for all x∈ Ω,

The easy proof is left to the reader

THEOREM1.21 If (f0) and (f1) hold then β±= infu ∈NΦ(u) 0 and every minimizer of

Φ on N+(resp N−) is a positive (resp negative) solution of (1.12).

We do not claim that β± is achieved This requires additional conditions on f and a;see Theorem 1.23 Since N is not, in general, a differentiable manifold, the Lagrangemultiplier rule is not applicable We shall use a general deformation lemma By definition

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(i) if t = 0 or if u /∈ Φ−1([c − 2ε, c + 2ε]) ∩ S2δthen η(t, u)= u;

(ii) η(1, Φc+ε∩ S) ⊂ Φc −ε;

(iii) η(t,·) is a homeomorphism of X ∀t ∈ [0, 1];

(iv) η(t, u) − u δ ∀u ∈ X, ∀t ∈ [0, 1];

(v) Φ(η(·, u)) is nonincreasing ∀u ∈ X;

(vi) Φ(η(t, u)) < c∀u ∈ Φc∩ Sδ,∀t ∈ [0, 1]

A proof can be found in [95], Lemma 1.4

PROOF OFTHEOREM1.21 The inequality β± 0 follows from the fact that the map ψu

from (1.13) is strictly decreasing Now let u∈ N+be such that Φ(u)= β+ We shall prove

that Φ′(u)= 0

It follows from assumption (f1) that

Φ(su) < Φ(u)= β+ for 0 < s= 1

If Φ′(u)= 0, then there exists δ > 0 and λ > 0 such that

v − u 3δ ⇒ Φ′(v) λ

Clearly β0 = max{Φ(u/2), Φ(3u/2)} < β+ For ε = min{(β+ − β0)/2, (λδ)/8} and

S= Bδ(u), Lemma 1.22 yields a deformation η such that

h

12

=12Φ′

u

the existence of s∈ (1/2, 3/2) with Φ′(h(s))h(s)= 0, i.e., h(s) ∈ N+, follows from the

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1.4 Existence of ground states

In order to prove the existence of a minimizer of Φ on N+, we assume

(f3) lim|t|→∞F (x, t )/t2= +∞, uniformly in x

THEOREM 1.23 Suppose that Ω is bounded and that f satisfies (f0) with p < 2∗,

(f1) and (f3) Moreover, suppose that − + a is positive Then there exists a minimizer

of Φ on N+and, hence, a positive solution of (1.12).

PROOF Let (un)⊂ N+be a minimizing sequence: Φ(un)→ β+ Let us define tn= un

and vn= un/tn We can assume that vn⇀ v in H01(Ω) Since, for every R > 0,

so that v= 0 If (tn) is unbounded, we can assume that tn→ +∞ We obtain, from (f3)

and Fatou’s lemma, the contradiction

It follows that (un) is bounded in H01(Ω) Going if necessary to a subsequence, we can

assume that un⇀ u in H01(Ω) By (f0), since (un)⊂ N+,

Theorem 1.23 is due to Liu and Wang [67] Note that (f3) is weaker than (f2) Efforts in

weakening (f2) have been made in [46,82] (see also the references therein)

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1.5 Symmetry of the ground state solution

In this section we assume that Ω is invariant by rotations: gΩ= Ω for every g ∈ SO(N).

Theorem 1.17 shows that, even when f is independent of x, the ground state is, in general,not radial However when f = f (|x|, u), a partial symmetry is always preserved by the

ground state

We arbitrarily choose a fixed direction P inRNwhich we will refer to as the north poledirection Let R > 0 and dσ denote the standard measure on ∂BR(0) The symmetriza-

tion A∗of a measurable set A⊂ ∂BR(0) is defined as the closed geodesic ball in ∂BR(0)

centered at the north pole and whose dσ -measure equals that of A The foliated Schwarzsymmetrization B∗of a Borel set B⊂ RNis defined on any sphere ∂BR(0) by

THEOREM1.24 We assume (f0), (f1) and

(A1) f : Ω× R → R is Hölder continuous on Ω × [−R, R] for every R > 0,

(A2) Ω is radially symmetric,

(A3) a and f ( ·, t) are radial functions for every t ∈ R.

Then every minimizer u of Φ on N+ or N− is a foliated Schwarz symmetric solution

of (1.12).

A related result can be found in [76], Theorem 3.1

1.6 Multiple positive solutions

Using a more topological argument Benci, Cerami and Passaseo [21,23] were able to lish the following result about the impact of the domain topology on the solution structure.Consider

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THEOREM 1.25 Suppose the Lusternik–Schnirelmann category of the domain satisfies

cat(Ω) 2

(a) If 2 < p < 2∗then for λ sufficiently large (1.14) has at least cat(Ω) + 1 solutions.

(b) If λ 0 then for p < 2∗ sufficiently close to 2∗ (1.14) has at least cat(Ω)+ 1

solutions.

The case p= 2∗ is of special interest It is closely related to the Yamabe problemfrom differential geometry This critical case is analytically more difficult because theembedding H01(Ω) ֒→ L2 ∗

(Ω) is not compact As discussed in Section 1.1, Brezis and

Nirenberg [32] obtained one solution for 0 < λ < λ1 if N 4, and, in the case N= 3,

showed that there exists λ∗∈ [0, λ1) so that (1.14) has a solution for λ∗< λ < λ1 Inthe case λ= 0 and p = 2∗, Bahri and Coron [8] obtained one solution if the domain hasnontrivial homology: Hk(Ω; Z2)= 0 for some k 1 The existence of multiple positive

solutions of (1.14) for p= 2∗is not known

REMARK1.26 More results about the effect of the topology and geometry of the domains

on the solutions structure have been given for the singularly perturbed nonlinear ellipticequation

turbed equations like (1.15) have been a very active area of research during the last fifteenyears and the number of papers abound A discussion of this topic goes beyond the scope

of our survey

1.7 The method of moving planes

This section is related to Section 1.5 We consider the problem

of minimal surfaces and was used by Serrin in 1971 and Gidas, Ni and Nirenberg [53]

in the study of semilinear elliptic equations The method was extended and simplified byBerestycki and Nirenberg [26] We describe a result of [26], following [29]

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We shall need the maximum principle for small domains Let w be a solution of

−w + c(x)w 0 in Ω,

w 0 on ∂Ω (1.17)The standard form of the maximum principle asserts that, if c(x) 0 in Ω then w(x) 0

in Ω In Stampacchia’s form we use a weaker assumption Let us recall that

S= inf

u ∈H 10(Ω)

u2∗ =1

∇u2 2

is independent of Ω

LEMMA1.27 Assume that w∈ H1(Ω) satisfies (1.17) with

c− N/2< S (1.18)

Then w 0 in Ω

PROOF It suffices to multiply (1.17) by w−, to integrate by parts and to use (1.18)

Assumption (1.18) is always satisfied ifc− ∞<∞ and |Ω| is sufficiently small

THEOREM 1.28 Let f be locally Lipschitz and let Ω be bounded, convex in some direction, say x1, and symmetric with respect to the plane x1= 0 Then any solution

u∈ C2(Ω)∩ C( Ω) of (1.16) is symmetric with respect to x1 and ∂u/∂x1< 0 for x1> 0

wλ(x)= u(2λ − x1, y)− u(x1, y), x∈ Ωλ

The function wλis well defined on Ωλsince Ω is convex in the direction x1and symmetricwith respect to the plane x1= 0 We shall prove that

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Since ˜u(x1, y)= u(−x1, y) is also a solution of (1.16), one finds that u(x1, y)= u(−x1, y).

It is easy to conclude that ∂u/∂x1< 0 for x1> 0 using Hopf’s lemma

REMARK1.29 (a) Theorem 1.14 follows directly from Theorem 1.28 It is interesting tonote that Theorem 1.28 is applicable to domains like cubes

(b) The method of moving planes is very flexible and has been adapted to a large variety

of problems It is not possible to give a bibliography within this survey The surveys byBerestycki [24] and by Brezis [29] contain many references

(c) With respect to Section 1.5 the assumptions on Ω and on f are somewhat stronger,but the results are applicable to any positive solution

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1.8 A priori bounds for positive solutions

Topological methods require the existence of a priori bounds for the set of all positive lutions In this section we briefly describe particular cases of three classical results Wedenote by λ1the first eigenvalue of− on H1

so-0(Ω) and by e1> 0 the corresponding

eigen-function Throughout this section we assume that Ω is a smooth bounded domain inRN

and that f : Ω× R+→ R+is continuous

The first result is due to Brezis and Turner [33] The proof uses Hardy’s inequality

THEOREM1.30 Assume that

for some t 0, we haveu∞ c

The second result is due to de Figueiredo, Lions and Nussbaum [51] The proof uses thePohozaev identity

THEOREM1.31 Assume that Ω is convex, f :R+→ R+is locally Lipschitz and

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2 Nodal solutions on bounded domains

In this section we report on recent results concerning nodal solutions of

In the third subsection we give a nonlinear version of Courant’s nodal domain theorem foreigenfunctions of the Laplace operator This gives an upper bound on the number of nodaldomains of a solutions of (2.1) related to the min–max description of the critical value Inthe Sections 2.4 and 2.5 we prove the existence and some properties of least energy nodalsolutions Finally, in Section 2.6 we study the existence of multiple nodal solutions

2.1 A natural constraint

In this subsection Ω may be unbounded We consider the problem (2.1) with f satisfying

(f0) and (f1) from Section 1.3 Recall the functional Φ : E= H1

0(Ω)→ R and the Nehari

manifold

N=u∈ E \ {0}: Φ′(u)u= 0

from Section 1.3 In Theorem 1.21 we showed that a minimizer of Φ on N+= {u ∈

N : u 0} is a positive solution of (2.1) In order to obtain nodal solutions, we consider

the nodal Nehari set

S=u∈ E: u+∈ N , u−∈ N

=u∈ E: u+= 0 = u−, Φ′(u)u+= 0 = Φ′(u)u−

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Clearly, S⊂ N contains the set of all nodal solutions

THEOREM2.1 Suppose (f0) and (f1) hold Then β := inf Φ(S) 0, and every minimizer

of Φ on S is a nodal solution of (2.1).

In Section 2.4 we shall prove the existence of a minimizer of Φ on S Since the maps

E→ E, u → u±are continuous but not differentiable, S is not a differentiable manifoldeven if N is one (as in Proposition 1.19)

PROOF OFTHEOREM2.1 Clearly β= inf Φ(S) inf Φ(N ) 0 by Theorem 1.21 Let

u∈ S be a minimizer and suppose Φ′(u)= 0 As a consequence of (f1), for any v∈ N ,

the functionR+∋ t → Φ(tv) ∈ R achieves its unique maximum at t = 1 Therefore

Φ su++ tu−= Φ su+

+ Φ(tu−) < Φ u+

+ Φ(u−)= Φ(u) (2.2)

for (s, t )∈ R2

+\{(1, 1)} By the continuity of Φ′there exist α, δ > 0 such thatΦ′(v) α

for v∈ U3δ(u) Setting

and observe that ψ (s, t )= (0, 0) is equivalent to h(s, t) ∈ S For (s, t) ∈ ∂D we have

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(b) In [17], Lemma 3.2, Bartsch and Weth prove that S∩ H2(Ω) is a co-dimension 2

submanifold of E∩ H2(Ω) if f satisfies the conditions (f0′) and (f1′)

(c) In [67] Liu and Wang give a slightly different proof of the above result without usingthe deformation lemma and therefore requiring less smoothness of the functional

2.2 Localizing critical points

A basic idea for localizing critical points can be formulated in a very general setting Let

X be a topological space and ϕ : G⊂ [0, ∞) × X → X be a continuous semiflow on X

Here G= {(t, u) ∈ [0, ∞) × X: 0 t < T (u)} is an open subset of [0, ∞) × X, where

T (u)∈ (0, ∞] is the maximal existence time of the trajectory t → ϕ(t, u) We often write

ϕt(u)= ϕ(t, u)

Given B⊂ A ⊂ X we call B positively invariant in A if, for u ∈ B and T > 0 with

ϕt(u)∈ A, 0 t T , it follows that ϕT(u)∈ B If even ϕT(u)∈ int B then B is said to

be strictly positively invariant In the case A= X, we simply call B (strictly) positively

invariant The notion of invariant sets has also been exploited in [65]

Recall that a continuous map γ : (C, D)→ (A, B) between pairs D ⊂ C, B ⊂ A of

topo-logical spaces is nullhomotopic if there exists a homotopy H : (C× [0, 1], D × [0, 1]) →(A, B) with H (x, 0)= γ (x) and H (x, 1) ∈ B for all x ∈ C

LEMMA 2.3 Let A ⊂ X be positively invariant, B ⊂ A strictly positively invariant Let

f : (C, D)→ (A, B) be not nullhomotopic, C a metric ( paracompact) space Then

A(B):=x∈ C ∃t 0: ϕt f (x)

∈ B = C

PROOF We argue by contradiction If A(B)= C then for each x ∈ C there exists τ (x) 0

with ϕτ (x)(f (x))∈ intAB Choose a neighborhood Vxof x in C with

ϕτ (x) f (y)

∈ intAB for all y∈ Vx

Let (πj)j∈J be a partition of unity subordinated to Vx: supp πj ⊂ Vxj Now we fine σ : C → [0, ∞), σ (x) := j ∈Jπj(x)τ (xj), and H : C × I → A, H (x, f ) :=

de-ϕt σ (x)(f (x)) This homotopy shows that f is nullhomotopic

There also exists an equivariant version of Lemma 2.3 when a group G acts on A and C.The extension is straightforward and therefore omitted

In a typical application, ϕ is the negative gradient flow of a functional Φ : X→ R, and

B contains a sublevel set Φb= {u ∈ X: J (u) b} A and B are closed and one wants to

find a critical point in A\ B For x ∈ C \ A(B) and u := f (x), one then has

ϕt(u) /∈ B, hence Φ ϕt(u)

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Thus un:= ϕt n(u), n∈ N, is a (PS)c-sequence in A\ B If un→ ¯u then ¯u /∈ B because

otherwise ¯u = ϕt(¯u) ∈ intAB, hence φtn(u)∈ intAB for some tn, a contradiction Thus

¯u ∈ A\ B

We shall now present examples of strictly positively invariant sets which can be used

to find nodal solutions of (2.1) Thus we are interested in finding critical points of theassociated functional Φ outside of

P±= {u ∈ E: ±u 0 a.e.} (2.3)Since P+and P−have empty interior one cannot use B= P+∪ P−

EXAMPLE2.4 Suppose Ω is bounded, and (f0′) and

(f4) there exists m > 0 so that t→ f (x, t) + mt is strictly increasing for all x ∈ Ω

hold We take

u, vm:= ∇u, ∇vL 2+ mu, vL 2

as scalar product in E and write · m for the corresponding norm Setting g(x, u)=

f (x, u)+ mu and G(x, t) =0tg(x, s) ds we can write Φ as

non-if v− u ∈ int(PX+) As a consequence of the strong maximum principle, K is strictly order

preserving, that is,

which induces a flow on X (and E) Since K is strictly order preserving the set PX±\ {0}

is strictly positively invariant More generally, if u∈ X is a subsolution, that is, u K(u),

then u+ (PX±\ {0}) is strictly positively invariant This follows from the fact that for v > 0

the vector field−∇Φ points at u + v into u + int PX+,

u+ v − ∇Φ(u + v) = K(u + v) ≫ K(u)

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Similarly, if u is a supersolution then u+ (PX±\ {0}) is strictly positively invariant In order

to find nodal solutions above some level α > 0 one can work with B:= Φα∪ PX+∪ PX−

which is strictly positively invariant for ϕ if there are no nodal solutions at the level α

EXAMPLE2.5 Suppose Ω is bounded, and (f0′) and

(f5) inft=0f (x, t )/t >−∞

hold Then we choose m > 0 so that f (x, t )+ mt > 0 for all t > 0, f (x, t) + mt < 0 for

all t < 0, and define·, ·m, X and the order relation≪ as in Example 2.4 The gradient

of Φ with respect to this metric is not necessarily order preserving but it does satisfy

u > 0 ⇒ K(u)≫ 0 and u < 0 ⇒ K(u)≪ 0

It follows as in Example 2.4 that the cones PX±\ {0} are strictly positively invariant

If Ω is unbounded or if f is only a Carathéodory function, the approach presented inExample 2.4 does not work because then one cannot work in X= C1( ∩ E Either the

cones PX±have empty interior or there is no flow on X due to a lack of regularity Here onecan often replace P±by their open neighborhoods in E

DEFINITION 2.6 Let K : E→ E be a continuous operator on a Banach space E A set

C⊂ E is said to be K-attractive if there exists ε0> 0 so that K(clos(Uε(C)))⊂ Uε(C)={u ∈ E: dist(u, C) < ε} for 0 < ε < ε0

LEMMA2.7 Let E be a Banach space, Φ∈ C1(E) and C= C1∪ · · · ∪ Cn⊂ E be a finite

union of convex sets Suppose ∇Φ = Id −K, and each Cj is K-attractive Then Φ has no critical points in clos(Uε0(C))\ C, where ε0is from Definition 2.6 Given ε∈ (0, ε0] there

exists a pseudo-gradient vector field V : E \ Fix(K) → E for Φ so that clos(Uε(C)) is

strictly positively invariant for the flow associated to−V

PROOF For ε∈ (0, ε0] and u ∈ ∂Uε(Cj), we have K(u)∈ Uε(Cj) Since C1, , Cnareconvex, a standard partition of unity argument yields a locally Lipschitz continuous map

Here are two examples of K-attractive sets

EXAMPLE 2.8 Suppose f satisfies (f0) and (f5) Thus there exists m > 0 as in

Exam-ple 2.5, and we define K(u)= (− + m)−1(f (·, u) + mu) If lim supt →0|f (x, t)|/|t| <

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λ1(Ω) uniformly in Ω then it has been proved in [12], Lemma 3.1, that P± = {u ∈E:±u 0 a.e.} is K-attractive We sketch the argument in a more complicated parameter

dependent situation (see Lemma 3.17)

EXAMPLE2.9 Suppose f satisfies (f0) and (f4) Set E= H1

0(Ω) and K(u)= (− +m)−1(f (·, u) + mu) as in Example 2.4 In [44] it is proved for bounded Ω that, for a strict

subsolution u∈ E ∩ W2,2, the cone u+ P+is K-attractive

EXAMPLE2.10 When Ω is unbounded, neighborhoods of shifted positive and negativecones in the directions of the first eigenfunctions have been proved to be K-attractive

in [66] The situation is more delicate here and some conditions on the spectrum of thelinear operator have be to assumed

The Examples 2.4, 2.5 and 2.8 together with Lemma 2.7 yield strictly positively invariantsets which can be used to find nodal solutions of (2.1) or (2.2) with the help of Lemma 2.3

We shall do this in Sections 2.4 and 2.6

2.3 Upper bounds on the number of nodal domains

In this subsection Ω⊂ RN may be unbounded A nodal domain of a continuous function

u : Ω→ R is a connected component of Ω \ u−1(0) We write nod(u)∈ N0∪ {∞} for the

number of nodal domains of u and set nod(u)= 0 for u = 0

LEMMA2.11 Suppose f satisfies (f0) Then every weak solution u ∈ E of (2.1) is

con-tinuous If Ω0⊂ Ω is a nodal domain of u then u · χΩ 0∈ H1

0(Ω)

PROOF First observe that f (·, u)/u ∈ LN/2loc (Ω) by (f0) and the Sobolev embedding

the-orem The Brezis–Kato theorem [30] implies u∈ Lqloc(Ω) for every 2 q <∞ and

there-fore f (·, u) ∈ Ls

loc(Ω) for s > N/2 Then u is continuous by elliptic regularity The last

Now we suppose that (f0) and (f1) hold Let N be the Nehari manifold and S⊂ N the

nodal Nehari set

PROPOSITION 2.12 Suppose (f0) and (f1) hold Let u be a critical point of Φ and fix

n∈ N

(a) If Φ(u) inf Φ(S) + n · inf Φ(N ) then nod(u) n + 1.

(b) If Φ(u) infv1, ,vnsup Φ(C(v1, , vn)) then nod(u) n The infimum

ex-tends over all n-tuples of linearly independent elements v1, , vn ∈ E, and

C(v1, , vn):= {ni =1λi· vi: λ1, , λn 0}

(c) If f is odd and Φ(u) infV⊂E,dim(V )=nsup Φ(V ) then nod(u) n Here the mum extends over all n-dimensional linear subspaces of E.

+ Φ(tu−) < Φ u+

+ Φ(u−)= Φ(u) (2. 2)

for (s, t )? ?? R2

+\{(1,... outside of

P±= {u ∈ E: ±u a.e.} (2. 3)Since P+and P−have empty interior one cannot use B= P+∪ P−

EXAMPLE2. 4 Suppose... that

h

12

=12Φ′

u

the existence of s∈ (1 /2, 3/ 2) with Φ′(h(s))h(s)= 0, i.e., h(s) ∈

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