Morse Index of Solutions of Nonlinear Elliptic Equations (De Gruyter Series in Nonlinear Analysis and Applications) 311053732X, 9783110537321

Table of contents :
Cover
De Gruyter Series in Nonlinear Analysis and Applications
Morse Index of Solutions of Nonlinear Elliptic Equations
© 2019
Preface
Contents
Notation
1 Preliminaries
2 Introduction to Morse theory
3 Morse theory for semilinear elliptic equations
4 Morse index of radial solutions of Lane–Emden
problems
5 Bifurcation from radial solutions
6 Morse index and symmetry for semilinear elliptic
equations in bounded domains
7 Morse index and symmetry for elliptic systems in
bounded domains
8 Some results in unbounded domains
Bibliography
Index

Citation preview

Lucio Damascelli and Filomena Pacella Morse Index of Solutions of Nonlinear Elliptic Equations

De Gruyter Series in Nonlinear Analysis and Applications

|

Editor-in Chief Jürgen Appell, Würzburg, Germany Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Mikio Kato, Tokyo, Japan Wojciech Kryszewski, Torun, Poland Umberto Mosco, Worcester, Massachusetts, USA Louis Nirenberg, New York, USA Simeon Reich, Haifa, Israel Alfonso Vignoli, Rome, Italy Vicenţiu D. Rădulescu, Krakow, Poland

Volume 30

Lucio Damascelli and Filomena Pacella

Morse Index of Solutions of Nonlinear Elliptic Equations |

Mathematics Subject Classification 2010 Primary: 3502, 35J61; Secondary: 35B99, 35B06 Authors Prof. Dr. Lucio Damascelli Università di Roma Tor Vergata Dipartimento di Matematica Via della Ricerca Scientifica 00133 Roma Italy [email protected] Prof. Dr. Filomena Pacella Università di Roma Sapienza Dipartimento di Matematica Piazzale Aldo Moro 2 00185 Roma Italy [email protected]

ISBN 978-3-11-053732-1 e-ISBN (PDF) 978-3-11-053824-3 e-ISBN (EPUB) 978-3-11-053743-7 ISSN 0941-813X Library of Congress Control Number: 2019937572 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2019 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface Morse theory owes its name to the mathematician M. Morse who analyzed the relationship between the topology of a manifold and the number and the type of critical points of a smooth function defined on it [180]. In some sense, it can be viewed as a beautiful and natural extension of the principle which asserts that every continuous function on a compact space has a minimum and a maximum point. The basic ideas of Morse theory can be summarized by the following two statements. Let M be a smooth finite dimensional manifold and g : M → ℝ a smooth real valued function. (I) If for some a < b, the set g −1 ([a, b]) is compact and does not contain any critical point of g, then the sublevel ga = {x ∈ M : g(x) ≤ a} is a deformation retract of gb = {x ∈ M : g(x) ≤ b} (II) If g has only one critical point p in g −1 ([a, b]) which is nondegenerate, and g −1 ([a, b]) is compact, then gb is homotopically equivalent to ga with a m(p)-cell attached, where m(p) is the number of directions where the Hessian matrix of g is negative definite. The number m(p) is defined as the Morse index of the critical point p, and the word nondegenerate means that the Hessian matrix of g at the point p is nonsingular. So, on one side the topology of the manifold, in particular its homology, influences the number and type of critical points of any smooth nondegenerate function defined on it, on the other side from the knowledge of the critical points of a smooth nondegenerate function it is possible to recover the topological properties of the manifold. Since the beginning, Morse theory has been proved useful in estimating the number and the type of critical points of functions related to many problems in analysis and differential geometry. Through the years and with the extension to an infinite dimensional setting, Morse theory has been successfully applied to many P. D. E. problems. In this book, we focus on semilinear elliptic equations of the type − Δu = f (x, u)

(1)

with Dirichlet or mixed boundary conditions. The Morse index m(u) of a solution u of (1) is the maximal dimension of suitable linear spaces X where a certain quadratic form Qu (ψ), related to the linearization of (1), is negative definite. By linearization, we mean the linear operator Lu = −Δ − f 󸀠 (x, u) where f 󸀠 denotes the derivative of the function f with respect to the second variable. The linear spaces X involved in the definition are prescribed by the boundary conditions assigned and the type of solutions considered. Typically, X will be subspaces of a suitable Sobolev space where the weak solutions of (1) belong. Since the weak solutions of (1) can be viewed as critical points https://doi.org/10.1515/9783110538243-201

VI | Preface of a functional J, the quadratic form Qu (ψ), corresponds to the one associated with the second derivative of J, so we are back to the original definition of the Morse index. However, let us observe that the definition of the Morse index by linearization can be given independently of the variational structure of (1). This means that the definition survives for problems for which there is not a “natural” variational formulation. This is, for example, the case when systems are considered, instead of scalar equations (see Chapter 7), or fully nonlinear problems are studied (cf. [35]). In this book, we study the Morse index of solutions of (1) having two purposes in mind. On one side, we want to describe some interesting applications, and on the other side we would like to focus on the computation or estimates of the Morse index itself. This last question is a delicate issue and it is obviously a prerequisite to the first one. When the domain where the problem is posed is a bounded domain of ℝN , it amounts to calculate or estimate the number of negative eigenvalues of the linear Schrödingertype operator Lu = −Δ − V, where V = V(x) = f 󸀠 (x, u). The spectral theory of this kind of linear operators is fascinating and difficult. A good knowledge of the properties of the potential function V is necessary to deduce information on the spectrum. Since V depends on the solution itself, this in turn means having precise information on solutions of (1). We present both some classical results for least energy solutions and some recent developments for radial solutions of Lane–Emden problems. Concerning applications, the ones included in this book reflect the taste of the authors and the work done by them through the years. In particular, we devote the last part of the book to the study of symmetry properties of solutions of (1) by Morse index. Symmetry naturally arises in many problems and it is often observed that symmetric solutions tend to minimize some energy associated. The connection between the Morse index of a solution and its symmetry was first shown in [186] driven from the idea that solutions with low Morse index should have “some symmetry.” In [186], this was proved for convex nonlinearities and Morse index one solutions, which are “almost” minima of the energy functional, since its Hessian matrix is negative definite only in one direction. Hence we are back to the principle of minimizing the energy by symmetry. It was subsequently shown that, for nonlinearities which are either convex or have convex derivative, solutions of (1) are symmetric as long as their Morse index is less than or equal to the dimension of the space ℝN . Later these results were extended to systems and unbounded domains. Morse theory has also strong connections with bifurcation, in particular, through degree theory. Plenty of results are available in the literature. We have chosen to present here a result on the existence of branches of nonradial solutions bifurcating from radial ones, when the domain is an annulus. This is achieved by detecting the change of Morse index of radial solutions, varying the bifurcation parameters. This is again a connection between Morse index and symmetry and shows that “break of symmetry” arises when the Morse index increases.

Preface | VII

The book is organized in eight chapters. We have made an effort to keep the exposition as simple and as self-contained as possible. Therefore, the first three chapters, in particular, Chapter 1 and Chapter 3, are somehow introductory, and could be used for a graduate course in P. D. E. Chapters 4–7 are more advanced and contain recent results. In the short Chapter 8, some developments for unbounded domains are outlined. The aim of Chapter 1 is to review different forms of weak and strong maximum principles as well as the classical eigenvalue theory for boundary value problems involving elliptic operators. We will consider mainly weak solutions and mixed Dirichlet–Neumann boundary conditions (with nonlinear Neumann boundary conditions on part of the boundary), therefore we include a section on Sobolev spaces of functions vanishing only on some portion of the boundary of the domain where they are defined. In particular, we will deal with functions defined in cylindrically symmetric domains. We give great emphasis on the variational characterization of the eigenvalues, which is crucial to study the Morse index. We also include a section on systems, describing the related maximum principles and spectral theory. This last one works well for symmetric systems to which the theory of compact self-adjoint operators can be applied, but not in the case of general systems. Nevertheless, we show how to exploit a symmetric system, naturally associated to a generic one, in order to get information and, in particular, to check the validity of the maximum principle. The material of this chapter concerning scalar equations is largely classical, but the proofs of many well-known results are not easy to be found or are spread in several textbooks. We feel that it is useful to present them all together. The results on systems are taken from recent papers and to present them together with the scalar case allows the reader to have a unified vision. In Chapter 2, we introduce the classical Morse theory for functions on finite dimensional manifolds as well as its extension to the infinite dimensional case. In particular, we will describe the steps to show that a mountain pass critical point of a functional in Banach spaces has Morse index less than or equal to one. This is a very important result for applications to semilinear elliptic equations, which has been proved in different ways and is reported and used many times but whose clear proof is not easy to be found. In Chapter 3, we introduce the Morse index of solutions of semilinear elliptic equations. We distinguish between the cases of positive or sign changing solutions. In the first case, we show the existence of a positive solution with Morse index one and present an application to an uniqueness result. In the second case, after proving the existence of a sign changing solution with Morse index two, we give some estimates for the Morse index of nodal radial solutions from which symmetry breaking results derive. The material of this chapter is not fully included in any book in nonlinear analysis but is mostly taken from several research articles. We believe that it could serve

VIII | Preface as a beginner reference for the study of solutions of semilinear elliptic equations and their Morse index. In Chapter 4, we review some recent results on sharp computations of the Morse index for radial solutions. To the purpose not to get distracted by too many technical details, and thus lose the main thread of the arguments, some proofs which are long and involve heavy computations have been omitted, giving precise references to the interested reader. We hope to have made clear the strategy for the evaluation of the negative eigenvalues of the linear operators involved. In Chapter 5, we describe one of the many applications of Morse theory to bifurcation. We describe some results about bifurcation from radial positive solutions of Lane–Emden Dirichlet problems in an annulus. As compared to the original paper [129], we simplify and detail some of the proofs. Finally, in Chapter 6 and Chapter 7, we present the results obtained by the authors about symmetry of solutions via Morse index bounds. Chapter 6 deals with the case of scalar equations. It is mostly taken from [186] and [190] but we present the results in an unified context simplifying some proofs and adding recent developments for nonlinear mixed boundary value problems from [78]. Chapter 7 deals with the case of systems. We present some results by the authors contained in [72] and [77] and we simplify some proofs of both papers. The bibliography that we have included is fairly extensive though, as is to be expected, far from exhaustive. Further references to a rich literature can be found in the works cited here. L. Damascelli and F. Pacella

Contents Preface | V Notation | XIII 1 1.1 1.2 1.2.1 1.2.2 1.3 1.4 1.5 1.5.1 1.5.2 1.5.3

Preliminaries | 1 Sobolev spaces | 1 Maximum principles | 14 Weak maximum principle | 15 Strong maximum principle | 20 Compact self-adjoint operators | 23 Eigenvalues of elliptic problems | 29 Systems of elliptic equations | 43 Spectral theory for symmetric systems | 47 Weak maximum principle for cooperative systems | 52 Comparison principles for semilinear elliptic systems | 55

2 2.1 2.1.1 2.1.2 2.1.3 2.2 2.2.1 2.2.2 2.2.3

Introduction to Morse theory | 59 Morse theory on finite dimensional manifolds | 59 Introduction | 59 Morse lemma and deformation theorems | 62 Morse inequalities | 68 Infinite dimensional Morse theory | 73 Critical groups and Morse lemma | 73 Morse inequalities | 76 Morse index of mountain pass critical points | 78

3 3.1 3.2 3.2.1 3.2.2 3.2.3

Morse theory for semilinear elliptic equations | 83 Introduction | 83 Positive solutions of Dirichlet problems | 86 Existence of a positive solution by the mountain pass theorem | 87 Existence of a positive solution by constrained minimization | 89 Uniqueness of solutions of Morse index one of Lane-Emden problems | 91 Sign changing solutions of Dirichlet problems | 97 Existence of a solution with Morse index two by constrained minimization | 97 Estimates of Morse index for symmetric sign changing solutions: the autonomous case | 104

3.3 3.3.1 3.3.2

X | Contents 3.3.3

4 4.1

Estimates of Morse index for symmetric sign changing solutions: the nonautonomous case | 110

4.4 4.5

Morse index of radial solutions of Lane–Emden problems | 121 Spectral decomposition for the linearized operator at a radial solution | 121 Asymptotic analysis of radial solutions | 128 The case N ≥ 3 | 128 The case N = 2 | 132 Computation of the Morse index of radial solutions in dimension N ≥ 3 | 136 Computation of the Morse index in dimension N = 2 | 143 A weighted eigenvalue problem in ℝN | 147

5 5.1 5.2 5.2.1 5.2.2 5.2.3 5.3 5.3.1 5.3.2 5.3.3

Bifurcation from radial solutions | 155 Preliminaries | 155 Bifurcation with respect to the exponent p | 158 Asymptotic analysis of the radial solution | 158 Asymptotic behavior of the radial eigenvalue β̃ 1 (p) | 160 Bifurcation results | 165 Bifurcation with respect to the radius of the annulus | 167 Asymptotic estimates for the radial solution | 167 Asymptotic analysis of the eigenvalue β̃ 1 (R) | 171 Bifurcation results | 175

6

6.3.1 6.3.2 6.4

Morse index and symmetry for semilinear elliptic equations in bounded domains | 177 Symmetry and monotonicity of positive solutions | 177 Moving planes and symmetry | 177 Monotonicity by the method of moving planes | 182 Counterexamples to radial symmetry | 184 Foliated Schwarz symmetry and related properties | 186 Foliated Schwarz symmetry of low Morse index solutions of elliptic Dirichlet problems | 196 Convex nonlinearities | 196 Nonlinearities with a convex derivative | 199 Symmetry of solutions of mixed boundary value problems | 205

7 7.1 7.2 7.2.1

Morse index and symmetry for elliptic systems in bounded domains | 213 Morse index of solutions of elliptic systems | 213 Symmetry results | 215 Moving and rotating planes | 215

4.2 4.2.1 4.2.2 4.3

6.1 6.1.1 6.1.2 6.1.3 6.2 6.3

Contents | XI

7.2.2 7.2.3 7.2.4 7.3 8 8.1 8.2 8.3

Foliated Schwarz symmetry | 217 Nonlinearities having convex components | 219 Nonlinearities with convex derivatives | 224 Examples | 229 Some results in unbounded domains | 235 Moving planes and symmetry in unbounded domains | 235 Foliated Schwarz symmetry | 239 Nonexistence and classification results | 242

Bibliography | 245 Index | 255

Notation – – – – – – – – – – –

𝕂 will denote either the field ℝ or the field C. ℕ = {0, 1, 2, . . . } is the set of natural numbers, ℕ+ = ℕ \ {0} = {1, 2, . . . }. ℝN = {x = (x1 , . . . , xN ) : xi ∈ ℝ} is the N-dimensional Euclidean space, N ≥ 1. ℝN+ = {x = (x1 , . . . , xN ) = (x󸀠 , xN ) ∈ ℝN = ℝN−1 × ℝ : xN > 0}. ℝN0 = {(x󸀠 , 0) : x󸀠 ∈ ℝN−1 } = 𝜕ℝN+ . It will be often identified with ℝN−1 . Q(x, r) = QN (x, r) = {y ∈ ℝN : |yi − xi | < r, i = 1, . . . , N} for r > 0, x ∈ ℝN . B(x, r) = BN (x, r) = Br (x) = {y ∈ ℝN : |y − x| < r} for r > 0, x ∈ ℝN . Br = Br (0). S(x, r) = SN−1 (x, r) = Sr (x) = {y ∈ ℝN : |y − x| = r} for r > 0, x ∈ ℝN . SN−1 = SN−1 (0, 1) = {y ∈ ℝN : |y| = 1} is the unit sphere or equivalently the set of directions in ℝN . Ω denotes an open set in ℝN , N ≥ 2 and, unless otherwise stated, it will be assumed that Ω is at least a Lipschitz domain, i. e., it is connected and has a locally Lipschitz boundary 𝜕Ω. By this, we mean the following: for any x = (x󸀠 , xN ) ∈ 𝜕Ω, there exists r > 0 and a Lipschitz function β : Q󸀠 = QN−1 (x󸀠 , r) → ℝ such that, up to rotations and relabelings of the coordinates, it holds Ω ∩ Q(x, r) = {y ∈ Q(x, r) : yN > β(y1 , . . . , yN−1 )},

𝜕Ω ∩ Q(x, r) = {y ∈ Q(x, r) : yN = β(y1 , . . . , yN−1 )}.

– – –

–

– –

Ω is a C k,α domain if the functions β belongs to the class C k,α (see below). If Ω1 and Ω2 are open sets, Ω1 ⊂⊂ Ω2 means that Ω1 is a compact subset of Ω2 . measN (A), or simply meas(A) or even |A| denotes the N-dimensional Lebesgue measure of a measurable set A ⊂ ℝN . If Ω is a bounded domain with Lipschitz boundary, we will consider the (N − 1) Hausdorff measure on the boundary 𝜕Ω, and denote by: measN−1 (D), the (N −1)-dimensional measure of a measurable set D ⊂ 𝜕Ω. A property holds almost everywhere (we will write shortly a. e.) in an open set Ω ⊂ ℝN , respectively on 𝜕Ω if it holds in Ω \ S with measN (S) = 0, respectively in 𝜕Ω \ T with measN−1 (T) = 0. C k (Ω) is the space of functions with continuous derivatives up to the order k in Ω. If Ω is bounded, the Banach space C k (Ω) consists of the functions u ∈ C k (Ω) such that for any multiindex β = (β1 , . . . , βN ) ∈ ℕN with |β| = β1 + ⋅ ⋅ ⋅ βN ≤ k the derivatives Dβ u =

𝜕|β| u

β β 𝜕x1 1 ...𝜕xNN

are (restrictions of functions) continuous in Ω, with the norm ‖u‖Ck (

Ω) https://doi.org/10.1515/9783110538243-202

= ∑ sup |Dβ u(x)|. |β|≤k x∈Ω

XIV | Notation – – –

Cck (Ω) is the subspace of C k (Ω) of the functions with continuous derivatives up to the order k in Ω whose support is a compact subset of Ω. C ∞ (Ω) is the space of functions with continuous derivatives of any order in Ω. For 0 < α ≤ 1, C k,α (Ω) is the subspace of C k (Ω) of the functions u whose derivatives of order k are α-Hölder continuous in Ω, and if Ω is bounded it is a Banach space with the norm sup

‖u‖Ck,α (Ω) = ‖u‖Ck (Ω) + ∑

|β|=k x,y∈Ω,x =y̸

– – – –

Cc∞ (Ω) is the space of functions having continuous derivatives of any order in Ω whose support is a compact subset of Ω. Lp (Ω), 1 ≤ p < ∞, denote the Lebesgue space of measurable functions u in Ω such that ‖u‖pp,Ω = ∫Ω |u(x)|p dx < ∞. L∞ (Ω) is the spaces of (essentially) bounded measurable functions u in Ω such that ‖u‖∞,Ω = ess supΩ |u(x)| < ∞. If Ω is a bounded domain and u ∈ L1 (Ω), the integral mean or average of u in Ω is 1 defined by the number meas(Ω) ∫Ω u(x) dx, and it is denoted by one of the symbols (u)Ω = ∫ − u(x) dx =

–

1 ∫ u(x) dx. meas(Ω) Ω

Ω

–

|Dβ u(x) − Dβ u(y)| . |x − y|α

If Ω is a bounded domain with Lipschitz boundary and Γ is a measurable subset of 𝜕Ω, we denote by Lp (Γ), 1 ≤ p ≤ ∞ the Lebesgue spaces with respect to the (N − 1)dimensional Hausdorff measure on Γ. W k,p (Ω), 1 ≤ k ≤ ∞, 1 ≤ p ≤ ∞ is the Sobolev space of functions u ∈ Lp (Ω) whose distributional derivatives Dβ u, |β| ≤ k, belong to Lp (Ω) with the norm k

1

β

p

‖u‖k,p,Ω = ( ∑ ∑ ∫ |D u| dx) j=0 |β|=j Ω

Notation | XV

– –

– –

–

H0k (Ω) = W0k,2 (Ω). If Γ is a relatively open subset of 𝜕Ω, W0k,p (Ω ∪ Γ), 1 ≤ p < ∞, is the closure of Cc∞ (Ω ∪ Γ) (or Cck (Ω ∪ Γ)) in the Sobolev space W k,p (Ω). We often consider the case when Γ is a smooth embedded (N − 1)-submanifold in ℝN . H0k (Ω ∪ Γ) = W0k,2 (Ω ∪ Γ). Trace (u) ∈ Lp (𝜕Ω) is the trace on 𝜕Ω of a function u ∈ W 1,p (Ω). It belongs to the 1− 1 ,p fractional space W p (𝜕Ω) and to other Lebesgue spaces, as we will recall. We will write sometimes u⌉𝜕Ω or simply u instead of Trace (u). For a function v : Ω → ℝ, we write v+ = max{v, 0} and v− = − min{v, 0} = max{−v, 0}

–

for the positive and negative part of u. If V is a Banach space, we will use the same notation that we use in ℝN for the ball centered at x ∈ V with radius r > 0, i. e., we write B(x, r) = Br (x) = {y ∈ V : ‖y − x‖ < r}

– –

for r > 0, x ∈ V and Br = Br (0).

V ∗ denotes the dual space of a Banach space V. We say that the Banach space W is continuously injected (or embedded) in the Banach space V and we write W 󳨅→ V if W ⊂ V and there exists a constant C > 0 such that ‖w‖V ≤ C‖w‖W for any w ∈ W. More generally, we use the same notation if there exists an injective continuous linear operator T : W → V.

1 Preliminaries Here we will state and/or prove some results needed to understand the content of the book. We will start by recalling the basic theory for Sobolev spaces of functions vanishing on some part of the boundary of a domain in Section 1.1. Next, we describe in Section 1.2 several forms of maximum principles. In Section 1.3 we recall the spectral theory of bounded self-adjoint operators in Hilbert spaces and apply it in Section 1.4 to some elliptic operators. We extend the classical theory for the Dirichlet boundary value problem to the case of mixed Dirichlet–Neumann problems, providing the construction and variational characterization of the eigenvalues and their main properties. A section on maximum principles and spectral theory for elliptic cooperative systems ends the chapter.

1.1 Sobolev spaces Limiting ourselves to first-order spaces, we recall some well-known properties of Sobolev spaces. For most of the proofs, we refer to [2, 41, 115, 116, 149, 174, 218, 227] or any book on Sobolev spaces and PDEs. We will detail a few proofs, in the case the proof itself is needed in the sequel or it is not easily available in textbooks. In the sequel, we refer to the notation for all the undefined symbols. Let Ω be an open set in ℝN , N ≥ 2. A useful characterization of the Sobolev spaces W 1,p (Ω) is the following. Theorem 1.1. A function v ∈ Lp (Ω) belongs to W 1,p (Ω) if and only if it has a representative v which, for any i = 1, . . . , N, is absolutely continuous on almost all segments in 𝜕v Ω parallel to the xi -axis that intersect Ω1 and whose (classical) partial derivatives 𝜕x i p belong to L (Ω). As a consequence, we have that: – if Ω is connected, v ∈ W 1,p (Ω) and ∇v = 0 in Ω, then v is constant in Ω; – if v ∈ W 1,p (Ω), then the positive and negative parts v+ = max{v, 0} and v− = − min{v, 0} = max{−v, 0}, belong to W 1,p (Ω), with ∇v(x) if v(x) > 0 ∇v+ (x) = { 0 if v(x) ≤ 0 0 −∇v(x)

∇v− (x) = {

if v(x) ≥ 0 if v(x) < 0

Moreover, ∇v = 0 a. e. on [v = 0] = {x ∈ Ω : v(x) = 0}. 1 By this we mean that if, e. g., i = N, the set of the points x 󸀠 ∈ ℝN−1 , such that the line Lx󸀠 = {(x 󸀠 , t) : t ∈ ℝ} intersects Ω in at least one compact interval {x 󸀠 } × [a, b] and v(x 󸀠 , .) is not absolutely continuous on [a, b], is a set of (N − 1)-measure zero. https://doi.org/10.1515/9783110538243-001

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2 | 1 Preliminaries We now recall some of the classical Sobolev inequalities. Theorem 1.2 (Sobolev inequalities in Cc1 (ℝN )). Let v ∈ Cc1 (ℝN ). Then: Np , then there exists a constant C, depending on p, N such 1. if 1 ≤ p < N, and p∗ = N−p that ‖v‖Lp∗ (ℝN ) ≤ C‖|∇v|‖Lp (ℝN ) 2.

if p = N and q ∈ [N, ∞), then there exists a constant C, depending on p, N, q such that ‖v‖Lq (ℝN ) ≤ C‖v‖W 1,N (ℝN )

3.

(1.1)

if p > N and γ = 1 −

N , p

(1.2)

then there exists a constant C, depending on p, N such that ‖v‖C0,γ (ℝN ) ≤ C‖v‖W 1,p (ℝN )

(1.3)

If Ω is an open set in ℝN , we denote by W01,p (Ω) the closure in W 1,p (Ω) of the set Cc∞ (Ω) (or Cc1 (Ω)) of the smooth functions whose support is a compact subset of Ω. We 1,p also denote by Wcomp (Ω) the set of the functions in W 1,p (Ω) whose support is a compact subset of Ω. Theorem 1.3 (Sobolev inequalities in W01,p (Ω)). i) If Ω = ℝN , then W01,p (ℝN ) = W 1,p (ℝN ), i. e., if v ∈ W 1,p (ℝN ) there exists a sequence vn ∈ Cc1 (ℝN ) that converges to v in the W 1,p norm. 1,p ii) If Ω ⊂ ℝN , then Wcomp (Ω) ⊂ W01,p (Ω), i. e., any v ∈ W 1,p (Ω) such that supp(v) is a compact subset of Ω is the limit in W 1,p (Ω) of a sequence vn ∈ Cc1 (Ω) and, therefore, 1,p Wcomp (Ω) is dense in W 1,p (Ω).

iii) If Ω ⊂ ℝN and v ∈ W01,p (Ω), then the trivial extension of v to ℝN belongs to W01,p (ℝN ) = W 1,p (ℝN ). As a consequence, we get Sobolev inequalities in W01,p (Ω): If Ω ⊆ ℝN and v ∈ W01,p (Ω) (in particular, if Ω = ℝN and v ∈ W 1,p (ℝN )), then: ∗ Np , and there exists a constant C, 1. if 1 ≤ p < N, then v ∈ Lp (Ω), where p∗ = N−p depending only on p, N such that ‖v‖Lp∗ (Ω) ≤ C‖|∇v|‖Lp (Ω)

(1.4)

for any v ∈ W01,p (Ω). Moreover, if q ∈ [p, p∗ ] then v ∈ Lq (Ω) and there exists a constant C > 0, depending on p, N, q, such that ‖v‖Lq (Ω) ≤ C‖v‖W 1,p (Ω)

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(1.5)

1.1 Sobolev spaces | 3

2.

if p = N, then v ∈ Lq (Ω) for any q ∈ [N, ∞) and, for any such q, there exists a constant C, depending on p, N, q such that ‖v‖Lq (Ω) ≤ C1 ‖v‖W 1,N (Ω)

3.

for any v ∈ W01,N (Ω) if p > N, then v ∈ L∞ (Ω) ∩ C 0,γ (Ω), where γ = 1 − depending on p, N such that

(1.6)

N , p

and there exists a constant C,

‖v‖C0,γ (Ω) ≤ C‖v‖W 1,p (Ω)

(1.7)

for any v ∈ W01,p (Ω). Note that only (1.4) is a “pure” Sobolev inequality, i. e., on the right-hand side only the Lp norm of the gradient appears. When Ω is bounded, we can substitute on the right-hand side the Lp norm of the gradient in all the previous inequalities, thanks to the fundamental Poincaré’s inequality, that we prove below using Sobolev’s inequalities. Theorem 1.4 (Poincaré’s inequality in W01,p (Ω)). Let Ω be a bounded domain, 1 ≤ p < ∞. Then there exists a constant C > 0, depending on p, N, such that for any v ∈ W01,p (Ω), 1

‖v‖Lp (Ω) ≤ C|Ωv | N ‖|∇v|‖Lp (Ω)

(1.8)

where Ωv = {x ∈ Ω : v(x) ≠ 0} ⊂ Ω and |Ωv | = measN (Ωv ). Proof. If 1 ≤ p < N, denoting by the same symbol the trivial extensions of functions ∗ ∗ to the whole ℝN , we have, by Hölder’s inequality with exponents pp and p∗p−p , and by Sobolev inequalities, 1− pp∗

∫ |v|p dx = ∫ |v|p dx ≤ |Ωv | Ω

(∫ |v|p dx) ∗

p p∗

p

≤ C|Ωv | N ∫ |∇v|p dx

Ω

Ωv

Ω

Np If instead p ≥ N ≥ 2 and q = N+p , then 1 ≤ q < N and p = q∗ , so that by Sobolev’s p inequality and Hölder’s inequality with exponents pq , p−q , we get (recalling that ∇v = 0 a. e. on [v = 0]) p

q∗

∫ |v| dx = ∫ |v| Ω

q

dx ≤ C(∫ |∇v| dx) Ω

Ω

q∗ p

q

= C(∫ |∇v| dx)

q∗ q

Ωv ∗

(1− pq ) qq

≤ C(∫ |∇v|p dx) |Ωv | Ω

q∗ q

p

= C|Ωv | N ∫ |∇v|p dx Ω

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4 | 1 Preliminaries If Ω is bounded (or more generally with finite measure), the Poincaré’s inequality and a “pure” Sobolev inequality are not true in the space W 1,p (Ω) as it can be easily observed considering the constant functions. Instead there are unbounded domains Ω where a pure Sobolev’s inequality holds true in W 1,p (Ω) (see [169] and the references therein for related questions). In particular, we consider the case of half-spaces. This in turn will give Poincaré’s type inequalities for a class of functions defined in cylindrically symmetric domains that will be extensively used in the sequel. Let us recall the notation: ℝN+ = {x = (x 󸀠 , xN ) ∈ ℝN : xN > 0} and ℝN0 = {(x󸀠 , 0) : 󸀠 x ∈ ℝN−1 } = 𝜕ℝN+ . The last one will be often identified with ℝN−1 . Let us denote by ṽ the reflection through the hyperplane xN = 0 of a function v defined in ℝN+ : v(x󸀠 , xN )

̃ 󸀠 , xN ) = { v(x

v(x󸀠 , −xN )

if xN ≥ 0 if xN < 0

It provides an extension operator in W 1,p (ℝN+ ), in the following sense. Let 1 ≤ p < ∞ and v ∈ W 1,p (ℝN+ ). Then ṽ ∈ W 1,p (ℝN ) with { 𝜕v (x󸀠 , xN ) 𝜕ṽ 󸀠 i (x , xN ) = { 𝜕x 𝜕v 󸀠 𝜕xi (x , −xN ) { 𝜕xi

if xN > 0 if xN < 0

{ 𝜕v (x󸀠 , xN ) 𝜕ṽ 󸀠 (x , xN ) = { 𝜕xN𝜕v 󸀠 𝜕xN − (x , −xN ) { 𝜕xN so that there exists a constant C such that

1 ≤ i ≤ N − 1,

,

if xN > 0

if xN < 0

,

‖v‖̃ W 1,p (ℝN ) ≤ C‖v‖W 1,p (ℝN+ )

(1.9)

As a consequence, it provides approximations in W 1,p (ℝN+ ) in the sense that Cc∞ (ℝN +) is dense in W 1,p (ℝN+ ). Moreover, Sobolev inequalities hold in W 1,p (ℝN+ ) and, by density, it is easy to prove trace Sobolev inequalities as well. Theorem 1.5 (Sobolev inequalities in W 1,p (ℝN+ )). Let 1 ≤ p < ∞ and v ∈ W 1,p (ℝN+ ). ∗ Np 1. If 1 ≤ p < N and p∗ = N−p , then v ∈ Lp (ℝN+ ) and there exists a constant C > 0, depending on p, N, such that ( ∫ |v|

Np N−p

dx)

N−p Np

ℝN+

≤ C( ∫ |∇v|p dx)

1 p

(1.10)

ℝN+

Moreover if q ∈ [p, p∗ ] then v ∈ Lq (ℝN+ ) and there exists a constant C > 0, depending on p, N, q, such that q

1 q

( ∫ |v| dx) ≤ C‖v‖W 1,p (ℝN+ ) ℝN+

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(1.11)

1.1 Sobolev spaces | 5

2.

If p = N and q ∈ [N, ∞), then v ∈ Lq (ℝN+ ) and there exists a constant C > 0, depending on p, N, q, such that 1 q

( ∫ |v|q dx) ≤ C‖v‖W 1,p (ℝN+ )

(1.12)

ℝN+

3.

If p > N, any function v ∈ W 1,p (ℝN+ ) is bounded and (has a representative) continuous in ℝN+ . Moreover, if γ = 1 − Np , there exists a constant C > 0, depending on p, N, such that ‖v‖C0,γ (ℝN+ ) ≤ C‖v‖W 1,p (ℝN+ )

(1.13)

Theorem 1.6 (Trace inequalities in Cc1 (ℝN+ ).). 1. If 1 ≤ p < N and p♯ = Np−p , there exists a constant C > 0, depending on p, N, such N−p that (

p♯

|v| dx )

∫

󸀠

1 p♯

p

≤ C( ∫ |∇v| dx)

1 p

(1.14)

ℝN+

ℝN0 =𝜕ℝN+

for any v ∈ Cc1 (ℝN+ ). Moreover, if q ∈ [p, p♯ ], there exists a constant C > 0, depending on p, N, q, such that (

∫

q

1 q

|v| dx ) ≤ C‖v‖W 1,p (ℝN+ ) 󸀠

(1.15)

ℝN0 =𝜕ℝN+

2.

for any v ∈ Cc1 (ℝN+ ). If p = N and q ∈ [N, ∞), there exists a constant C > 0, depending on p, N, q, such that 1 q

(

∫

|v|q dx󸀠 ) ≤ C‖v‖W 1,N (ℝN+ )

(1.16)

ℝN0 =𝜕ℝN+

for any v ∈ Cc1 (ℝN+ ). Proof. Let 1 ≤ p ≤ N. If v ∈ Cc1 (ℝN+ ) and q ≥ 1, we have +∞

𝜕 󵄨󵄨 󸀠 󵄨󵄨 󸀠 󵄨󵄨q 󵄨q (󵄨v(x , xN )󵄨󵄨󵄨 ) dxN 󵄨󵄨v(x , 0)󵄨󵄨 = − ∫ 𝜕xN 󵄨 0 +∞

𝜕v 󸀠 󵄨 󵄨q−2 = −q ∫ 󵄨󵄨󵄨v(x󸀠 , 0)󵄨󵄨󵄨 v(x 󸀠 , 0) (x , xN ) dxN 𝜕xN 0

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6 | 1 Preliminaries so that integrating in the x󸀠 -variable and using Hölder’s inequality we get 󵄨 󵄨q−1 󵄨 󵄨 󵄨q 󵄨 ∫ 󵄨󵄨󵄨v(x󸀠 , 0)󵄨󵄨󵄨 dx󸀠 ≤ q ∫ 󵄨󵄨󵄨v(x)󵄨󵄨󵄨 󵄨󵄨󵄨∇v(x)󵄨󵄨󵄨 dx ℝN

ℝN−1

1 󸀠

1

p p 󵄨(q−1)p󸀠 󵄨 󵄨p 󵄨 dx) ( ∫ 󵄨󵄨󵄨∇v(x)󵄨󵄨󵄨 dx) ≤ q( ∫ 󵄨󵄨󵄨v(x)󵄨󵄨󵄨

ℝN

ℝN

where p󸀠 is the conjugate exponent of p, i. e.,

1 p

+

1 p󸀠

= 1.

If 1 ≤ p < N, then (q − 1)p ∈ [p, p ] if and only if q ∈ [p, p♯ ], while if p = N then (q − 1)p󸀠 ∈ [p, ∞) if and only if q ∈ [p, ∞), and, if this is the case, by the Sobolev inequalities we obtain 󸀠

∗

󵄨 󵄨q 󵄨 󵄨p ∫ 󵄨󵄨󵄨v(x󸀠 , 0)󵄨󵄨󵄨 dx 󸀠 ≤ C( ∫ 󵄨󵄨󵄨∇v(x)󵄨󵄨󵄨 dx)

ℝN−1

ℝN

q−1 p

1

p 󵄨 󵄨p ( ∫ 󵄨󵄨󵄨∇v(x)󵄨󵄨󵄨 dx) = C‖|∇v|‖qLp

ℝN

If p > N, by considering the reflection through the hyperplane xN = 0, it is immediate to see that any function v ∈ W 1,p (ℝN+ ) is bounded and (has a representative) Hölder continuous in ℝN+ , so that we can consider the trace or restriction of a function v to the boundary hyperplane xN = 0. In the case when 1 ≤ p ≤ N, we can also define the trace of a Sobolev function on the boundary hyperplane xN = 0 using the previous theorem. Indeed, thanks to the previous inequalities, the restriction operator T which maps the function v = v(x󸀠 , xN ) ∈ Cc1 (ℝN+ ) (endowed with the W 1,p norm) to the function Tv = Tv(x󸀠 ) with Tv(x󸀠 ) = v(x󸀠 , 0) ∈ Lp (ℝN−1 ), extends by density to a bounded linear operator T : W 1,p (ℝN+ ) → Lp (ℝN−1 ) which coincides with the pointwise trace on the boundary for any v ∈ W 1,p (ℝN+ ) ∩ C 0 (ℝN+ ). We will refer to Tv as the trace of v on ℝN0 (which we identify with ℝN−1 ) and denote it by – Trace(v) or simply v⌉ℝN or even v if its meaning is clear from the context. 0

The previous inequalities extend immediately to traces in W 1,p (ℝN+ ) and we get the following theorem. Theorem 1.7 (Trace inequalities in W 1,p (ℝN+ )). Denoting by the same symbol v both a function and its trace on the boundary hyperplane ℝN0 , we have:

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1.1 Sobolev spaces | 7

1.

If 1 ≤ p < N and p♯ = such that

Np−p N−p

(

then there exists a constant C > 0, depending on p, N,

p♯

|v| dx )

∫

󸀠

1 p♯

p

≤ C( ∫ |∇v| dx)

1 p

(1.17)

ℝN+

ℝN0 =𝜕ℝN+

for any v ∈ W 1,p (ℝN+ ). Moreover, for any q ∈ [N, p♯ ], there exists a constant C > 0, depending on p, N, q, such that (

∫

q

1 q

|v| dx ) ≤ C‖v‖W 1,p (ℝN+ ) 󸀠

(1.18)

ℝN0 =𝜕ℝN+

2.

for any v ∈ W 1,p (ℝN+ ). If p = N, then for any q ∈ [N, ∞) there exists a constant C > 0, depending on p, N, q, such that (

∫

q

1 q

|v| dx ) ≤ C‖v‖W 1,N (ℝN+ ) 󸀠

(1.19)

ℝN0 =𝜕ℝN+

3.

for any v ∈ W 1,N (ℝN+ ). If p > N and v ∈ W 1,p (ℝN+ ), then v has an Hölder continuous representative and is bounded in ℝN+ . Moreover, if γ = 1 − Np , there exists a constant C > 0, depending on p, N, such that ‖v‖C0,γ (ℝN ) ≤ C‖v‖W 1,p (ℝN+ ) 0

(1.20)

for any v ∈ W 1,p (ℝN+ ). Using partitions of unity and local flattenings, many of the previous results can be used to deduce corresponding inequalities in the spaces W 1,p (Ω) when Ω is a bounded Lipschitz domain. Theorem 1.8 (Extensions, approximations and traces). Let Ω be a bounded Lipschitz domain, and 1 ≤ p < ∞. Then: 1. C ∞ (Ω) ∩ W 1,p (Ω) is dense in W 1,p (Ω). 2. There exists a bounded linear operator E : W 1,p (Ω) → W 1,p (ℝN ) such that Ev(x) = v(x) a. e. in Ω for any v ∈ W 1,p (Ω) (and ‖Ev‖W 1,p (ℝN ) ≤ C‖v‖W 1,p (Ω) for some constant C). We will refer to it as an extension operator. 3. There exists a bounded linear operator T : W 1,p (Ω) → Lp (𝜕Ω) such that Tv(x) = v(x) for any x ∈ 𝜕Ω if v ∈ W 1,p (Ω) ∩ C 0 (Ω) (and ‖Tv‖Lp (𝜕Ω) ≤ C‖v‖W 1,p (Ω) for some constant C).

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8 | 1 Preliminaries As before, we will refer to the operator T as the trace operator and denote by Trace(v) ∈ Lp (𝜕Ω) the trace Tv of a function v ∈ W 1,p (Ω). Note that the trace of v ∈ 1− 1 ,p W 1,p (Ω) actually belongs to the fractional space W p (𝜕Ω) and to other Lebesgue spaces, as we will recall. In the sequel, we will write sometimes v⌉𝜕Ω or simply v instead of Trace(v). Using the extension operator E, it is easy to see that the embeddings theorems for Sobolev spaces hold in every Lipschitz domain. Moreover, some embeddings are compact. Theorem 1.9 (Sobolev embeddings and Rellich–Kondrachov theorem). Let Ω be a bounded Lipschitz domain. Then ∗ Np – If 1 ≤ p < N and p∗ = N−p , then W 1,p (Ω) 󳨅→ Lp (Ω), i. e., there exists a constant C > 0 depending on p, N, Ω, such that ‖v‖Lp∗ (Ω) ≤ C‖v‖W 1,p (Ω)

– –

(1.21)

for any v ∈ W 1,p (Ω). Moreover, if 1 ≤ q < p∗ then W 1,p (Ω) is continuously and compactly embedded in Lq (Ω), i. e., W 1,p (Ω) 󳨅→ Lq (Ω), and any bounded sequence vn in W 1,p (Ω) has a ∗ subsequence vkn that converges in Lq (Ω) to a function v ∈ Lp (Ω) (with v ∈ W 1,p (Ω) if p > 1). If p = N, then W 1,N (Ω) is continuously and compactly embedded in Lq (Ω) for any q ≥ 1. If p > n, then W 1,p (Ω) is continuously and compactly embedded in C 0 (Ω). Analogously, the following embeddings involving the traces hold.

Theorem 1.10 (Trace embeddings in bounded domains). Let Ω be a bounded Lipschitz domain. Then ♯ – If 1 ≤ p < N and p♯ = Np−p , then W 1,p (Ω) 󳨅→ Lp (𝜕Ω), i. e., there exists a constant N−p C > 0 depending on p, N, such that ‖v‖Lp♯ (𝜕Ω) ≤ C‖v‖W 1,p (Ω)

– –

(1.22)

for any v ∈ W 1,p (Ω). Moreover, if 1 < p < N and 1 ≤ q < p♯ then W 1,p (Ω) is continuously and compactly embedded in Lq (𝜕Ω). If p = N, then W 1,N (Ω) is continuously and compactly embedded in Lq (𝜕Ω) for any q ≥ 1. If p > n, then W 1,p (Ω) is continuously and compactly embedded in C 0 (Ω) and, therefore, in C 0 (𝜕Ω).

As recalled before, if Ω is a bounded domain the Poincaré’s inequality and a pure Sobolev inequality are not true in the space W 1,p (Ω), but they do hold in subspaces

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1.1 Sobolev spaces | 9

bigger than W01,p (Ω), in particular in the space of functions whose trace vanishes on a part of the boundary with positive (N − 1)-dimensional measure. Theorem 1.11 (More general Poincaré’s, pure Sobolev and trace inequalities). Let Ω be a bounded Lipschitz domain, 1 < p < ∞, and let us denote by [Trace(v) = 0] the set {x ∈ 𝜕Ω : Trace(v)(x) = 0}. For any a > 0, there exists a constant C > 0, depending on p, N, a, such that for any v ∈ W 1,p (Ω) with measN−1 ([Trace(v) = 0]) ≥ a > 0 the following statements hold: i) If 1 < p < N, ‖v‖Lp∗ (Ω) ≤ C‖|∇v|‖Lp (Ω)

‖v‖Lp♯ (𝜕Ω) ≤ C‖|∇v|‖Lp (Ω)

(1.23) (1.24)

ii) If 1 < p < ∞, 1

‖v‖Lp (Ω) ≤ C|Ωv | N ‖|∇v|‖Lp (Ω)

(1.25)

where, as before, Ωv = {x ∈ Ω : v(x) ≠ 0} ⊂ Ω and |Ωv | = measN (Ωv ). The same inequalities hold – for any v ∈ W 1,p (Ω) with measN ([v = 0]) ≥ a > 0 with a constant C depending on p, N, a. – for any v ∈ W 1,p (Ω) with ∫ −Ω v dx = 0 with a constant C depending on p, N. Proof. Step 1. We start by proving for any p > 1 a first Poincaré’s inequality with a constant depending on p and N. Namely, we prove that there exists a constant C > 0 such that ‖v‖Lp (Ω) ≤ C‖|∇v|‖Lp (Ω)

(1.26)

for any v ∈ W 1,p (Ω) with measN−1 ([Trace(v) = 0]) ≥ a > 0. Suppose by contradiction that this is not true. Then there exists a sequence vn with measN−1 ([Trace(vn ) = 0]) ≥ a > 0 and ‖vn ‖Lp ≥ n‖|∇vn |‖Lp

(1.27)

and by homogeneity we can assume that ‖vn ‖Lp = 1. If this is the case, then the sequence is bounded in W 1,p (Ω), so that, up to a subsequence, vn converges weakly in W 1,p (Ω) to a function v ∈ W 1,p (Ω). Thanks to the compact embedding into Lp (Ω), a subsequence converges to v a. e. and strongly in Lp (Ω), so that ‖v‖Lp = 1, and v ≢ 0. On the other hand by (1.27) and by the weak semicontinuity of the norm, we have that ‖|∇v|‖Lp ≤ lim inf ‖|∇vn |‖Lp = 0, so that ∇v = 0 a. e. in Ω, and v is equal to a

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10 | 1 Preliminaries constant c. The constant c cannot be zero, since ‖v‖Lp = 1 and this is a contradiction. Indeed by the compact trace embeddings Trace(vn ) → Trace(v) (N − 1)-a. e. in 𝜕Ω and strongly in Lp (𝜕Ω); if Trace(vn ) = 0 on Sn , with measN−1 (Sn ) ≥ a > 0, then v = 0 a.e on ∞ S = ⋂∞ j=1 ⋃n=j Sn with measN−1 (S) ≥ a, which is impossible since v ≡ c ≢ 0. Step 2. By the Sobolev and Trace embeddings for W 1,p (Ω) and (1.26), it follows that the pure Sobolev inequality (1.23) and the pure Trace inequality (1.24) hold. Step 3. Proceeding exactly as in the proof of Theorem 1.4 we get the Poincaré’s inequality (1.25) with the dependence of the constant on the measure of Ω. The proof for functions with vanishing mean or vanishing in a set of positive measure in Ω is exactly the same. Remark 1.12. The best constant for Sobolev embedding has been studied; see [17, 36, 213] for the case of ℝN and [164] for the case of functions vanishing on a part of the boundary, and the references therein. The previous theorem applies in particular to the subspace of W 1,p (Ω) of the functions whose trace vanishes on a fixed part of the boundary having positive (N − 1)-measure. This class of functions will be considered in the next chapters, therefore, we state explicitly the results relative to it. Let Ω be a Lipschitz domain in ℝN , and let Γ ⊂ 𝜕Ω a relatively open subset of the boundary. We denote by W01,p (Ω ∪ Γ) the closure in W 1,p (Ω) of the subspace Cc1 (Ω ∪ Γ). It coincides with the subspace of the functions in W 1,p (Ω) whose trace vanishes on 𝜕Ω \ Γ and obviously W01,p (Ω ∪ Γ) = W01,p (Ω) if Γ = 0, while W01,p (Ω ∪ Γ) = W 1,p (Ω) if Γ = 𝜕Ω. In particular, we will consider the space H01 (Ω ∪ Γ) = W01,2 (Ω ∪ Γ) which is the closure of Cc∞ (Ω ∪ Γ) in the Hilbert space H 1 (Ω), and coincides with the space of functions v ∈ H 1 (Ω) such that Trace(v) = 0 on 𝜕Ω \ Γ. Setting Γ0 = 𝜕Ω \ Γ we have also Γ = 𝜕Ω \ Γ0 since, in the relative topology of 𝜕Ω, Γ is open and Γ0 is its exterior. We require the following conditions: Γ is either empty or a smooth (N − 1) − submanifold

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(1.28)

1.1 Sobolev spaces | 11

𝜕Ω \ (Γ0 ∪ Γ) = Γ0 ∩ Γ = 𝜕Γ0 ∩ 𝜕Γ

is either empty or a smooth (N − 2) − submanifold

(1.29)

Let us summarize the previous results in the following statement. Theorem 1.13 (Sobolev, Trace and Poincaré’s inequalities in H01 (Ω ∪ Γ)). Let Ω be a bounded Lipschitz domain in ℝN , N ≥ 2, Γ and Γ0 subsets of the boundary, as described in (1.28), (1.29). Moreover, let us assume that Γ0 has positive (N − 1)-Hausdorff measure. If N ≥ 3, then there exist constants C1 , C2 > 0 such that (∫ |v|

2N N−2

dx)

Ω

(∫ |v|

2N−2 N−2

dσ)

N−2 2N

N−2 2N−2

2

≤ C1 (∫ |∇v| dx)

1 2

Ω 2

≤ C2 (∫ |∇v| dx)

Γ

(1.30) 1 2

(1.31)

Ω

for any v ∈ H01 (Ω ∪ Γ). If N = 2, the norm in the left-hand side can be replaced by any Lq norm, q > 1. Moreover, if N ≥ 2, then there exists a constant C depending on N and meas(N−1) (𝜕Ω\ Γ) such that 2

1 2

1 N

2

(∫ |v| dx) ≤ C|Ωv | (∫ |∇v| dx) Ω

1 2

(1.32)

Ω

for any v ∈ H01 (Ω ∪ Γ), where Ωv = {x ∈ Ω : v(x) ≠ 0} ⊂ Ω and |Ωv | is its N-dimensional measure. Remark 1.14. Note that the constants which appear in the previous theorem depend on the (N − 1)-measure of the portion of the boundary where the functions involved vanish. In some cases, considering subsets D of a given domain Ω, it could be important to have the previous inequalities with constants independent of the specific subset D. When dealing with some domains with cylindrical symmetry, we can use the basic inequalities that hold in the half spaces and get further Poincaré’s inequalities with fixed constants, as we show below. Denoting by x = (x󸀠 , xN ) a point x = (x1 , . . . , xN−1 , xN ) ∈ ℝN , the domains we consider will be subsets of the half-space ℝN+ = {x = (x1 , . . . , xN ) ∈ ℝN : xN > 0} defined as follows. Definition 1.15. We say that a bounded Lipschitz domain Ω in ℝN , N ≥ 2, has cylindrical symmetry or is a cylindrically symmetric domain if assuming that inf{t ∈ ℝ : (x󸀠 , t) ∈ Ω} = 0,

sup{t ∈ ℝ : (x󸀠 , t) ∈ Ω} = b > 0

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12 | 1 Preliminaries then, for every h ∈ [0, b), the set Ωh = Ω ∩ {xN = h} is either a closed (N −1)-dimensional ball or a closed (N −1)-dimensional annulus with the center on the xN axis. For such domains, we will always set Γ0 = 𝜕Ω ∩ ℝN+ ;

Γ = 𝜕Ω \ Γ0 = Ω0 = int(Ω ∩ ℝN0 )

(1.33)

Thus Γ is a relatively open flat part of the boundary at the height xN = 0, which, by our assumptions, is either a (N − 1)-dimensional ball or a (N − 1)-dimensional annulus. Examples 1.16. Cylindrically symmetric domains are the following: – a half-ball (BNR )+ = BR ∩ ℝN+ = {x = (x1 , . . . , xN ) ∈ ℝN : |x| < R; xN > 0}, –

a half-annulus (ANR1 ,R2 )+ = {x = (x1 , . . . , xN ) ∈ ℝN : R1 < |x| < R2 ; xN > 0},

–

a cylinder 󵄨 󵄨 CR,b = {x = (x󸀠 , xN ) ∈ ℝN : 󵄨󵄨󵄨x 󸀠 󵄨󵄨󵄨 < R; 0 < xN < b},

–

an annular cylinder 󵄨 󵄨 CR1 ,R2 ,b = {x = (x󸀠 , xN ) ∈ ℝN : R1 < 󵄨󵄨󵄨x 󸀠 󵄨󵄨󵄨 < R; 0 < xN < b},

–

a cone 󵄨 󵄨 R KR,b = {x = (x󸀠 , xN ) ∈ ℝN : 󵄨󵄨󵄨x 󸀠 󵄨󵄨󵄨 < (b − xN ); 0 < xN < b} . b

Note that Γ0 is smooth in the first two examples, and is not in the other cases (the cone is not smooth at the vertex, and the cylinders at height b). When considering cylindrically symmetric domains, we will take advantage of their simple geometry to prove all the relevant inequalities that we need, starting from (1.10), (1.14), no matter how big is the part of the boundary where the functions vanish. Note that if v ∈ H01 (Ω∪Γ) then the trivial extension of v to ℝN+ belongs to H 1 (ℝN+ ) and has vanishing trace on ℝN0 \ Γ. As a consequence, using Hölder’s inequality, we obtain Poincaré’s type inequalities both in Ω and on the flat boundary Γ. More precisely, we have the following.

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1.1 Sobolev spaces | 13

Theorem 1.17 (Poincaré’s inequalities in cylindrically symmetric domains). Let N ≥ 2 and let Ω be a bounded cylindrically symmetric domain. There exist constants C1 , C2 > 0, depending only on N, such that for any v ∈ H01 (Ω ∪ Γ) 󵄨2 󵄨 ∫ |v|2 dx ≤ C1 󵄨󵄨󵄨[v ≠ 0]󵄨󵄨󵄨 N ∫ |∇v|2 dx

(1.34)

󵄨1 󵄨 ∫ |v| dx ≤ C2 󵄨󵄨󵄨[v ≠ 0]󵄨󵄨󵄨 N ∫ |∇v|2 dx

(1.35)

Ω

Ω

2

󸀠

Γ

Ω

where [v ≠ 0] = {x ∈ Ω : v(x) ≠ 0} and |[v ≠ 0]| denotes its measure. Proof. Let us first consider the case N ≥ 3. By density, we can assume that v ∈ Cc∞ (Ω∪Γ) and we denote by v also the trivial extension to ℝN+ . By Hölder’s and Sobolev inequalities, we have that 2

2

∫ |v| dx = ∫ |v| 1 dx ≤ ( ∫ |v| Ω

2N N−2

dx)

ℝN+

[v=0] ̸

󵄨 󵄨 ≤ C1 󵄨󵄨󵄨[v ≠ 0]󵄨󵄨󵄨

2 N

N−2 N

󵄨󵄨 󵄨2 󵄨󵄨[v ≠ 0]󵄨󵄨󵄨 N

󵄨 󵄨2 ∫ |∇v|2 dx = C1 󵄨󵄨󵄨[v ≠ 0]󵄨󵄨󵄨 N ∫ |∇v|2 dx Ω

ℝN+

so that we get (1.34). If instead N = 2 then p∗ = 2 for p = 1 and, since ∇v = 0 a. e. on [v = 0], by Sobolev and Hölder’s inequalities we get 2

∗ 󵄨 󵄨 ∫ |v|2 dx = ∫ |v|1 dx ≤ C( ∫ |∇v| dx) ≤ C 󵄨󵄨󵄨[v ≠ 0]󵄨󵄨󵄨 ∫ |∇v|2 dx

Ω

[v=0] ̸

[v=0] ̸

[v=0] ̸

and (1.34) follows. To get (1.35), we observe that for any x󸀠 = (x1 , . . . , xN−1 ) ∈ ℝN−1 we have that +∞

v2 (x 󸀠 , 0) = − ∫ 0

𝜕v 󸀠 𝜕v2 󸀠 (x , t) dt = −2 ∫ v(x󸀠 , t) (x , t) dt 𝜕xN 𝜕xN +∞ 0

Integrating over Γ and using the Poincaré’s inequality (1.34) and Hölder’s inequality, we get 2

2

1 2

2

1 2

∫ v (x , 0) dx ≤ 2 ∫ |v||∇v| dx ≤ 2(∫ |v| dx) (∫ |∇v| dx) dx Γ

󸀠

󸀠

Ω

ℝN+

Ω

1 N

󵄨 󵄨 ≤ C2 󵄨󵄨󵄨[v ≠ 0]󵄨󵄨󵄨 ∫ |∇v|2 dx Ω

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14 | 1 Preliminaries

1.2 Maximum principles Here, we consider standard semilinear elliptic equations of the type − Δu = f (x, u)

in Ω

(1.36)

where Ω is a Lipschitz domain in ℝN , N ≥ 2 and f is a continuous function in Ω × ℝ. 1 A weak solution of (1.36) is a function u ∈ Hloc (Ω) such that the function x 󳨃→ 1 f (x, u(x)) belongs to Lloc (Ω) and ∫ ∇u ⋅ ∇φ dx = ∫ f (x, u)φ dx Ω

(1.37)

Ω

Cc∞ (Ω).

for any φ ∈ When N ≥ 3 and u ∈ H 1 (Ω), we assume that the function x 󳨃→ f (x, u(x)) belongs 2N 2N to L N+2 (Ω) (note that N+2 is the conjugate exponent of the critical Sobolev exponent 2N ∗ 2 = N−2 ). Then, by density, (1.37) is satisfied for any φ ∈ H01 (Ω). We say that u ∈ H 1 (Ω) (weakly) satisfies the inequality u ≤ 0 on 𝜕Ω

(u ≥ 0 on 𝜕Ω)

if u+ ∈ H01 (Ω) (u− ∈ H01 (Ω)). Moreover, we write − Δu ≥ (≤) f (x, u) in Ω

(1.38)

∫ ∇u ⋅ ∇φ dx ≥ (≤) ∫ f (x, u)φ dx

(1.39)

(in a weak sense) if

Ω

Ω

for any φ ∈ H01 (Ω) with φ ≥ 0 in Ω. A weak solution of the Dirichlet problem −Δu = f (x, u) in Ω { u=0 on 𝜕Ω

(1.40)

is a function u ∈ H01 (Ω) such that (1.37) holds for any φ ∈ H01 (Ω). Let Ω be a bounded Lipschitz domain, Γ a relatively open portion of its boundary, Γ0 = 𝜕Ω \ Γ, f : Ω × ℝ and g : Γ × ℝ → ℝ continuous functions and let us denote by 𝜐 the outer normal on 𝜕Ω. A weak solution of the Dirichlet–Neumann problem (with nonlinear boundary conditions) −Δu = f (x, u) { { { {u = 0 { { { { 𝜕u = g(x, u) { 𝜕𝜐

in Ω on Γ0 on Γ

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(1.41)

1.2 Maximum principles | 15

is a function u ∈ H01 (Ω ∪ Γ) such that ∫ ∇u ⋅ ∇φ dx = ∫ f (x, u)φ dx + ∫ g(x󸀠 , u|Γ )φ|Γ dx󸀠 Ω

(1.42)

Γ

Ω

for any φ ∈ H01 (Ω ∪ Γ). Analogously, we say that u ∈ H 1 (Ω) weakly satisfies the inequality −Δu ≤ f (x, u) { { { {u ≤ 0 { { { { 𝜕u ≤ g(x, u) { 𝜕𝜐

in Ω on Γ0

(1.43)

on Γ

if u+ ∈ H01 (Ω ∪ Γ) and ∫ ∇u ⋅ ∇φ dx ≤ ∫ f (x, u)φ dx + ∫ g(x 󸀠 , u)φ dx 󸀠 Ω

Ω

(1.44)

Γ

for any φ ∈ H01 (Ω ∪ Γ), φ ≥ 0, with the obvious modification in the case of the reversed inequality. 1.2.1 Weak maximum principle Definition 1.18. Let D be a bounded domain in ℝN and c ∈ L∞ (D). We say that the operator L = −Δ + c satisfies the maximum principle in D (or the maximum principle holds for the operator L in D) if for any v ∈ H 1 (D) such that −Δv + c(x)v ≤ 0 in D { v≤0 on 𝜕D it holds that v ≤ 0 in D. If c ≥ 0, then the maximum principle holds in any bounded domain Ω, namely the following classical (weak) form of the weak maximum principle holds (see, e. g., [115, 126, 193, 194, 218] among others for many generalizations). Theorem 1.19 (First case for weak maximum principle). Let N ≥ 2, Ω a bounded domain in ℝN and let v ∈ H 1 (Ω) weakly satisfy −Δv + c(x)v ≤ 0 { v≤0

in Ω on 𝜕Ω

(1.45)

with c ∈ L∞ (Ω) and c ≥ 0 a. e. in Ω. Then v ≤ 0 in Ω.

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16 | 1 Preliminaries Proof. By hypothesis, the nonnegative function v+ belongs to H01 (Ω) and can be used as a test function in (1.45), yielding 2 󵄨 󵄨2 󵄨 󵄨2 ∫󵄨󵄨󵄨∇v+ 󵄨󵄨󵄨 dx ≤ ∫󵄨󵄨󵄨∇v+ 󵄨󵄨󵄨 + c(x)(v+ ) dx ≤ 0 Ω

Ω

so that v+ = 0, since ∫Ω |∇v+ |2 , which by Poincaré’s inequality is an equivalent norm on H01 (Ω), vanishes. In general, the maximum principle holds in domains with small measure, as we prove now using the Poincaré inequality, regardless of the sign of c(x). Theorem 1.20 (Weak maximum principle in small domains). Let N ≥ 2, Ω a bounded domain in ℝN and M ≥ 0. Assume that c ∈ L∞ (Ω), ‖c‖L∞ (Ω) ≤ M. Then there exists δ > 0, depending on M, such that the maximum principle holds for the operator −Δ + c in Ω󸀠 ⊂ Ω provided |Ω󸀠 | = measN (Ω󸀠 ) < δ. More generally, there exists δ > 0, depending on M, such that the following holds. If Ω󸀠 ⊆ Ω is a subdomain of Ω and v ∈ H 1 (Ω󸀠 ), then −Δv + c(x)v ≤ 0

{

v≤0

in Ω󸀠 on 𝜕Ω󸀠

󳨐⇒

v ≤ 0 in Ω󸀠

(1.46)

provided |[v > 0]| < δ, where [v > 0] = {x ∈ Ω󸀠 : v(x) > 0} and |[v > 0]| = measN ([v > 0]). (In particular, this condition is satisfied if |Ω󸀠 | < δ.) Proof. By hypothesis, the nonnegative function v+ belongs to H01 (Ω󸀠 ) and can be used as a test function in (1.46), yielding 2 󵄨 󵄨2 ∫ 󵄨󵄨󵄨∇v+ 󵄨󵄨󵄨 ≤ M ∫ (v+ )

Ω󸀠

Ω󸀠

On the other hand by the Poincaré’s inequality (1.8), we get 2

2

M ∫ (v+ ) ≤ MC|[v > 0]| N ∫ |∇v+ |2 Ω󸀠

Ω󸀠

If the measure of [v > 0] = {x ∈ Ω󸀠 : v(x) > 0} is sufficiently small, precisely when 2 MC|[v > 0]| N < 1, then necessarily ∫Ω󸀠 |∇v+ |2 dx = 0, which implies that v+ ≡ 0 in Ω󸀠 1

(since (∫Ω󸀠 |∇v+ |2 dx) 2 is an equivalent norm on H01 (Ω󸀠 )). As a corollary, we have the following.

Theorem 1.21 (Weak comparison principle in small domains). Let Ω be a bounded domain in ℝN , f = f (x, s) : Ω × ℝ → ℝ a continuous function which is locally Lipschitz continuous in s ∈ ℝ uniformly w. r. t. x ∈ Ω, and u, v ∈ H 1 (Ω) ∩ L∞ (Ω) such that ‖u‖L∞ (Ω) , ‖v‖L∞ (Ω) ≤ A.

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1.2 Maximum principles | 17

Then there exists δ > 0, depending on f and A (more precisely on the Lipschitz constant of f for s in [−A, A]) such that the following holds: if Ω󸀠 ⊆ Ω is a bounded subdomain of Ω with |[u > v] ∩ Ω󸀠 | < δ and −Δu ≤ f (x, u), −Δv ≥ f (x, v) in Ω󸀠 { u≤v on 𝜕Ω󸀠

(1.47)

then u ≤ v in Ω󸀠 . Proof. If Ω󸀠 ⊆ Ω and u, v satisfy (1.47), the difference u − v satisfies the linear inequality −Δ(u − v) + c(x)(u − v) ≤ 0

{

u−v ≤0

in Ω󸀠 on 𝜕Ω󸀠

where c : Ω → ℝ is defined by f (x,u(x))−f (x,v(x)) u(x)−v(x)

−c(x) = { 0

if u(x) ≠ v(x) if u(x) = v(x)

Since f is Lipschitz continuous w. r. t. s ∈ [−A, A], we get that c ∈ L∞ (Ω) with ‖c‖L∞ (Ω) ≤ M, where M is a Lipschitz constant of f for s ∈ [−A, A]. So the result is a consequence of the previous maximum principle. Remark 1.22. Of course, the choice of the value c(x) in points where u(x) = v(x) is arbitrary, but note that if f is a C 1 -function, then we can write a linear inequality for the difference u − v with a continuous coefficient, namely u − v satisfies the inequality ̃ −Δ(u − v) + c(x)(u − v) ≤ 0 where c̃ is the continuous function given by ̃ −c(x) ={

f (x,u(x))−f (x,v(x)) u(x)−v(x) 󸀠

f (x, u(x))

if u(x) ≠ v(x) if u(x) = v(x)

(1.48)

1

= ∫ f 󸀠 (x, tu(x) + (1 − t)v(x)) dt 0

For what concerns mixed boundary value problems, analogous properties hold. Let us fix Ω and the subsets Γ and Γ0 of 𝜕Ω that satisfy (1.28), (1.29). As before, we do not exclude the case of Γ = 0 (i. e., when the Dirichlet problem is studied), in which case Γ0 = 𝜕Ω. Definition 1.23. Let Ω be a bounded Lipschitz domain in ℝN , with Γ, Γ0 ⊆ 𝜕Ω satisfying 𝜕 (1.28), (1.29), c ∈ L∞ (Ω), d ∈ L∞ (Γ). We say that the operator L = (−Δ+c; 𝜕𝜐 +d) satisfies

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18 | 1 Preliminaries the maximum principle in Ω ∪ Γ if for any v ∈ H 1 (Ω) such that −Δv + cv ≤ 0 { { { {v ≤ 0 { { { { 𝜕v + dv ≤ 0 { 𝜕𝜐

in Ω on Γ0 on Γ

it holds that v ≤ 0 in Ω. As before, it is easy to see that for any bounded domain Ω: if c, d ≥ 0 then the maximum principle holds in Ω ∪ Γ. Definition 1.24. Let Ω be a bounded Lipschitz domain in ℝN , with Γ, Γ0 ⊆ 𝜕Ω satisfying (1.28), (1.29). If D ⊂ Ω is an open subdomain (i. e., it is open and connected), let us set ΓD = Γ ∩ 𝜕D, (Γ0 )D = 𝜕D \ ΓD . In particular when Γ = 0 then ΓD = 0 and (Γ0 )D = 𝜕D. We say that D is a regular subdomain of Ω if one of the following conditions holds: 1. Γ = 0 (then D can be any open connected subset of Ω) 2. Γ ≠ 0 and ΓD = Γ ∩ 𝜕D = 0 3. D is a Lipschitz domain, ΓD ≠ 0, and both ΓD and (Γ0 )D satisfy the conditions analogous to (1.28), (1.29). In cylindrically symmetric domains, a maximum principle in regular subdomains with small measure, analogous to Theorem 1.20, holds. Theorem 1.25 (Weak maximum principle in small cylindrical domains). Let N ≥ 2, Ω a cylindrically symmetric domain, α ∈ L∞ (Ω), β ∈ L∞ (Γ) with ‖α‖L∞ (Ω) ≤ M, ‖β‖L∞ (Γ) ≤ M. Then there exists δ > 0, depending on M, such that the following holds: if Ω󸀠 ⊆ Ω is a regular subdomain of Ω and v ∈ H 1 (Ω󸀠 ) satisfies −Δv + α(x)v ≤ 0 { { { {v ≤ 0 { { { { 𝜕v { 𝜕𝜐 + β(x)v ≤ 0

in Ω󸀠 on Γ󸀠0 = 𝜕Ω󸀠 ∩ ℝN+

(1.49)

on Γ = 𝜕Ω \ Γ󸀠0 󸀠

󸀠

then v ≤ 0 in Ω󸀠 , provided |[v > 0]| < δ, where [v > 0] = {x ∈ Ω󸀠 : v(x) > 0} and |[v > 0]| = measN ([v > 0]). In particular, this condition is satisfied if |Ω󸀠 | < δ. Proof. By hypothesis, the nonnegative function v+ belongs to H01 (Ω󸀠 ∪ Γ󸀠 ) and can be used as a test function, yielding 2 2 󵄨 󵄨2 ∫ 󵄨󵄨󵄨∇v+ 󵄨󵄨󵄨 ≤ M(∫ (v+ ) + ∫(v+ ) )

Ω󸀠

Ω󸀠

Γ󸀠

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1.2 Maximum principles | 19

On the other hand, by the Poincaré’s inequalities (1.34), (1.35), we get 2

2

M(∫ (v+ ) + ∫(v+ ) ) Γ󸀠

Ω󸀠

󵄨2 󵄨 󵄨 󵄨1 󵄨 󵄨2 ≤ MC(󵄨󵄨󵄨[v > 0]󵄨󵄨󵄨 N +󵄨󵄨󵄨[v > 0]󵄨󵄨󵄨 N ) ∫ 󵄨󵄨󵄨∇v+ 󵄨󵄨󵄨 Ω󸀠

2

1

If the measure of ([v > 0]) is sufficiently small, then MC(|[v > 0]| N + |[v > 0]| N ) < 1, which implies that v+ ≡ 0 in Ω󸀠 . As before, a corollary is the following Theorem 1.26 (Weak comparison principle in small cylindrical domains). Let Ω be a cylindrically symmetric domain, f (|x|, s) and g(|x󸀠 |, s) two functions defined in [0, +∞) × ℝ which are locally Lipschitz continuous functions w. r. t. the second variable, and u, v ∈ H 1 (Ω) ∩ L∞ (Ω) with their trace belonging to L∞ (𝜕Ω), such that ‖u‖L∞ (Ω) , ‖v‖L∞ (Ω) ≤ A. Then there exists δ > 0, depending on A, f , g such that the following holds: if Ω󸀠 ⊆ Ω is a regular subdomain of Ω, |[u > v] ∩ Ω󸀠 | < δ and −Δu ≤ f (|x|, u), −Δv ≥ f (|x|, v) { { { {u ≤ v { { { { 𝜕u 𝜕v { 𝜕𝜐 ≤ g(|x|, u), 𝜕𝜐 ≥ g(|x|, v)

in Ω󸀠 on Γ󸀠0 = 𝜕Ω󸀠 ∩ ℝN+

(1.50)

on Γ = 𝜕Ω \ Γ󸀠0 󸀠

󸀠

then u ≤ v in Ω󸀠 . Proof. Since f is Lipschitz continuous, the difference u − v satisfies a linear inequality, so the result is a consequence of the previous maximum principle, but we also give a direct short proof. Since u, v ∈ L∞ and f , g are locally Lipschitz continuous in the second variable, there exists Λ > 0 such that the functions fΛ̃ (|x|, s) = f (|x|, s) − Λs, g̃Λ (|x|, s) = g(|x|, s) − Λs, are nonincreasing in s ∈ [−A, A] for any x ∈ Ω (so that if u > v then f (|x|, u) − Λu ≤ f (|x|, v), −Λv). By hypothesis, the nonnegative function (u − v)+ belongs to H01 (Ω ∪ Γ) and can be used as a test function in the inequalities, yielding 2 2 󵄨 󵄨2 ∫󵄨󵄨󵄨∇(u − v)+ 󵄨󵄨󵄨 dx − ∫ Λ((u − v)+ ) dx − ∫ Λ((u − v)+ ) dx 󸀠 ≤ 0

Ω

Ω

Γ

On the other hand, by Poincaré’s inequalities (1.34), (1.35), we get that 2 2 󵄨 󵄨2 ∫󵄨󵄨󵄨∇(u − v)+ 󵄨󵄨󵄨 dx − ∫ Λ((u − v)+ ) dx − ∫ Λ((u − v)+ ) dx󸀠

Ω

Ω

Γ

󵄨1 󵄨 󵄨2 󵄨 󵄨2 󵄨 ≥ (1 − ΛC(󵄨󵄨󵄨[u > v] ∩ Ω󸀠 󵄨󵄨󵄨 N +󵄨󵄨󵄨[u > v] ∩ Ω󸀠 󵄨󵄨󵄨 N )) ∫󵄨󵄨󵄨∇(u − v)+ 󵄨󵄨󵄨 dx Ω

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20 | 1 Preliminaries and the last quantity is strictly positive if the measure of ([u > v]) is sufficiently small and (u − v)+ does not vanish. Remark 1.27. For general elliptic operators in nondivergence form, a maximum principle in domains with small measure has been proved in [31, 32], using the Aleksandrov–Bakelman–Pucci inequality (see [126]). 1.2.2 Strong maximum principle Let us recall the following versions of the strong maximum principle and Hopf’s lemma (see, e. g., [126, 193, 194] and the references therein for many generalizations). We say that Ω satisfies an interior sphere condition at x0 ∈ 𝜕Ω if there exists a ball B ⊂ Ω which has x0 on its boundary. Theorem 1.28 (Strong maximum principle and Hopf’s lemma). Let Ω be a (bounded or 1 unbounded) domain in ℝN , and let v ∈ Hloc (Ω) ∩ C 1 (Ω) be a weak solution of −Δv + c(x)v ≥ 0 in Ω

{

v≥0

(1.51)

in Ω

with c ∈ L∞ loc (Ω). Then either v ≡ 0 in Ω or v > 0 in Ω. Moreover if c ∈ L∞ (Ω), v ∈ C 0 (Ω ∪ {x0 }), v > 0 in Ω and x0 ∈ 𝜕Ω is a point where v(x0 ) = 0 and the interior sphere condition is satisfied then 𝜕v (x ) > 0 𝜕s 0

(1.52)

for any inward directional derivative (if v does not belong to the class C 1 (Ω ∪ {x0 }) this means that if s is a inward direction, i. e. there exists δ > 0 such that x0 + ts ∈ Ω for any v(x +ts)−v(x0 ) > 0). t ∈ (0, δ), then lim inft→0+ 0 t The proof of this theorem is based on the following lemma, which proves Hopf’s lemma in a particular case. Lemma 1.29. Let B = B(y, r1 ) a ball, x0 ∈ 𝜕B and let v ∈ H 1 (B) ∩ C 0 (B ∪ {x0 }) be a weak solution of −Δv + c(x)v ≥ 0 in B with c ∈ L∞ (B). Assume further that v > 0 in B and v(x0 ) = 0. Then 𝜕v (x ) > 0 for any inner direction s (this means that if (y − x0 ) ⋅ s > 0 then 𝜕s 0 lim inft→0+

v(x0 +ts)−v(x0 ) t

> 0).

Proof. Since we know that v ≥ 0, we can suppose that c ≥ 0 by substituting c with its positive part c+ . In fact, by hypothesis, we have that −Δv + c+ (x)v ≥ −Δv + c(x)v ≥ 0

in B

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1.2 Maximum principles | 21

Then we consider the annulus A = B \ B󸀠 where B = B(y, r1 ), B󸀠 = B(y, r21 ), and the function 2

2

z(x) = e−αr1 − e−α|x−y|

with α > 0 to be determined. The function z satisfies z < 0 in A, z = 0 on 𝜕B, z < 0 on 𝜕B󸀠 . Moreover, if x ∈ A then 2

Δz = −e−α|x−y| (4α2 |x − y|2 − 2αN) so that 2

−Δz + c(x)z = e−α|x−y| (4α2 |x − y1|2 − 2αN + c(x) 2

≥ e−α|x−y| (4α2

2

e−αr1

2

e−α|x−y|

− c(x)) 2

r12 r12 r2 − 2αN − ‖c‖∞ ) ≥ e 4 (4α2 ( 1 ) − 2αN − ‖c‖∞ ) > 0 4 4

if α is sufficiently large. Since v > 0 in B, it has a positive miminum on 𝜕B󸀠 , and if ε is sufficiently small then v + εz > 0 on 𝜕B󸀠 . By the weak maximum principle, since w = v + εz ≥ 0 on 𝜕A and −Δw + c(x)w ≥ ε(−Δz + c(x)z) ≥ 0 in A,we get that w = v + εz ≥ 0 in A, which in turn implies that 𝜕v 𝜕z 𝜕w (x ) = (x ) + ε ≥0 𝜕s 0 𝜕s 0 𝜕s Since

𝜕z (x ) 𝜕s 0

2

= 2α(x0 − y) ⋅ se−α|x0 −y| < 0, we get that

𝜕v (x ) 𝜕s 0

> 0.

Proof of Theorem 1.28. Assume that v ≢ 0 in Ω and let Ω+ = {x ∈ Ω : v(x) > 0}. Since v ≥ 0 and v ≢ 0 in Ω, then Ω+ ≠ 0. Suppose by contradiction that there exist points in Ω where v vanishes, i. e., Ω \ Ω+ ≠ 0. Then 𝜕Ω+ ∩ Ω ≠ 0 and we can find a point x1 ∈ Ω+ which is closer to 𝜕Ω+ than to 𝜕Ω. If r1 = sup{r > 0 : B(x1 , r) ⊂ Ω+ }, then B = B(x1 , r1 ) ⊂ Ω+ , B ⊂ Ω and there exists x0 ∈ 𝜕Ω+ ∩ Ω such that v(x0 ) = 0. By Lemma 1.29 𝜕v (x ) > 0 for any inward directional 𝜕s 0 derivative, and this contradicts the fact that ∇v(x0 ) = 0, since x0 is a minimum point in Ω for the function v. So if v ≢ 0 in Ω necessarily v > 0 in Ω. The last part of the assertion, regarding the sign of the inner derivative, has already been proved in Lemma 1.29. Let us remark that a consequence of Hopf’s lemma for Dirichlet–Neumann problems, under a global C 1 regularity hypothesis, is the following result.

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22 | 1 Preliminaries Theorem 1.30 (Strong maximum principle in H01 (Ω ∪ Γ)). Let Ω be a bounded domain in ℝN with the subsets Γ, Γ0 of 𝜕Ω satisfying (1.28), (1.29) and let v ∈ H01 (Ω∪Γ)∩C 1 (Ω∪Γ) be a weak solution of −Δv + c(x)v ≥ 0 { { { { { { {v ≥ 0 { v≥0 { { { { { { 𝜕v + d(x)v ≥ 0 { 𝜕𝜐

in Ω in Ω

(1.53)

on Γ0 on Γ

with c ∈ L∞ (Ω), d ∈ C 0 (Γ). Assume further that Ω satisfies an interior sphere condition at every point in Γ. Then either v ≡ 0 in Ω or v > 0 in Ω ∪ Γ. Proof. By the previous strong maximum principle if v ≢ 0 in Ω, then v > 0 in Ω and hence, by continuity, v ≥ 0 on Γ. Let x0 ∈ Γ and assume by contradiction that v(x0 ) = 0. Then by Hopf’s lemma 𝜕v (x ) < 0 where 𝜐 is the exterior normal, since v is positive 𝜕𝜐 0 in Ω and vanishes in x0 . This contradicts the Neumann condition 𝜕v (x ) + d(x)v(x0 ) = 𝜕𝜐 0 𝜕v (x ) ≥ 0. Hence v is positive on Γ. 𝜕𝜐 0 The strong maximum principle implies the following. Theorem 1.31 (Strong comparison principle). Let Ω be a (bounded or unbounded) do1 main in ℝN , and let u, v ∈ Hloc (Ω) ∩ C 1 (Ω) satisfy weakly −Δu ≤ f (x, u); −Δv ≥ f (x, v) in Ω { u≤v in Ω

(1.54)

where f = f (x, u) : Ω × ℝ → ℝ is a continuous function which is locally Lipschitz continuous in u, uniformly w.r.t x ∈ Ω. Then either u ≡ v in Ω or u < v in Ω. In the latter case, let x0 ∈ 𝜕Ω a point where u(x0 ) = v(x0 ) and the interior sphere condition is satisfied. Assume that B ⊂ Ω is a ball which has x0 on its boundary and u, v ∈ C 1 (B). Then 𝜕u (x ) < 𝜕v (x ) for any inward 𝜕s 0 𝜕s 0 directional derivative. Proof. It is a consequence of the strong maximum principle (Theorem 1.28). Indeed, as in Theorem 1.21, it suffices to observe that by hypothesis the difference w = u − v is nonpositive in Ω, and satisfies the linear inequality −Δ(u − v) + c(x)(u − v) ≤ 0 where

f (x,u(x))−f (x,v(x)) u(x)−v(x)

−c(x) = { 0

if u(x) ≠ v(x) if u(x) = v(x)

∈ L∞ loc (Ω)

Moreover, if B ⊂ Ω is a ball which has x0 on its boundary and u, v ∈ C 1 (B), then c ∈ L∞ (B).

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1.3 Compact self-adjoint operators | 23

1.3 Compact self-adjoint operators Here, we recall some basic properties of compact self-adjoint operators in Hilbert spaces and the related spectral theory (see [41, 151] or any text in linear functional analysis for more details). Let H be an Hilbert space on the field 𝕂 (= ℝ or C) with scalar product (. , .) and let us denote by ℬ(H) the space of bounded linear operators A : H → H with the norm ‖A‖ℬ(H) = sup

x =0, ̸ x∈H

‖Ax‖H = sup ‖Ax‖ ‖x‖H x∈H,‖x‖=1

If h ∈ H, let us denote by h⊥ = {k ∈ H : (k, h) = 0} the set of the vectors which are orthogonal to h, which is a closed subspace of H. Analogously, if S ⊂ H then S⊥ is the subspace of the vectors k ∈ H orthogonal to every h ∈ S. Definition 1.32. An operator A ∈ B(H) is self-adjoint if (Ax, y) = (x, Ay)

for any x, y ∈ H

Some properties of self-adjoint operators are summarized in the following proposition. Proposition 1.33. Let H be an Hilbert space and A ∈ ℬ(H) a self-adjoint operator. 1. (Au, u) is real for any u ∈ H. 2. Let us define (Au, u) ‖u‖2 (Au, u) M = MA = sup (Au, u) = sup ‖u‖2 u=0 ̸ ‖u‖=1 m = mA = inf (Au, u) = inf

u=0 ̸

‖u‖=1

so that m‖u‖2 ≤ (Au, u) ≤ M‖u‖2 ∀u ∈ H, and let λ ∈ {mA , MA } be such that |λ| = max{|m|, M}. Then |(Au, u)| 󵄨 󵄨 |λ| = sup 󵄨󵄨󵄨(Au, u)󵄨󵄨󵄨 = sup = ‖A‖ ‖u‖2 ‖u‖=1

3.

u=0 ̸

A = 0 if and only if m = M = 0.

Proof. The proof of statement 1. is straightforward, and (3) follows immediately from (1) and (2). Let us prove statement 2. It is immediate to see that |λ| = sup‖u‖=1 |(Au, u)| = supu=0̸ |(Au,u)| and we have to ‖u‖2 prove that |λ| = ‖A‖. If u ∈ H with ‖u‖ = 1, then 󵄨󵄨 󵄨 2 󵄨󵄨(Au, u)󵄨󵄨󵄨 ≤ ‖Au‖‖u‖ ≤ ‖A‖‖u‖ = ‖A‖ so that |λ| ≤ ‖A‖.

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24 | 1 Preliminaries Let us show the opposite inequality, namely ‖Au‖ ≤ |λ| = sup‖u‖=1 |(Au, u)| for any u ∈ H with ‖u‖ = 1. This is obvious if Au = 0, while if Au ≠ 0 it derives from the following identity: ‖Au‖ =

Au Au 1 Au Au (Au, Au) 1 = (A(u + ), u + ) − (A(u − ), u − ) ‖Au‖ 4 ‖Au‖ ‖Au‖ 4 ‖Au‖ ‖Au‖

which follows easily using the bilinearity of the scalar product and the self-adjointness of A. From this identity, the parallelogram identity and the fact that ‖u‖ = 1 we get ‖Au‖ ≤

2 2 ‖Au‖2 1 󵄩󵄩󵄩󵄩 Au 󵄩󵄩󵄩󵄩 1 󵄩󵄩󵄩 Au 󵄩󵄩󵄩󵄩 1 2 |λ|󵄩󵄩u + ) = |λ| 󵄩󵄩 + |λ|󵄩󵄩󵄩u − 󵄩󵄩 = |λ|(2‖u‖ + 2 4 󵄩󵄩 ‖Au‖ 󵄩󵄩 4 󵄩󵄩 ‖Au‖ 󵄩󵄩 4 ‖Au‖2

Let A ∈ ℬ(H) and μ ∈ 𝕂. We consider the operator Aμ = (A − μI) : H → H where I : x ∈ H 󳨃→ x is the identity map. Definition 1.34. The spectrum σ(A) of A is the set of μ ∈ 𝕂 such that Aμ = (A − μI) does not have a continuous inverse (A − μI)−1 : H → H. The resolvent set of A is ρ(A) = 𝕂 \ σ(A), i. e., it is the set of μ ∈ 𝕂 such that Aμ = (A − μI) is bijective with bounded inverse (A − μI)−1 : H → H. In particular, the point spectrum of A is the set of μ ∈ σ(A) such that Ker(Aμ ) ≠ {0}. If μ belongs to the point spectrum, then it is called an eigenvalue of A and the subspace Ker(Aμ ) is called the eigenspace corresponding to the eigenvalue μ. Any nonzero element of the eigenspace, namely any x ≠ 0 such that Ax = μx, is called an eigenvector of A corresponding to the eigenvalue μ. The dimension of the eigenspace Ker(Aμ ), if finite, is called the multiplicity of the eigenvalue μ, otherwise we say that μ has infinite multiplicity. If an eigenvalue μ has multiplicity 1, we will call μ a simple eigenvalue. It is easy to see that the eigenvalues of a self-adjoint operators are real. Moreover, eigenvectors u1 , u2 corresponding to different eigenvalues μ1 ≠ μ2 are orthogonal, since μ1 (u1 , u2 ) = (Au1 , u2 ) = (u1 , Au2 ) = μ2 (u1 , u2 ). Definition 1.35. An operator A ∈ ℬ(H) is said to be compact if it maps bounded sets into precompact sets, i. e., any bounded sequence {xn } in H has a subsequence {xkn } such that ykn = A(xkn ) converges in H to some y ∈ H. The collection of compact linear operators on H will be denoted by 𝒦(H). It is easy to see from the definition that if A ∈ 𝒦(H) and B ∈ ℬ(H), then the compositions AB and BA are compact. As a consequence, if A is a compact operator on an infinite dimensional Hilbert space H, then 0 ∈ σ(A) (for if A is invertible in ℬ(H) then I = AA−1 is compact, but the unit sphere is never compact if H is infinite dimensional). The following proposition shows a fundamental property of compact self-adjoint operators.

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1.3 Compact self-adjoint operators | 25

Proposition 1.36. Let A : H → H be a nonzero compact self-adjoint operator and let μ ∈ {mA , MA } be such that |μ| = max{|m|, M}. Then μ is an eigenvalue of A. Proof. By Proposition 1.33, |μ| = sup‖u‖=1 |(Au, u)| = ‖A‖ ≠ 0. Let un be a sequence, with ‖un ‖ = 1, such that (Aun , un ) → μ. Then ‖μun − Aun ‖2 = (μun − Aun , μun − Aun ) = μ2 ‖un ‖2 + ‖Aun ‖2 − 2μ(Aun , un ) ≤ μ2 + ‖A‖2 − 2μ(Aun , un ) → μ2 + μ2 − 2μ2 = 0

as n → ∞

Hence μun − Aun → 0. By the compactness of A, there exists a subsequence ukn such that Aukn → z ∈ H as n → ∞. This implies that μukn = μukn − Aukn + Aukn → 0 + z = z. Setting v = μ1 z we deduce that ukn → v. By the continuity of the norm and of the operator A, it follows that ‖v‖ = lim ‖ukn ‖ = 1, in particular v ≠ 0, and Av = lim Aukn = z = μv. Thus v is an eigenvector corresponding to the nonzero eigenvalue μ. By iterating Proposition 1.36, we get a fundamental result about compact selfadjoint operators in Hilbert spaces. Theorem 1.37 (Spectral theorem for compact self-adjoint operators). Let H be a Hilbert space and A : H → H a nonzero compact self-adjoint operator. Then there exists a (finite or infinite) sequence of eigenvalues {μn }n∈J , J ⊆ ℕ+ , and a corresponding sequence of eigenvectors {un } which form an orthonormal basis of (Ker A)⊥ . Moreover: 1. Ax = ∑n∈J μn (x, un )un for any x ∈ H. 2. Any nonzero eigenvalue μ of A belongs to {μn }, and the dimension of the eigenspace of μ is finite and it is the cardinality of the set {i ∈ ℕ : μi = μ}. 3. |μ1 | = ‖A‖, |μ1 | ≥ |μ2 | ≥ . . . . If the sequence is infinite, then limn→∞ μn = 0. Proof. The proof is based on an iteration of the previous proposition. We know that if μ1 ∈ {mA , MA } with |μ1 | = max{|m|, M} then |μ1 | = ‖A‖ ≠ 0, μ1 is an eigenvalue of A, and there exists a corresponding eigenvector u1 with ‖u1 ‖ = 1. Let us set X1 = H, A1 = A, m1 = mA , M1 = MA , X2 = [u1 ]⊥ = {x ∈ H : (x, u1 ) = 0}, and A2 = A⌉X2 : X2 → H. Since X2 is a closed subspace of H, it is a complete Hilbert space. Moreover, X2 is invariant with respect to A, i. e., A(X2 ) ⊆ X2 . Indeed, if x⊥u1 , then (Ax, u1 ) = (x, Au1 ) = μ1 (x, u1 ) = 0. Hence A2 : X2 → X2 is a compact self-adjoint operator on the Hilbert space X2 . If A2 = 0, we stop and obtain that Ax = μ1 (x, u1 )u1 for any x ∈ H, because any x ∈ H can be written as x = (x, u1 )u1 + y with y ∈ X2 = [u1 ]⊥ , so that Ax = (x, u1 )Au1 + Ay = μ1 (x, u1 )u1 + 0. If instead A2 is a nonzero compact self-adjoint operator in X2 , by the previous Proposition 1.36 we obtain that if we set m2 = mA2 , M2 = MA2 and μ2 ∈ {m2 , M2 } with |μ2 | = max{|m2 |, M2 }, then μ2 is an eigenvalue of A2 . Hence there exists a corresponding eigenvector u2 with ‖u2 ‖ = 1, and it is orthogonal to u1 because u2 ∈ X2 . Note that m1 ≤ m2 ≤ 0 ≤ M2 ≤ M1 , so that |μ2 | ≤ |μ1 |.

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26 | 1 Preliminaries Continuing in this way, we have two alternatives. Either we stop at the nth step with Xn = {u1 , . . . , un−1 }⊥ , i. e., the operator is zero on Xn+1 = {u1 , . . . , un }⊥ , or we construct an infinite sequence of subspaces Xn and operators An with the above properties. In the first case, since any x ∈ H can be written as x = (x, u1 )u1 + ⋅ ⋅ ⋅ + (x, un )un + y with y ∈ Xn+1 , we have that (1) holds. In the second alternative, we obtain infinite sequences of subspaces X1 = H ⊃ X2 ⊃ . . . Xn ⊃ . . . with Xn = {u1 , . . . , un−1 }⊥ , of operators An = An = A⌉Xn : Xn → Xn , of eigenvalues μn with |μ1 | ≥ |μ2 | ≥ . . . and of mutually orthogonal corresponding unit eigenvectors un . Let us show that limn→∞ μn = 0. Since |μn | is nonincreasing, there exists ‖u ‖ u limn→∞ |μn | = ε ≥ 0. Suppose by contradiction that ε > 0. Then ‖ μn ‖ = |μn | ≤ ε1 ,

and the sequence u

un μn

ukn with A( μkn ) converging in H. kn

n

n

is bounded. By the compactness of A, there exists a subsequence u

This is impossible, since A( μkn ) = ukn , and ukn is an orthonormal sequence. So kn

limn→∞ μn = 0 if the sequence is infinite. If x ∈ H and we define yn = x − ∑ni=1 (x, ui )ui , then yn is orthogonal to u1 , . . . , un , so that yn ∈ Xn+1 , and by the Pythagorean theorem, ‖x‖2 = ‖yn ‖2 + ∑ni=1 |x, ui |2 . It follows that ‖yn ‖ ≤ ‖x‖, hence the sequence yn is bounded. Moreover, Ax = Ayn +∑ni=1 μi (x, ui )ui , and (1) is equivalent to show that Ayn → 0 as n → ∞. Since Ayn = An+1 yn we have that ‖Ayn ‖ ≤ ‖An+1 ‖‖yn ‖ ≤ |μn+1 |‖x‖ → 0 as n → ∞. Thus (1) is proved. The sequence of the eigenvectors (ui ) form an orthonormal basis of the subspace (Ker A)⊥ . Indeed, if y ∈ Ker A, then μi (ui , y) = (Aui , y) = (ui , Ay) = 0, so that (ui , y) = 0, and every ui belongs to (Ker A)⊥ . Moreover, for any x ∈ H we have that x−∑i∈J (x, ui )ui ∈ Ker A by (1), since A(x − ∑i∈J (x, ui )ui ) = Ax − ∑i∈J (x, ui )Aui = Ax − ∑i∈J (x, ui )μi ui = 0. It follows that if x ∈ (Ker A)⊥ then x − ∑i∈J (x, ui )ui ∈ Ker A ∩ (Ker A)⊥ = {0}. Therefore, x = ∑i∈J (x, ui )ui and the sequence (ui ) form an orthonormal basis of the closed subspace (Ker A)⊥ . Let μ be a nonzero eigenvalue, and suppose by contradiction that it is different from any μn previously obtained. Then there exists a corresponding eigenvector u ≠ 0 orthogonal to all the eigenvectors un , and Au = μu ≠ 0. On the other hand, Au = ∑j∈J μj (u, uj )uj = 0, which gives a contradiction. A nonzero eigenvalue μ can appear only a finite number of times in the sequence, since lim μn = 0. If it appears p times with p associate mutually orthogonal unit eigenvectors, these eigenvectors generate its eigenspace. Indeed, if there exists a unit eigenvector u corresponding to the eigenvalue μ and orthogonal to the previous ones, it is also orthogonal to all the eigenvectors un obtained in the previous construction, and as before we would obtain that 0 ≠ μu = Au = ∑j∈J (u, uj )uj = 0. Corollary 1.38 (Spectral theorem for positive compact self-adjoint operators). Let H be a Hilbert space and A : H → H a nonzero compact self-adjoint operator which is

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1.3 Compact self-adjoint operators | 27

positive definite, i. e., (Af , f ) > 0

for any f ∈ H, f ≠ 0

Then the previous sequence of eigenvectors {un } form an orthonormal basis of H, and all the eigenvalues are positive and form a nonincreasing sequence μ1 = ‖A‖ ≥ μ2 ≥ . . . . Proof. It is enough to observe that since A is positive definite then Ker(A) = {0}. Let H be an infinite dimensional Hilbert space and A ∈ ℬ(H) a nonzero self-adjoint compact operator. We know that the value 0 belongs to the spectrum of A and it can be an eigenvalue (if A is not injective) or not (in that case A is injective but either it is not surjective or the inverse is not continuous). The previous theorem gives the sequence of nonzero eigenvalues satisfying the properties 1–3. We now show that any other number μ ≠ 0, μ ≠ μk , belongs to the resolvent set ρ(A) = 𝕂 \ σ(A), i. e., there exists the continuous inverse (A − μI)−1 : H → H. Theorem 1.39 (Fredholm alternative). Let A : H → H a nonzero compact self-adjoint operator on the Hilbert space H, and {μn }n∈J⊆ℕ+ , {un }n∈J⊆ℕ+ , the sequences of nonzero eigenvalues and corresponding mutually orthogonal eigenvectors of A. 1. If μ ≠ 0 and μ ≠ μk ∀k ∈ J, then μ ∈ ρ(A) = 𝕂 \ σ(A). More precisely, for any h ∈ H there exists a unique f ∈ H such that Af = μf + h, and f = (A − μI)−1 h =

2.

μn −1 [h + ∑ (h, un )un ] μ μ − μn n∈J

(1.55)

(the series converges and defines a continuous linear operator). If μ = μk for some k ∈ J, then Im(A − μI) = [Ker(A − μI)]⊥ , i. e., the equation (in the unknown f ) Af = μk f + h has a solution if and only if (h, u) = 0 ∀u ∈ Ker(A − μI). If this is the case, then the solutions of the equation are given by f + Ker(A − μI) = {f + u : u ∈ Ker(A − μI)}, with f =

μn −1 [h + ∑ (h, un )un ] μ μ − μn n∈J,μ =μ ̸

(1.56)

n

Proof. 1. Let us first show that given h ∈ H, if μ ≠ μk ∀k ∈ J and there exists f ∈ H such that Af = μf + h, then f is necessarily given by (1.55) where the series converges. In this way, as we show below we obtain the continuous inverse of A − μI.

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28 | 1 Preliminaries If μf + h = Af = ∑n∈J μn (f , un )un , then μf = −h + ∑n∈J μn (f , un )un . Taking the scalar product with ui , i ∈ J, we get (μf , ui ) = −(h, ui ) + μi (f , ui ), so that (μ − μi )(f , ui ) = 1 (−h, ui ). Inserting this value in the previous equality, we −(h, ui ) and (f , ui ) = (μ−μ ) i

1 get that μf = −h + ∑n∈J μn (μ−μ (−h, un )un and (1.55) follows. Let us show that the n) series converges for any h ∈ H and the convergence is uniform with respect to any unit vector h. This will show that (A − μI)−1 is bounded, and inserting the value of f given by (1.55) in the formula (A − μI)f = ∑n∈J μn (f , un )un − μf it is easy to see that one gets h. |μ | If we set α = supk∈J |μ−μk | , then α < ∞ because μ ≠ μk for any k ∈ J and μk → 0 as k k → ∞. If m < n, thanks to the Pythagorean theorem, we have that

󵄩󵄩2 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 μk 󵄩 (h, uk )uk 󵄩󵄩󵄩 ‖sn − sm ‖ = 󵄩󵄩 ∑ 󵄩󵄩 󵄩󵄩 μ − μ k 󵄩 󵄩k=m+1 2 n 󵄨󵄨 ∞ 󵄨 󵄨 μ 󵄨󵄨 󵄨 󵄨2 󵄨 󵄨2 = ∑ 󵄨󵄨󵄨 k 󵄨󵄨󵄨 󵄨󵄨󵄨(h, uk )󵄨󵄨󵄨 ≤ α2 ∑ 󵄨󵄨󵄨(h, uk )󵄨󵄨󵄨 󵄨󵄨 μ − μk 󵄨󵄨 2

k=m+1

k=m+1

2 2 and the last term tends to 0 as m → ∞ because the series ∑∞ k=1 |(h, uk )| ≤ ‖h‖ converges. Moreover,

(A − μI)−1 h = with Bh = 2

2.

1 α |μ|

−1 μ 2

∑n∈J 2

μn μ−μn

μn −1 −1 (h, un )un ] = [h + ∑ Ih + Bh μ μ − μ μ n n∈J

bounded, since an analogous computation shows that

‖h‖ . ‖Bh‖ ≤ Considering the case when μ = μk for some k ∈ J, the previous computations show that the series (1.56) converges and defines a bounded linear operator, because the terms with μn = μk are omitted. Finally, if (A − μk I)f = h and u is an eigenvector corresponding to the eigenvalue μk , then (h, u) = ((A − μk I)f , u) = (f , (A − μk I)u) = 0 so that (h, u) = 0. Vice versa, it is easy to verify that if (h, u) = 0 for any u ∈ Ker(A − μI), then the series defines a vector f such that Af = h, and any other vector g satisfying Ag = h has the form g = f + u : u ∈ Ker(A − μI).

Remark 1.40. The name Fredholm alternative depends on the fact that by the previous theorem the following alternative holds. Either (i) the homogeneous equation Af − μf = 0

(1.57)

admits only the trivial solution f = 0 and then the inhomogeneous equation Af − μf = h

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(1.58)

1.4 Eigenvalues of elliptic problems | 29

has a unique solution for any h ∈ H or (ii) the homogeneous equation (1.57) has nontrivial solutions, and then the nonhomogeneous equation (1.58) has a solution if and only if h ∈ (Ker(A − μI))⊥ .

1.4 Eigenvalues of elliptic problems In this section, we review the classical theory of eigenvalues for boundary value problems involving elliptic operators. We consider linear elliptic mixed boundary value problems in a general bounded Lipschitz domain Ω ⊂ ℝN , N ≥ 2, and we denote by Γ0 , Γ relatively open disjoint subsets of the boundary 𝜕Ω that satisfy (1.28), (1.29). We construct (as in [78]) a sequence of eigenvalues and prove their variational characterization adapting the standard methods used for Dirichlet problems (based on the theory of positive compact self-adjoint operators) by working in the product space V = L2 (Ω) × L2 (Γ). Of course, the case of a Dirichlet boundary problem is a particular case of Dirichlet–Neumann problems when Γ = 0. We only observe that in the case of Dirichlet problems obviously we do not work in the product space L2 (Ω) × L2 (Γ) but instead only in L2 (Ω). As in Section 1.1, we denote by the same symbol a function belonging to H01 (Ω ∪ Γ) and its trace on the boundary. Let us consider the bilinear form in H01 (Ω ∪ Γ) defined by B(u, φ) = ∫(∇u ⋅ ∇φ + cuφ) dx + ∫ du φ dσ

(1.59)

Γ

Ω

with the associated quadratic form Q(u) = B(u, u) = ∫(|∇u|2 + c|u|2 ) dx + ∫ d|u|2 dσ

(1.60)

Γ

Ω

where we assume that c ∈ L∞ (Ω),

d ∈ L∞ (Γ)

(1.61)

Remark 1.41. In the spectral theory that follows, it would be enough to assume the following hypotheses on c, d: c− ∈ L∞ (Ω),

d− ∈ L∞ (Γ)

(1.62)

d ∈ LN−1 (Γ)

(1.63)

and N

c ∈ L 2 (Ω),

if N ≥ 3, while c and d can belong to any Lq space with q > 1, if N = 2. However, in order to make the presentation not too heavy, in the sequel we will consider only the case when c and d are bounded.

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30 | 1 Preliminaries If Λ ≥ 0 let us define, together with (1.59), the bilinear form BΛ (u, φ) = ∫(∇u ⋅ ∇φ + (c + Λ)uφ) dx + ∫ (d + Λ)uφ dσ Ω

(1.64)

𝜕Ω

Since (1.61) holds, B and BΛ are continuous symmetric bilinear forms on H01 (Ω ∪ Γ). Moreover, again by (1.61), a number Λ0 ≥ 0 ∈ ℝ can be chosen in such a way that for any Λ ≥ Λ0 the bilinear form BΛ is coercive in H01 (Ω ∪ Γ), i. e., it gives an equivalent scalar product in H01 (Ω ∪ Γ). Hence we fix any Λ ≥ Λ0 . Let us consider the Hilbert space V = L2 (Ω) × L2 (Γ)

(1.65)

with the scalar product ((f1 , f2 ) ⋅ (g1 , g2 )) = ∫ f1 g1 dx + ∫ f2 g2 dσ

(1.66)

Γ

Ω

We can identify f = (f1 , f2 ) ∈ V with the continuous linear functional φ ∈ H01 (Ω ∪ Γ) 󳨃→ (f1 , φ)L2 (Ω) + (f2 , φ)L2 (Γ)

(1.67)

By the Riesz representation theorem, for any f = (f1 , f2 ) ∈ V there exists a unique u =: T(f ) ∈ H01 (Ω ∪ Γ) such that BΛ (u, φ) = (f , φ)V = (f1 , φ)L2 (Ω) + (f2 , φ)L2 (Γ)

∀φ ∈ H01 (Ω ∪ Γ)

i. e., u = T(f ) is the unique weak solution of the problem −Δu + (c(x) + Λ)u = f1 { { { {u = 0 { { { { 𝜕u + (d(x) + Λ)u = f2 { 𝜕𝜐

in Ω on Γ0

(1.68)

on Γ

The solution u belongs to H01 (Ω ∪ Γ) and ‖u‖H 1 (Ω∪Γ) ≤ C‖f ‖V 0

for some C > 0

If we identify a function u ∈ H01 (Ω∪Γ) with the pair (u, Trace(u)) ∈ H01 (Ω∪Γ)×L2 (Γ) ⊂ V, we can consider T as a continuous linear operator T : V → V defined by f = (f1 , f2 ) 󳨃→ (u, Trace(u)), which maps V into V. We observe that T is compact because of the compact embedding of H01 (Ω∪Γ) in V. Moreover, T is a positive self-adjoint operator. Indeed (recall that BΛ is an equivalent scalar product in H01 (Ω ∪ Γ)) if f = (f1 , f2 ) ≠ 0, and hence u ≠ 0, we have that (T(f ), f )V = (u, f )V = BΛ (u, u) > 0

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1.4 Eigenvalues of elliptic problems | 31

so that T is positive. If T(f ) = u, T(g) = v, i. e., BΛ (u, ϕ) = (f , ϕ)V , BΛ (v, ϕ) = (g, ϕ)V , then (T(f ), g)V = (u, g)V = (g, u)V = BΛ (v, u) = BΛ (u, v) = (f , v)V = (f , T(g))V which means that T is self-adjoint. Thus, by the spectral theory of positive compact self-adjoint operators in Hilbert spaces there exist a nonincreasing sequence {μΛj } of positive eigenvalues with limj→∞ μΛj = 0 and a corresponding sequence {wj } ⊂ H01 (Ω ∪ Γ) of eigenvectors such that T(wj ) = μΛj wj and wj is an orthonormal basis of V. If we set λjΛ = μ1Λ , then wj solve the problem j

−Δwj + (c(x) + Λ)wj = λjΛ wj { { { { {w = 0 j { { { 𝜕w { { j + (d(x) + Λ)w = λΛ w j j j { 𝜕𝜐

in Ω on Γ0

(1.69)

on Γ

Translating, and denoting by λj the differences λj = λjΛ − Λ, we get the existence of a sequence {λj } of eigenvalues, with −∞ < λ1 ≤ λ2 ≤ . . . , limj→+∞ λj = +∞, and of a corresponding sequence of eigenfunctions {wj } that weakly solve the following mixed boundary eigenvalue problem: −Δw + c(x)w = λw { { { {w = 0 { { { { 𝜕w + d(x)w = λw { 𝜕𝜐

in Ω on Γ0

(1.70)

on Γ

Moreover, since c ∈ L∞ (Ω), by standard elliptic regularity theory the eigenfunctions wj belong at least to C 1 (Ω) (and they are far more regular if the coefficients and the domain are smooth). In the next theorem, we show the variational characterization of the eigenvalues λj of (1.70) as well as the main properties of the first eigenfunctions, i. e., the eigenfunctions corresponding to the first eigenvalue. Theorem 1.42. Assume that Ω is a bounded Lipschitz domain in ℝN , N ≥ 2, Γ0 and Γ are relatively open disjoint subsets of the boundary 𝜕Ω that satisfy (1.28), (1.29) and (1.61) holds. Then there exist sequences of eigenvalues {λj }j∈ℕ ⊂ ℝ, with limj→∞ λj = +∞, and eigenfunctions {wj }j∈ℕ ⊂ H01 (Ω ∪ Γ) which satisfy (1.70) and such that the sequence {(wj , Trace(wj ))} is an orthonormal basis of the space V = L2 (Ω) × L2 (Γ). Moreover, defining for v ∈ H01 (Ω ∪ Γ), v ≠ 0, the Rayleigh quotient R(v) =

Q(v) Q(v) = (v, v)L2 (Ω) + (v, v)L2 (Γ) ‖v‖2V

(1.71)

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32 | 1 Preliminaries with Q as in (1.60), and denoting by Hk a k-dimensional subspace of H01 (Ω ∪ Γ), the following properties hold: (i) λ1 =

min

v∈H01 (Ω∪Γ) v=0 ̸

R(v) =

min

v∈H01 (Ω∪Γ) (v,v)V =1

Q(v)

(ii) If m ≥ 2, then λm =

min

R(v) =

v∈H01 (Ω∪Γ) v=0 ̸ v⊥w1 ,...,v⊥wm−1

min

v∈H01 (Ω∪Γ) (v,v)V =1 v⊥w1 ,...,v⊥wm−1

Q(v)

where the orthogonality conditions are meant in the space V. (iii) If m ≥ 2, then λm = min max R(v) = min max Q(v) Hm v∈Hm ‖v‖V =1

Hm v∈Hm v=0 ̸

(iv) If m ≥ 2, then2 λm = max min R(v) = max min Q(v) Hm−1 v⊥Hm−1 v=0 ̸

Hm−1 v⊥Hm−1 ‖v‖V =1

where the orthogonality condition is meant in the space V. (v) If w ∈ H01 (Ω ∪ Γ), w ≠ 0 and R(w) = λ1 , then w is an eigenfunction corresponding to λ1 . (vi) Any first eigenfunction does not change sign in Ω and the first eigenvalue is simple, i. e., up to scalar multiplication there is only one eigenfunction corresponding to the first eigenvalue. Proof. We have already proved the existence of the sequences of eigenvalues and eigenfunctions. Moreover, we observe that the second equality in i)–iv) is a consequence of the homogeneity of R: for any k ∈ ℝ \ {0}, v ∈ H01 (Ω ∪ Γ) \ {0} it holds R(kv) = R(v) = Q( ‖v‖v ). V

As in (1.64), we define RΛ (v) =

BΛ (v,v) (v,v)V

=

BΛ (v,v) (v,v)L2 (Ω) +(v,v)L2 (Γ)

for v ∈ H01 (Ω ∪ Γ) v ≠ 0.

Since RΛ (v) = R(v) + Λ, once the properties are proved for the eigenvalues λjΛ of (1.69) we recover the results for λj by translation. Therefore, for simplicity of notation, we make the proof for Λ = 0, and we assume that B = B0 is coercive, and hence gives an equivalent scalar product in H01 (Ω ∪ Γ). 2 In the sequel, we will refer to this result also when m = 1, meaning that H0 = {0}, and λ1 = minv∈H 1 (Ω∪Γ),v=0̸ R(v), proved in (i). 0

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1.4 Eigenvalues of elliptic problems | 33

(i) The sequence {wj } is an orthonormal basis of V. By definition for every u ∈ H01 (Ω ∪ Γ), we have B(wk , u) = λk (wk , u)V

(1.72)

∀k ∈ ℕ+ 1

In particular, B(wk , wj ) = λk (wk , wj )V ∀k, j ∈ ℕ+ , hence the sequence {(λj )− 2 wj } is an orthonormal basis of H01 (Ω ∪ Γ) with respect to the scalar product given by the bilinear form B. If v ∈ H01 (Ω ∪ Γ) and (v, v)V = 1, using the Fourier expansions in V and in H01 (Ω ∪ Γ) we have v = ∑ βk wk , k∈J

(v, v) = ∑ |βk |2 = 1 k∈J

and +∞

1

v = ∑ αk ((λk )− 2 wk ), k=1

+∞

B(v, v) = ∑ |αk |2 k=1

1

It is easy to see, using that βk = (v, wk )V , αk = B(v, (λk )− 2 wk ) and (1.72) that αk = 1 (λk ) 2 βk . Thus +∞

+∞

k=1

k=1

B(v, v) = ∑ λk βk2 ≥ λ1 ∑ βk2 = λ1 = B(w1 , w1 ) and (i) follows. (ii) If v⊥{w1 , . . . , wm−1 } and (v, v)V = 1, then v = ∑∞ k=m βk wk and as before B(v, v) ≥ λm . Since B(wm , wm ) = λm , the claim (ii) follows. (iii) If dim(Hm ) = m and {v1 , . . . , vm } is a basis of Hm , there exists a linear combination 0 ≠ v = ∑m i=1 αi vi which is orthogonal to all w1 , …wm−1 (m unknown coefficients and (m − 1) equations), so that by (ii) we obtain that maxv∈Hm R(v) ≥ λm . On the other v=0 ̸

hand, if Hm = span(w1 , . . . , wm ) then maxv∈Hm R(v) = λm . Indeed if v = ∑m k=1 βk wk , as in the proof of (i) we have R(v) =

v=0 ̸ 2 ∑m k=1 λk βk m ∑k=1 βk2

≤ λm and R(wm ) = λm . Hence (iii)

follows. (iv) The proof is similar to the previous one. If {v1 , . . . , vm−1 } is a basis of an m − 1-dimensional subspace Hm−1 , there exists a linear combination 0 ≠ w = ∑m i=1 αi wi of the first m eigenfunctions which is orthogonal to Hm−1 , and R(w) ≤ λm . Hence minv⊥Hm−1 R(v) ≤ λm for every (m − 1)-dimensional subspace Hm−1 . On the other v=0 ̸

hand, by taking Hm−1 = span(w1 , . . . , wm−1 ) then minv⊥Hm−1 R(v) = λm by (ii), so that (iv) follows.

v=0 ̸

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34 | 1 Preliminaries (v) By normalizing, we can suppose that (w, w)V = 1. Let v ∈ H01 (Ω ∪ Γ), t > 0. Then by (i) R(w + tv) = B(w+tv,w+tv) ≥ λ1 , i. e., B(w, w) + t 2 B(v, v) + 2tB(w, v) ≥ λ1 [(w, w)V + (w+tv,w+tv) V

t 2 (v, v)V + 2t(w, v)V ] = λ1 + λ1 t 2 (v, v) + 2tλ1 (w, v). Since B(w, w) = λ1 , dividing by t and letting t → 0 we obtain that B(w, v) ≥ λ1 (w, v)V and changing v with −v we deduce that B(w, v) = λ1 (w, v)V for any v ∈ H01 (Ω∪Γ), i. e., w is a first eigenfunction. (vi) Let w1 be a first eigenfunction. Multiplying (1.70) by w1+ and integrating we deduce that B(w1+ , w1+ ) = λ1 (w1+ , w1+ )V , so that by (v) w1+ is a first eigenfunction if it is not identically zero. The same applies to w1− . Suppose now that w1+ ≢ 0. Then it is a first eigenfunction and solves (1.70) and, by the strong maximum principle (Theorem 1.28), w1+ = w1 > 0 in Ω. If w1 , w2 are two eigenfunctions corresponding to λ1 , they do not change sign in Ω, so that they cannot be orthogonal in V. This implies that the first eigenvalue is simple.

Remark 1.43. 1. In statement (iii) of Theorem 1.42, we write max instead of sup since the unit sphere in a finite dimensional space is compact and, therefore, sup v∈Hm Q(v) = ‖v‖V =1

max v∈Hm Q(v). Instead, the fact that ‖v‖V =1

inf max R(v) = min max R(v) Hm v∈Hm v=0 ̸

2.

Hm v∈Hm v=0 ̸

is part of the proof of statement (iii). Let us explain why we write min instead of inf in statement (iv) of Theorem 1.42. If Hm−1 is a (m − 1) dimensional subspace of H01 (Ω ∪ Γ), then S = H⊥ m−1 (where the 2 2 1 orthogonality is in V = L (Ω) × L (Γ)) is a subspace of H0 (Ω ∪ Γ) which is closed in V and, therefore, in H01 (Ω ∪ Γ). So S is a Hilbert space, and thanks to the compact embedding of H01 (Ω ∪ Γ) into V the unit sphere of V ∩ S is weakly closed in S. Moreover, Q(v) (which by the hypothesis made at the beginning of the proof is in fact the square of a norm in H01 (Ω ∪ Γ)) is weakly lower semicontinuous. Therefore, by standard arguments of calculus of variations we have that infv⊥Hm−1 Q(v) ‖v‖V =1

is attained and it is in fact a minimum (note that the same argument can be applied to say that in (i) and (ii) the minimum is achieved, though it has been proved directly). Instead, the fact that sup min R(v) = max min R(v) Hm−1 v⊥Hm−1 v=0 ̸

Hm−1 v⊥Hm−1 v=0 ̸

is part of the proof of statement (iv). Next, we prove some monotonicity and continuity results on the eigenvalues that will be needed in the sequel.

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1.4 Eigenvalues of elliptic problems | 35

For any regular subdomain D of Ω (see Definition 1.24), consider the eigenvalue problem 󸀠 {−Δw + c (x)w = λw { { {w = 0 { { { { 𝜕w + d󸀠 (x)w = λw { 𝜕𝜐

in D on (Γ0 )D = 𝜕D \ ΓD

(1.73)

on ΓD = Γ ∩ 𝜕D

with c󸀠 ∈ L∞ (D), d󸀠 ∈ L∞ (ΓD ). We use the notation λk (−Δ + c󸀠 ;

𝜕 + d󸀠 ; D ∪ ΓD ) 𝜕𝜐

(1.74)

for the eigenvalues of problem (1.73). In particular, when ΓD = 0, we write λk (−Δ + c󸀠 ; D)

(1.75)

for the eigenvalues for the Dirichlet problem, with corresponding eigenfunctions wk in H01 (D) that solve the problem −Δw + c󸀠 (x)w = λw

{

w=0

in D on 𝜕D

We keep the simple notation λk for the eigenvalues of (1.70), namely λk = λk (−Δ + c;

𝜕 + d; Ω ∪ Γ) 𝜕𝜐

Finally, if v ∈ H01 (D ∪ ΓD ) we write Q(u; D ∪ ΓD ) = ∫(|∇u|2 + c󸀠 |u|2 ) dx + ∫ d󸀠 u2 δσ D

(1.76)

ΓD

Q(u; D ∪ ΓD ) Q(u; D ∪ ΓD ) = R(v; D ∪ ΓD ) = (v, v)L2 (Ω) + (v, v)L2 (Γ) ‖v‖2V

(1.77)

for the quadratic form and the Rayleigh quotient corresponding to the choices of D, c󸀠 , d󸀠 . We will write Q(v) and R(v) for the quadratic form and the Rayleigh quotient given by (1.76), (1.77) when D, c󸀠 , d󸀠 are fixed. Let us recall that V = L2 (Ω) × L2 (Γ) and denote by Hk a k-dimensional subspace of H01 (Ω ∪ Γ). Moreover, we denote by X a dense subspace of H01 (Ω ∪ Γ) and by Xk a k-dimensional subspace of X. In particular, we will consider spaces of smooth functions with compact support in Ω ∪ Γ, e. g., X = Cc1 (Ω ∪ Γ) (or X = Cc∞ (Ω ∪ Γ)), which is a dense subspace of H01 (Ω ∪ Γ).

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36 | 1 Preliminaries Theorem 1.44. Assume that Ω is a bounded Lipschitz domain in ℝN , N ≥ 2, Γ0 and Γ are relatively open disjoint subsets of the boundary 𝜕Ω that satisfy (1.28), (1.29) and that (1.61) holds. (i) If X is a dense subspace of the space H01 (Ω ∪ Γ) (in particular if X = Cc1 (Ω ∪ Γ)) and we denote by Xk a k-dimensional subspace of X then for any m ∈ ℕ+ λm = inf max R(v) Xm v∈Xm v=0 ̸

(ii)

(Monotonicity w. r. t. subdomains) If Ω1 ⊂ Ω2 ⊂ Ω are regular subdomains of the domain Ω and Γi = ΓΩi = Γ ∩ Ωi , i = 1, 2, then 1 2 λm ≥ λm

for any m ∈ ℕ+

𝜕 i + d; Ωi ∪ Γi ), i = 1, 2. where λm = λm (−Δ + c; 𝜕𝜐 1 2 Moreover, if Ω ⊂ Ω ⊂ Ω and 𝜕Ω1 ∩ Ω2 ≠ 0 then the strict inequality λ11 > λ12 holds for the first eigenvalue. (iii) (Continuity w. r. t. subdomains) For any m ∈ ℕ+ , it holds

λm = inf{λm (−Δ + c;

𝜕 + d; D ∪ ΓD )} 𝜕𝜐

the infimum being among the regular subdomains D such that D ∪ ΓD is a compact subset of Ω ∪ Γ. (iv) (Monotonicity w. r. t. the coefficients) If c󸀠 ∈ L∞ (Ω), d󸀠 ∈ L∞ (Γ) and c ≥ c󸀠 a. e. in 𝜕 󸀠 Ω, d ≥ d󸀠 a. e. on Γ, then λm ≥ λ󸀠 m for any m ∈ ℕ+ , where λm = λm (−Δ + c󸀠 ; 𝜕𝜐 + 󸀠 d ; Ω ∪ Γ). (v) (Continuity w. r. t. the coefficients) If cn is a sequence in L∞ (Ω), dn is a sequence in n L∞ (Γ) and cn → c in L∞ (Ω), dn → d in L∞ (Γ), then limn→∞ λm = λm ∀m ∈ ℕ+ , n n 𝜕 n where λm = λm (−Δ + c ; 𝜕𝜐 + d ; Ω ∪ Γ) (vi) Let Γ = 0 and set λ1 (D) = λ1 (−Δ + c; D) if D is an open subset of Ω. Then lim

meas(D)→0

λ1 (D) = +∞

(vii) Let Ω be a cylindrically symmetric domain (see Definition 1.15), and let us consider a regular subdomain Ω󸀠 ⊂ Ω (see Definition 1.24). Setting Γ󸀠0 = 𝜕Ω󸀠 ∩ ℝN+ , Γ󸀠 = 𝜕 𝜕Ω󸀠 \ Γ󸀠0 , λ1 (Ω󸀠 ∪ Γ󸀠 ) = λ1 (−Δ + c; 𝜕𝜐 + d; Ω󸀠 ∪ Γ󸀠 ), we have lim

meas(Ω󸀠 )→0

λ1 (Ω󸀠 ∪ Γ󸀠 ) = +∞

In the proof, we will exploit the following simple result. Lemma 1.45. Let X be a normed space, and let us consider for m ∈ ℕ+ a set of linearly independent vectors z1 , . . . , zm . Assume that φni are sequences of vectors converging to zi in X for i = 1, . . . , m. Then the vectors φn1 , . . . , φnm are linearly independent if n is sufficiently large.

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1.4 Eigenvalues of elliptic problems | 37

Proof. Suppose by contradiction that (up to a subsequence) for any n ∈ ℕ there exist tkn n n n n t1n , . . . , tm not all vanishing such that ∑m 1 we have that k=1 tk φk = 0. Setting sk = m n 2 (∑i=1 |ti | ) 2

m n n n 2 n ∑m k=1 sk φk = 0, ∑k=1 |sk | = 1 and the sequences sk are bounded for any k = 1, . . . , m. 2 Then (up to a subsequence) the coefficients snk → sk ∈ ℝ as n → ∞, with ∑m k=1 |sk | = 1, m m n n so that letting n → ∞ we get that 0 = ∑k=1 sk φk → ∑k=1 sk zk , i. e., the vectors z1 , . . . , zm are linearly dependent, which is a contradiction.

Proof of Theorem 1.44. (i) Let us recall that λm = minHm maxv∈Hm R(v) and set v=0 ̸

λ̃m = inf max R(v) Xm v∈Xm v=0 ̸

Obviously, λm ≤ λ̃m , since X ⊂ H01 (Ω ∪ Γ), so we have to show the opposite inequality. Let w1 , . . . , wm the eigenfunctions corresponding to the eigenvalues λk = λk (−Δ + 𝜕 c; 𝜕𝜐 + d; Ω ∪ Γ), k = 1, . . . , m, normalized in V = L2 (Ω) × L2 (Γ), i. e., such that ‖wi ‖V = 1, i = 1, . . . , m. By the density of X in H01 (Ω ∪ Γ) and by Lemma 1.45 there exist sequences φni , i = 1, . . . , m, such that φn1 , . . . , φnm are linearly independent and φni converges to wi in H01 (Ω ∪ Γ), as n → ∞. Let us set X n = span(φn1 , . . . , φnm ), and denote by ψn the point of X n such that maxv∈X n ,v=0̸ R(v) = R(ψn ). Then m

ψn = ∑ snk φnk k=1

for some sn1 , . . . , snm ∈ ℝ. Since the function R(v) is homogeneous of degree zero, without loss of generality and arguing as in Lemma 1.45, we can assume that n 2 n ∑m k=1 |sk | = 1 and the sequences sk are bounded. Hence, up to a subsequence m n sk → sk ∈ ℝ as n → ∞, with ∑k=1 |sk |2 = 1 and setting m

ψ = ∑ sk wk k=1

we have ψn → ψ Therefore, R(ψn ) → R(ψ).

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38 | 1 Preliminaries Since w1 , . . . , wm are the first m eigenfunctions, ‖wk ‖V = 1, k = 1, . . . , m and they are orthogonal in V and H01 (Ω ∪ Γ) as well, we have that m

‖ψ‖2V = ∑ |sk |2 = 1 k=1

m

m

k=1

k=1

R(ψ) = Q(ψ) = ∑ |sk |2 λk ≤ ∑ |sk |2 λm = λm Hence R(ψn ) → R(ψ) ≤ λm , which implies that λ̃m ≤ λm , and we conclude that λm = λ̃m . (ii) Since Cc1 (Ω1 ∪ Γ1 ) ⊂ Cc1 (Ω2 ∪ Γ2 ) any m-dimensional subspace Xm of Cc1 (Ω1 ∪ Γ1 ) is an m-dimensional subspace of Cc1 (Ω2 ∪ Γ2 ), m ≥ 1. Hence the monotonicity of the eigenvalues follows by the variational characterization of the eigenvalues proved in (i). Moreover, if 𝜕Ω1 ∩ Ω2 ≠ 0 and m = 1 then the equality λ11 = λ12 cannot hold. Indeed if λ11 = λ12 then the trivial extension of the first eigenfunction in Ω1 ∪ Γ1 to Ω2 ∪ Γ2 would be a first eigenfunction in Ω2 by Theorem 1.42 (v), vanishing in some point of Ω2 , contradicting the property (vi) stated in the same theorem. (iii) By (i), for any ε > 0 there exists an m-dimensional subspace Xm ⊂ Cc1 (Ω ∪ Γ) such that λm ≤ maxv∈Xm R(v) < λm + ε. v=0 ̸

If we consider a regular subdomain D such that D ∪ ΓD contains the support of m functions which span the subspace Xm , using (ii) we get that λm ≤ λm (−Δ + 𝜕 + d(x); D ∪ ΓD ) ≤ maxv∈Xm v=0̸ R(v) ≤ λm + ε. c(x); 𝜕𝜐 (iv) If Hm−1 is a (m − 1)-dimensional subspace of H01 (Ω ∪ Γ) and v⊥Hm−1 , we have that Q(v) = ∫[|∇v|2 + c|v|2 ] dx + ∫ d|v|2 dσ Γ

Ω 2

≥ ∫[|∇v| + c |v| dx] + ∫ d󸀠 |v|2 = Q󸀠 (v) 󸀠

2

(1.78)

Γ

Ω

Hence, denoting by R󸀠 (v) the Rayleigh quotient with coefficients c󸀠 , d󸀠 , we have that R(v) ≥ R󸀠 (v). By the arbitrarity of v, we get min

v⊥Hm−1 ,v=0 ̸

R(v) ≥

min

v⊥Hm−1 ,v=0 ̸

R󸀠 (v)

󸀠 Then we easily conclude recalling that λm = maxHm−1 minv⊥Hm−1 R(v) (and λm =

maxHm−1 minv⊥Hm−1 R󸀠 (v)) by Theorem 1.42(iv). (v)

v=0 ̸

v=0 ̸

Let Hm−1 be an (m − 1)-dimensional subspace of H01 (Ω ∪ Γ) and v⊥Hm−1 . Denoting n by λm , Qn and Rn the eigenvalues, the quadratic form and the Raylegh quotient

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1.4 Eigenvalues of elliptic problems | 39

associated to the coefficients cn and dn , we have 󵄨 󵄨󵄨 n 2 n 2 n 󵄨󵄨Q(v) − Q (v)󵄨󵄨󵄨 = ∫ |c − c ||v| dx + ∫ |d − d ||v| Γ

Ω

󵄩 󵄩 󵄩 󵄩 ≤ (󵄩󵄩󵄩c − cn 󵄩󵄩󵄩L∞ (Ω) + 󵄩󵄩󵄩d − dn 󵄩󵄩󵄩L∞ (Γ) )(∫ |v|2 + ∫ |v|2 ) Ω

Γ

It follows that 󵄨󵄨 󵄨 󵄩 󵄩 n n󵄩 n󵄩 󵄨󵄨R(v) − R (v)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩c − c 󵄩󵄩󵄩L∞ (Ω) + 󵄩󵄩󵄩d − d 󵄩󵄩󵄩L∞ (Γ) From this inequality, it is not difficult to deduce that 󵄨󵄨 󵄨 󵄨󵄨 min R(v) − min Rn (v)󵄨󵄨󵄨 󵄨󵄨v⊥H 󵄨󵄨 v⊥H m−1 m−1 v=0 ̸

v=0 ̸

󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩c − cn 󵄩󵄩󵄩L∞ (Ω) + 󵄩󵄩󵄩d − dn 󵄩󵄩󵄩L∞ (Γ)

and finally 󵄨󵄨 󵄨󵄨 󵄨󵄨 n󵄨 n 󵄨󵄨λm − λm 󵄨󵄨󵄨 = 󵄨󵄨󵄨max min R(v) − max min R (v)󵄨󵄨󵄨 󵄨 H v⊥H 󵄨 v⊥H H m−1

m−1

m−1

m−1

v=0 ̸

v=0 ̸

󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩c − cn 󵄩󵄩󵄩L∞ (Ω) + 󵄩󵄩󵄩d − dn 󵄩󵄩󵄩L∞ (Γ) → 0 as n → ∞ (vi) If v ∈ H01 (D), since c ∈ L∞ (Ω), there exists C ≥ 0 such that Q(v; D) ≥ ∫ |∇v|2 dx − C ∫ |v|2 dx, D

D

while by Poincaré’ s inequality 2

∫ |v|2 dx ≤ C 󸀠 |D| N ∫ |∇v|2 dx. D

D

It follows that R(v; D) =

2 1 Q(v; D) ∫D |∇v| dx − C → +∞ if |D| → 0. ≥ −C ≥ 2 2 2 󸀠 ∫D |v| ∫D |v| dx C |D| N

(vii) If v ∈ H01 (Ω󸀠 ∪ Γ󸀠 ), since c, d ∈ L∞ there exists C ≥ 0 such that Q(v; Ω󸀠 ∪ Γ󸀠 ) ≥ ∫ |∇v|2 dx − C(∫ |v|2 dx + ∫ |v|2 dσ), Ω󸀠

Ω󸀠

Γ󸀠

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40 | 1 Preliminaries while by Poincaré’s inequalities (Theorem 1.17) we have that there exists a constant C 󸀠 such that 󵄨 󵄨2 󵄨 󵄨1 ∫ |v|2 dx + ∫ |v|2 dσ ≤ C 󸀠 (󵄨󵄨󵄨Ω󸀠 󵄨󵄨󵄨 N + 󵄨󵄨󵄨Ω󸀠 󵄨󵄨󵄨 N ) ∫ |∇v|2 dx.

Ω󸀠

Γ󸀠

Ω󸀠

It follows that R(v; Ω󸀠 ∪ Γ󸀠 ) = ≥

∫Ω󸀠 |∇v|2 dx Q(v; Ω󸀠 ∪ Γ󸀠 ) ≥ −C (∫Ω󸀠 |v|2 dx + ∫Γ󸀠 |v|2 dσ) (∫Ω󸀠 |v|2 dx + ∫Γ󸀠 |v|2 dσ) 2 N

1

1

C 󸀠 (|Ω󸀠 | + |Ω󸀠 | N )

󵄨 󵄨 − C → +∞ if 󵄨󵄨󵄨Ω󸀠 󵄨󵄨󵄨 → 0

In Theorem 1.44, the space X can be any dense subspace of H01 (Ω ∪ Γ), and in particular we consider the space X = Cc1 (Ω ∪ Γ). In some applications (see Chapter 4), when Ω = BR is a ball and we consider the Dirichlet problem (i. e., when Γ = 0), we will need to approximate the eigenvalues in the ball with the eigenvalues in the annulus BR \ B 1 with h ∈ ℕ large. This can be done h using Theorem 1.44 and the following result. Proposition 1.46. Let N ≥ 2 and BR = BR (0) a ball in ℝN centered at the origin, with radius R > 0. Then Cc∞ (BR \ {0}) is dense in H01 (BR ), i. e., for any function v ∈ H01 (BR ) there exists a sequence {wn } ⊂ Cc∞ (BR \ {0}) that converges to v in H 1 (BR ). Proof. Let us denote by H01 (BR \ {0}) the closure of the subspace Cc∞ (BR \ {0}) in H 1 (BR \ {0}) (or in H 1 (BR ) which is the same space). Since by definition Cc∞ (BR ) is dense in H01 (BR ) and Cc∞ (BR \ {0}) is dense in H01 (BR \ {0}), it suffices to prove that for any v ∈ Cc∞ (BR ) there exists a sequence vn ∈ H01 (BR \ {0}) that converges to v in H 1 (BR ). In particular, we will consider functions vn ∈ H 1 (BR ) with compact support in BR \ {0}, therefore belonging to H01 (BR \ {0}) by (ii) of Theorem 1.3. Let us first assume that N ≥ 3, v ∈ Cc∞ (BR ) and consider the functions: 1 0 if 0 ≤ t ≤ 2n { { { 1 gn (t) = {2nt − 1 if 2n < t < n1 , { { if t ≥ n1 {1

t ∈ [0, +∞), n ∈ ℕ+

1 Then gn ∈ C 0,1 ([0, +∞)) satisfies gn (t) = 0 for 0 ≤ t ≤ 2n , gn (t) = 1 for t ≥ n1 , 0 ≤ gn (t) ≤ 󸀠 1 ∀t ∈ [0, +∞) and |gn (t)| ≤ 2n. Let us set vn (x) = gn (|x|)v(x). Then, since the support of vn is a compact subset of BR \ {0}, it follows that vn ∈ H01 (BR \ {0}) with ∇vn = gn (|x|)∇v(x) + ∇gn (|x|)v(x), and we

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1.4 Eigenvalues of elliptic problems |

41

claim that vn → v in H 1 (BR ). Indeed, it is immediate to see that as n → ∞ we have 2

∫ |vn − v|2 dx = ∫ (1 − gn (|x|)) |v|2 dx ≤ ∫ |v|2 dx → 0 BR

BR

B1

n

󵄨2

󵄨 󵄨2 󵄨 ∫ |∇vn − ∇v|2 dx ≤ C(∫ 󵄨󵄨󵄨1 − gn (||x)󵄨󵄨󵄨 |∇v|2 dx + ∫ 󵄨󵄨󵄨gn󸀠 (|x|)󵄨󵄨󵄨 |v|2 dx)

BR

BR

BR

≤ C( ∫ |∇v|2 dx + n2 ∫ |v|2 dx) B1

B1

n

n

The last term converges to 0 as n → ∞ since ∫B |∇v|2 dx → 0 and 1 n

󵄨 󵄨2 ∫ 󵄨󵄨󵄨gn󸀠 (|x|)󵄨󵄨󵄨 |v|2 dx ≤ Cn2 ∫ |v|2 dx

BR

B1

n 2

≤ n2 (measN (B 1 )) N ( ∫ |v|2 ) ∗

n

N−2 N

≤ C( ∫ |v|2 ) ∗

B1

If instead N = 2 and v ∈

1 , 2n

(1.79)

n

we consider the functions

0 { { { { log(t) gn (t) = {2 + { log(n) { { {1 Then gn (t) = 0 if 0 ≤ t ≤ 1 gn󸀠 (t) = t log(n) if n12 < t < n1 .

→0

B1

n

Cc1 (BR ),

N−2 N

1 n2 < n1

if 0 ≤ t ≤ if

1 n2

0 and c ∈ L∞ (BR ). Let us denote, for h ∈ ℕ+ , by Ah = {x ∈ ℝN : h1 < |x| < R} the annulus obtained by removing a small ball around the origin, and let λm = λm (−Δ + c; BR ) be the eigenvalues of the Dirichlet problem in BR . Then λm = inf {λm (−Δ + c; Ah )} = lim λm (−Δ + c; Ah ) h∈ℕ

h→∞

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42 | 1 Preliminaries Proof. By Proposition 1.46, we know that Cc1 (BR \ {0}) is dense in H01 (BR ). Then by (i) of Theorem 1.44, for any ε > 0 there exists an m dimensional subspace Xm ⊂ Cc1 (BR \ {0}) such that maxv∈Xm ,v=0̸ R(v) < λm + ε. If h is large, then Ah will contain all supports of m functions spanning Xm . Hence, using also the monotonicity of the eigenvalues given by (ii) of Theorem 1.44, we get λm ≤ λm (−Δ + c; Ah ) ≤ max R(v) ≤ λm + ε. v∈Xm ,v=0 ̸

By the arbitrarity of ε > 0, it follows that λm = limh→∞ {λm (−Δ + c; Ah )} and since the sequence αh = λm (−Δ + c; Ah ) is nonincreasing, we also get λm = infh∈ℕ {λm (−Δ + c; Ah )}. Remark 1.48. In Proposition 1.46 and Corollary 1.47, we can substitute the ball with any bounded domain Ω and the origin 0 with any interior point. Remark 1.49. If N ≥ 3 adapting the proof of Theorem 1.46 for the two-dimensional case, and using cylindrical coordinates, it is not difficult to show that Cc∞ (BR \ F) is dense in H01 (BR ) if F = {x = (x1 , . . . , xn ) ∈ ℝN : x1 = x2 = 0}. Let us end the section with a result that gives a necessary and sufficient condition for the validity of the maximum principle that depends on the spectrum of the corresponding operator. Recalling Definition 1.23 for an operator satisfying the maximum principle, we have the following. Theorem 1.50. Let Ω be a bounded domain in ℝN , with Γ, Γ0 ⊆ 𝜕Ω satisfying (1.28), (1.29), c ∈ L∞ (Ω), d ∈ L∞ (Γ). 𝜕 The operator L = (−Δ + c; 𝜕𝜐 + d) satisfies the maximum principle in Ω ∪ Γ if and 𝜕 only if λ1 (−Δ + c; 𝜕𝜐 + d; Ω) > 0. Proof. If λ1 (Ω) ≤ 0, then a positive first eigenfunction φ1 satisfies −Δφ1 + c(x)φ1 = λ1 φ1 ≤ 0 { { { {φ = 0 1 { { { { 𝜕φ1 + dφ1 = λ1 φ1 { 𝜕𝜐

in Ω on Γ0

(1.81)

on Γ

and hence the maximum principle cannot hold, otherwise φ1 should be nonpositive. 𝜕 If instead λ1 (−Δ + c; 𝜕𝜐 + d; Ω) > 0, by the variational characterization of the first eigenvalue it holds λ1 = minv∈H 1 (Ω∪Γ),v=0̸ R(v) > 0, so that Q(v) > 0 for any v ≠ 0 in H01 (Ω ∪ Γ). Suppose that v ∈ H 1 (Ω) satisfies −Δv + cv ≤ 0 { { { {v ≤ 0 { { { { 𝜕v + dv ≤ 0 { 𝜕𝜐

in Ω on Γ0 on Γ

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1.5 Systems of elliptic equations | 43

Then v+ ∈ H01 (Ω ∪ Γ) can be used as a test function, and we obtain that Q(v+ ) ≤ 0, which implies v+ ≡ 0 in Ω. Remark 1.51. The maximum principle in small domains (Theorem 1.20) can be seen also as a consequence of Theorem 1.50 and (vi) in Theorem 1.44. Analogously, if Ω is a cylindrically symmetric domain, Theorem 1.25 follows from Theorem 1.50 and (vii) in Theorem 1.44.

1.5 Systems of elliptic equations In this section, we extend some results stated for elliptic equations in previous sections to the case of elliptic linear and semilinear systems. For simplicity, only the case of Dirichlet problems is considered. Let Ω be any bounded domain in ℝN , N ≥ 2, and D a m × m matrix with bounded entries: D = (dij )m i,j=1 ,

dij ∈ L∞ (Ω)

(1.82)

We consider the linear elliptic system −ΔU + D(x)U = F

{

U=0

in Ω

(1.83)

on 𝜕Ω

i. e., −Δu1 + d11 u1 + ⋅ ⋅ ⋅ + d1m um = f1 { { { { { {. . . . . . { {−Δum + dm1 u1 + ⋅ ⋅ ⋅ + dmm um = fm { { { { {u1 = ⋅ ⋅ ⋅ = um = 0

in Ω ...

in Ω on 𝜕Ω

where F = (f1 , . . . , fm ) ∈ (L2 (Ω))m , U = (U1 , . . . , Um ). Definition 1.52. The matrix D or the associated system (1.83) is said to be – cooperative or weakly coupled in Ω if dij ≤ 0 a. e. in Ω, –

whenever i ≠ j

(1.84)

fully coupled in Ω if it is weakly coupled in Ω and the following condition holds: ∀I, J ⊂ {1, . . . , m}, I, J ≠ 0, I ∩ J = 0, I ∪ J = {1, . . . , m} ∃i0 ∈ I, j0 ∈ J : meas({x ∈ Ω : di0 j0 < 0}) > 0

(1.85)

Before going on we fix some notation and definitions.

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44 | 1 Preliminaries – –

Inequalities involving vectors should be understood to hold componentwise, e. g., if Ψ = (ψ1 , . . . , ψm ), Ψ nonnegative means that ψj ≥ 0 for any index j = 1, . . . , m. If m ≥ 2 and 1 ≤ p ≤ ∞, we will consider the Banach spaces m

m

Lp (Ω) = (Lp (Ω)) ,

W1,p (Ω) = (W 1,p (Ω))

If p = 2, we consider in particular the Hilbert spaces m

m

L2 (Ω) = (L2 (Ω)) ,

H1 (Ω) = (H 1 (Ω))

and the space H10 (Ω) = (H01 (Ω))m , the closure in H1 (Ω) of the subspace Cc1 (Ω; ℝm ). If f = (f1 , . . . , fm ), g = (g1 , . . . , gm ), the scalar products are defined by m

m

i=1

i=1 Ω

(f , g)L2 (Ω) = ∑(fi , gi )L2 (Ω) = ∑ ∫ fi gi dx m

m

(f , g)H1 (Ω) = ∑(fi , gi )H 1 (Ω) = ∑ ∫ ∇fi ⋅ ∇gi dx 0

–

0

i=1

i=1 Ω

(1.86)

If U = (u1 , . . . , um ), Ψ = (ψ1 , . . . , ψm ) ∈ H10 (Ω), and the matrix D satisfies (1.82), we set m

∇U ⋅ ∇Ψ = ∑ ∇ui ⋅ ∇ψi

(1.87)

D(x)(U, Ψ) = ∑ dij (x)ui ψj

(1.88)

i=1 m

i,j=1

B(U, Φ) = ∫[∇U ⋅ ∇Φ + D(U, Φ)] dx Ω m

m

i=1

i,j=1

= ∫[∑ ∇ui ⋅ ∇ϕi + ∑ dij ui ϕj ] dx Ω

–

i. e., D(x)(U, Ψ) is the action of the bilinear form associated to the matrix D on the pair (U, Ψ), and B is the bilinear form in H10 (Ω) associated to the operator −Δ + D and to system (1.83). If U = (u1 , . . . , um ) ∈ H1 (Ω), we say that U weakly satisfies U ≤ 0 on 𝜕Ω

–

(1.89)

(U ≥ 0 on 𝜕Ω)

if U + ∈ H10 (Ω) (U − ∈ H10 (Ω)), i. e., if u+i ∈ H01 (Ω) (u−i ∈ H01 (Ω)) for any i = 1, . . . , m. If U = (u1 , . . . , um ) ∈ H1 (Ω) and D satisfies (1.82), we say that U weakly satisfies the inequality − ΔU + D(x)U ≥ 0

in Ω

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(1.90)

1.5 Systems of elliptic equations | 45

if for any i = 1, . . . , m the inequalities m

−Δui + ∑ dij uj ≥ 0 j=1

in Ω

are satisfied in a weak sense, i. e., m

∫(∇ui ⋅ ∇ψ + ∑ dij uj ψ) dx ≥ 0 j=1

Ω

(1.91)

for any ψ ∈ H01 (Ω) with ψ ≥ 0 and any i = 1, . . . , m. This implies that ∫ ∇U ⋅ ∇Ψ + D(x)(U, Ψ) Ω m

m

i=1

i,j=1

= ∫[∑ ∇ui ⋅ ∇ψi + ∑ dij (x)ui ψj ] dx ≥ 0 Ω

(1.92)

for any nonnegative Ψ = (ψ1 , . . . , ψm ) ∈ H10 (Ω). As a matter of fact, (1.92) is equivalent to (1.90), as it is easy to see considering test functions Ψ with all components vanishing except one. It is well known that either condition (1.84) or conditions (1.84) and (1.85) together are needed in the proofs of maximum principles for systems (see [94, 97, 201] and the references therein). In particular, if both are fulfilled the strong maximum principle holds as it is shown in the next theorem. Theorem 1.53 (Strong maximum principle and Hopf’s lemma). Suppose that (1.82) and (1.84) hold and U = (u1 , . . . , um ) ∈ C 1 (Ω; ℝm ) is a weak solution of the inequalities −ΔU + D(x)U ≥ 0 in Ω

and

U ≥ 0 in Ω

Then: 1. for any k ∈ {1, . . . , m} either uk ≡ 0 or uk > 0 in Ω; in the latter case if uk ∈ C 1 (Ω; ℝm ), 𝜕u Ω satisfies the interior sphere condition at P ∈ 𝜕Ω and uk (P) = 0 then 𝜕𝜐k (P) < 0, where 𝜐 is the unit exterior normal vector at P; 2. if in addition (1.85) holds, then the same alternative holds for all k = 1, . . . , m, i. e. either U ≡ 0 in Ω or U > 0 in Ω. In the latter case if U ∈ C 1 (Ω; ℝm ), Ω satisfies the interior sphere condition at P ∈ 𝜕Ω and U(P) = 0 then 𝜕U (P) < 0, where 𝜐 is the unit 𝜕𝜐 exterior normal vector at P.

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46 | 1 Preliminaries Proof. It is enough to apply the scalar strong maximum principle using (1.84) to obtain that each component uk is either identically equal to zero or positive. Moreover, if (1.85) holds, then the same alternative holds for all the components. Indeed, since dij ≤ 0 if i ≠ j and ui ≥ 0, for any equation we have −Δuj + djj uj ≥ ∑ −dij ui ≥ 0 i=j̸

This implies that, for any j = 1, . . . , m, either uj ≡ 0 or uj > 0 and in the latter case by Hopf’s lemma we have the sign of the normal derivative on a point of the boundary where the function uj vanishes. Suppose now that (1.85) holds and U does not vanish identically in Ω and let J ⊂ {1, . . . , m} the set of indexes j such that uj > 0 in Ω. Assume by contradiction that J is a proper subset of {1, . . . , m}, and let I = {1, . . . , m} \ J the set of indexes i such that ui ≡ 0. If i0 , j0 are as in (1.85) then −Δui0 + di0 i0 ui0 ≥ − ∑ −di0 j uj ≥ −di0 ,j0 uj0 ≢ 0 j=i̸ 0

Hence, by the scalar strong maximum principle, ui0 > 0, which is a contradiction. Remark 1.54. Maximum principles, Harnack inequalities and other estimates for systems have been studied in many papers with general conditions and also for elliptic operators not in divergence form (see [201] and the references therein). We consider now the bilinear form and the quadratic form associated to the system (1.83), namely m

m

i=1

i,j=1

B(U, Φ) = ∫[∇U ⋅ ∇Φ + C(U, Φ)] = ∫[∑ ∇ui ⋅ ∇ϕi + ∑ cij ui ϕj ] Ω

Ω

(1.93)

for U = (u1 , . . . , um ), Φ = (ϕ1 , . . . , ϕm ) ∈ H10 (Ω) and Q(Ψ) = B(Ψ, Ψ) = ∫(|∇Ψ|2 + D(x)(Ψ, Ψ))dx Ω m

m

i=1

i,j=1

= ∫(∑ |∇ψi |2 + ∑ dij (x)ψi ψj ) dx Ω

(1.94)

for Ψ = (ψ1 , . . . , ψm ) ∈ H10 (Ω). Sometimes we will also write Q(Ψ; Ω) instead of Q(Ψ) specifying the domain. It is easy to see that this quadratic form coincides with the quadratic form associated to the symmetric system −ΔU + C(x)U = F

{

U=0

in Ω on 𝜕Ω

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(1.95)

1.5 Systems of elliptic equations | 47

i. e., −Δu1 + c11 u1 + ⋅ ⋅ ⋅ + c1m um { { { ...... { { { {−Δum + cm1 u1 + ⋅ ⋅ ⋅ + cmm um

= f1 ...

= fm

where C = 21 (D + Dt ) and Dt is the transpose of D, i. e., C = (cij ),

1 cij = (dij + dji ) 2

(1.96)

So, to study the sign of the quadratic form Q, we can also use the properties of the symmetric system and this will be crucial when studying the Morse index of a solution of a semilinear system in Chapter 7. Therefore, we review briefly the spectral theory for these kinds of symmetric systems, and use it to prove some results that we need for the possible nonsymmetric system (1.83). Remark 1.55. If system (1.83) is cooperative, respectively fully coupled, so is the associate symmetric system (1.95).

1.5.1 Spectral theory for symmetric systems Let Ω be a bounded domain in ℝN , N ≥ 2, and let C = C(x) = (cij (x))m i,j=1 be a symmetric matrix whose elements are bounded functions: cij ∈ L∞ (Ω),

cij = cji

a. e. in Ω

(1.97)

Let us consider the bilinear forms m

m

i=1

i,j=1

B(U, Φ) = ∫[∇U ⋅ ∇Φ + C(U, Φ)] = ∫[∑ ∇ui ⋅ ∇ϕi + ∑ cij ui ϕj ] Ω

Ω

(1.98)

and, for Λ > 0 BΛ (U, Φ) = ∫[∇U ⋅ ∇Φ + (C + ΛI)(U, Φ)] Ω m

m

i=1

i,j=1

= ∫[∑(∇ui ⋅ ∇ϕi + Λui ϕi ) + ∑ cij ui ϕj ] Ω

(1.99)

Since cij ∈ L∞ and cij = cji , B and BΛ are continuous symmetric bilinear forms, and, since | ∫Ω cij ui ϕj | ≤ C ∫Ω (u2i + ϕ2j ), there exists Λ ≥ 0 such that BΛ is coercive in H10 (Ω), i. e., it is an equivalent scalar product in H10 (Ω).

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48 | 1 Preliminaries By the Riesz representation theorem, identifying F ∈ L2 (Ω) with the linear functional U ∈ H10 (Ω) 󳨃→ (U, F)L2 (Ω) , for any F ∈ L2 there exists a unique U =: T(F) ∈ H10 (Ω)

such that BΛ (U, Φ) = (F, U)L2 (Ω) for any Φ ∈ H10 (Ω) and ‖U‖H1 (Ω) ≤ C‖F‖L2 (Ω) for some 0 constant C > 0. In other words, U is the unique weak solution of the system −ΔU + (C(x) + ΛI)U = F

in Ω

{

U=0

(1.100)

on 𝜕Ω

i. e., −Δu1 + (c11 + Λ)u1 + ⋅ ⋅ ⋅ + c1m um = f1 { { { { { {. . . { { −Δum + cm1 u1 + ⋅ ⋅ ⋅ + (cmm + Λ)um = fm { { { { {u1 = ⋅ ⋅ ⋅ = um = 0

in Ω ...

in Ω

on 𝜕Ω

Moreover, T : F 󳨃→ U, maps L2 (Ω) into L2 (Ω) and is compact because of the compact embedding of H10 (Ω) in L2 (Ω). It is also a positive operator, since (T(F), F)L2 (Ω) = (U, F)L2 (Ω) = BΛ (U, U) > 0 if F ≠ 0 which implies U ≠ 0 (recall that BΛ is an equivalent scalar product in H10 (Ω)), and since C is symmetric, it is also self-adjoint. Indeed, if T(F) = U, T(G) = V, i. e., BΛ (U, Φ) = (F, U)L2 (Ω) , BΛ (V, Φ) = (G, V)L2 (Ω) for any Φ ∈ H10 (Ω), then (T(F), G)L2 (Ω) = (U, G)L2 (Ω) = (G, U)L2 (Ω) = BΛ (V, U) =

BΛ (U, V) = (F, V)L2 (Ω) = (F, T(G))L2 (Ω) . Thus, by the spectral theory of positive compact self-adjoint operators in Hilbert spaces there exists a nonincreasing sequence {μΛj } of eigenvalues with limj→∞ μΛj = 0 and a corresponding sequence {W j } ⊂ H10 (Ω) of eigenvectors such that G(W j ) = μΛj W j . Setting λjΛ =

1 , then W j solves the system −ΔW j +(C+ΛI)W j = λjΛ W j and W j = 0 on 𝜕Ω. μΛj and denoting by λj the differences λj = λjΛ − Λ, we conclude that there

Translating, exists a sequence {λj } of eigenvalues, with −∞ < λ1 ≤ λ2 ≤ . . . , limj→+∞ λj = +∞, and a corresponding sequence of eigenfunctions {W j } which weakly solve the systems −ΔW j + CW j = λj W j { j W =0

in Ω on 𝜕Ω

(1.101)

i. e., if W j = (w1 , . . . , wm ) −Δw1 + c11 w1 + ⋅ ⋅ ⋅ + c1m wm { { { ...... { { { {−Δwm + cm1 w1 + ⋅ ⋅ ⋅ + cmm wm

= λj w1 ...

= λj wm

Moreover, by (scalar) elliptic regularity theory applied iteratively to each equation, the eigenfunctions W j belong at least to C 1 (Ω; ℝm ). We now collect the variational formulation and some properties of eigenvalues and eigenfunctions.

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1.5 Systems of elliptic equations | 49

Theorem 1.56. Let Ω be a bounded domain in ℝN , N ≥ 2. Suppose that C = (cij )m i,j=1

satisfies (1.97), and let {λj }, {W j } be the sequences of eigenvalues and eigenfunctions satisfying (1.101). Define the Rayleigh quotient R(V) =

Q(V) (V, V)L2 (Ω)

for V ∈ H10 (Ω),

V ≠ 0

(1.102)

with Q(V) = B(V, V) and B as in (1.98). Then the following properties hold, where Hk denotes a k-dimensional subspace of H10 (Ω) and the orthogonality conditions V⊥W k or V⊥Hk stand for the orthogonality in L2 (Ω). (i) λ1 = min R(V) = min Q(V) V∈H10 (Ω) V =0 ̸

V∈H10 (Ω) (V,V)L2 =1

(ii) If k ≥ 2, then λk =

min

V∈H10 (Ω) V =0 ̸ V⊥W 1 ,...,V⊥W k−1

R(V) =

min

V∈H10 (Ω) (V,V)L2 (Ω) =1

Q(V)

V⊥W 1 ,...,V⊥W k−1

(iii) If k ≥ 2, then λk = min max R(V) Hk V∈Hk V =0 ̸

(iv) If k ≥ 2, then λk = max min R(V) Hk−1 V⊥Hk−1 V =0 ̸

(v) If W ∈ H10 (Ω), W ≠ 0 and R(W) = λ1 , then W is an eigenfunction corresponding to λ1 . (vi) If the system is fully coupled in Ω, then the first eigenfunction does not change sign in Ω and the first eigenvalue is simple, i. e., up to scalar multiplication there is only one eigenfunction corresponding to the first eigenvalue. Proof. The proofs of (i), . . . , (v) do not depend on cooperativeness and they are exactly the same as in Theorem 1.42, so we omit them. To prove (vi), let us first show that if the system is cooperative and W is a first eigenfunction, then W + and W − are eigenfunctions, if they do not vanish. + We multiply (1.101) by W + = (w1+ , . . . , wm ) and integrate.

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50 | 1 Preliminaries If in the ith equation, multiplied by wi+ we write wj = wj+ − wj− for j ≠ i and recall that by cooperativeness −cij wj− wi+ ≥ 0, we deduce that B(W + , W + ) ≤ λ1 (W + , W + ) so that, by (v), W + is a first eigenfunction if it does not vanish. The same argument applies to W − . If the system is also fully coupled, then the assertion (vi) follows from the strong maximum principle, which is valid under the fully coupling hypothesis. In fact if, e. g., W + does not vanish, it is a nonnegative first eigenfunction and by the strong maximum principle (Theorem 1.53) it is strictly positive in Ω, i. e., W − ≡ 0 and W > 0 in Ω if W it is positive somewhere. If W 1 , W 2 are two eigenfunctions corresponding to λ1 , they do not change sign in Ω, so that they cannot be orthogonal in L2 (Ω). This implies that the first eigenvalue is simple. When Ω󸀠 is a subdomain of Ω and C 󸀠 = C 󸀠 (x) = (cij󸀠 (x))m i,j=1 satisfies the analogous of (1.97) in Ω󸀠 (i. e., cij󸀠 = cji󸀠 and they are bounded), we will denote by λk (−Δ + C 󸀠 ; Ω󸀠 )

(1.103)

the eigenvalues of the Dirichlet problem, with corresponding eigenfunctions W k in H01 (Ω󸀠 ) which solve the problem −Δw + C 󸀠 (x)w = λW

{

W =0

in Ω󸀠 on 𝜕Ω󸀠

We keep the simple notation λk for the eigenvalues λk = λk (−Δ + C; Ω). Properties analogous to those of Theorem 1.44 also hold for systems. Theorem 1.57. Assume that Ω is a bounded domain in ℝN , N ≥ 2 and C = (cij )m i,j=1 satisfies (1.97). (i) If X is a dense subspace of the space H10 (Ω) (in particular, if X = (Cc1 (Ω))m ) and we denote by Xk a k-dimensional subspace of X then λk = inf max R(v) Xk v∈Xk v=0 ̸

(ii) (Monotonicity w. r. t. subdomains) If Ω1 ⊂ Ω2 ⊂ Ω are subdomains of the domain Ω, then λk (−Δ + C; Ω1 ) ≥ λk (−Δ + C; Ω2 ) Moreover, if Ω1 ⊂ Ω2 ⊂ Ω and 𝜕Ω1 ∩ Ω2 ≠ 0, then the strict inequality λ1 (−Δ + C; Ω1 ) > λ1 (−Δ + C; Ω2 ) holds (i. e., only for k = 1).

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1.5 Systems of elliptic equations | 51

(iii) (Continuity w. r. t. subdomains) For any k ∈ ℕ+ , it holds λk = inf{λk (−Δ + C; Ω󸀠 )} where the infimum is taken among the subdomains Ω󸀠 such that Ω󸀠 is a compact subset of Ω. (iv) (Monotonicity w. r. t. the coefficients) a) Let C 󸀠 (x) be a symmetric matrix in L∞ (Ω), and assume that C ≥ C 󸀠 , i. e., the symmetric matrix C(x) − C 󸀠 (x) is positive semidefinite a. e. in Ω or equivalently m

m

i,j=1

i,j=1

∀ξ ∈ ℝm , a. e. in Ω

∑ cij ξi ξj ≥ ∑ cij󸀠 ξi ξj

Then, setting λk󸀠 = λk (−Δ + C 󸀠 ; Ω), for any k ≥ 1, we have λk ≥ λ 󸀠 k

(1.104)

b) Assume that the system associated to the matrix C is fully coupled and let C 󸀠󸀠 (x) be a symmetric matrix in L∞ (Ω) such that cij ≥ cij󸀠󸀠 a. e. in Ω, for any i, j ∈ {1, . . . , m}. Then, setting λ1󸀠󸀠 = λ1 (−Δ + C 󸀠󸀠 ; Ω), the following inequality holds for the first eigenvalues: λ1 ≥ λ1󸀠󸀠

(1.105)

(v) (Continuity w. r. t. the coefficients) If C n is a sequence of symmetric matrices in L∞ (Ω), and C n → C in L∞ (Ω), then the corresponding eigenvalues converge, i. e., for any k ∈ ℕ+ we have λk (−Δ + C n ; Ω) → λk (−Δ + C; Ω) (vi) Let us consider an open subset Ω󸀠 ⊂ Ω and set λ1 (Ω󸀠 ) = λ1 (−Δ + C; Ω󸀠 ). Then lim

meas(Ω󸀠 )→0

λ1 (Ω󸀠 ) = +∞

Proof. The proofs of all the claims, except for part b) of claim (iv), which depends on the fully coupling of the system, are exactly the same as those in Theorem 1.44 and we omit them. Note that the analogous of claim (iv) in Theorem 1.44 is just part a) of claim (iv) of Theorem 1.57, which can be proved in the same way. To prove part b) of claim (iv), let us observe that if W 1 = (w1 , . . . , wm ) is a first eigenfunction for the system (1.101), then it does not change sign in Ω, since the system is fully coupled. By normalizing it, we can assume that (W 1 , W 1 )L2 (Ω) = 1 and W 1 > 0 in Ω. Denoting by Q󸀠󸀠 the bilinear form corresponding to the matrix C 󸀠󸀠 , since wi wj ≥ 0 and cij ≥ cij󸀠󸀠 we get that m

m

i=1

i,j=1

λ1 = Q(W 1 ) = ∫[∑ |∇wi |2 + ∑ cij wi wj ] dx Ω

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52 | 1 Preliminaries m

m

i=1

i,j=1

≥ ∫[∑ |∇wi |2 + ∑ cij󸀠󸀠 wi wj ] dx = Q󸀠󸀠 (W 1 ) ≥ λ1󸀠󸀠 Ω

(1.106)

Remark 1.58. Note that (1.105) only holds for the first eigenvalue, unlike (1.104) which requires the ordering of the matrices and not of the single entries. 1.5.2 Weak maximum principle for cooperative systems Let us turn back to the (possibly) nonsymmetric cooperative system (1.83): −ΔU + D(x)U = F { U=0

in Ω on 𝜕Ω

and assume that the matrix D = (dij )m i,j=1 satisfies dij ∈ L∞ (Ω),

dij ≤ 0,

whenever i ≠ j

(1.107)

Then we consider the symmetric system (1.95) associated to (1.83) and we denote by λj(s) = λj(s) (−Δ+D; Ω) the eigenvalues of the corresponding linear operator −Δ+C (where C = 21 (D + Dt ) i. e. cij =

1 2 The eigenvalues λj(s)

(dij + dji )) and by Wj(s) the corresponding eigenfunctions.

will be called symmetric eigenvalues of the (nonsymmetric)

operator −Δ + D. The bilinear form corresponding to the symmetric operator will be denoted by Bs (U, Φ), i. e., if U, Φ ∈ H10 (Ω) we set m

m

i=1

i,j=1

Bs (U, Φ) = ∫[∇U ⋅ ∇Φ + C(U, Φ)] = ∫[∑ ∇ui ⋅ ∇ϕi + ∑ cij ui ϕj ] Ω

Ω

As already remarked, the quadratic form (1.94) associated to the system (1.83) coincides with that associated to the system (1.95), i. e., Q(Ψ; Ω) = ∫ |∇Ψ|2 + D(x)(Ψ, Ψ) = ∫ |∇Ψ|2 + C(x)(Ψ, Ψ) = Bs (Ψ, Ψ) Ω

Ω

for Ψ ∈ H01 (Ω; ℝm ). Definition 1.59. We say that the maximum principle holds for the operator −Δ + D in an open set Ω󸀠 ⊆ Ω if for any U ∈ H1 (Ω󸀠 ) such that U ≤ 0 on 𝜕Ω󸀠 (i. e., U + ∈ H10 (Ω󸀠 )) and −ΔU +D(x)U ≤ 0 in Ω󸀠 (i. e., ∫ ∇U ⋅∇Φ+D(x)(U, Φ) ≤ 0 for any nonnegative Φ ∈ H10 (Ω󸀠 )) it holds that U ≤ 0 a. e. in Ω. Let us denote by λj(s) (Ω󸀠 ), j ∈ ℕ+ , the sequence of the eigenvalues of the symmetric

system (1.95) in an open set Ω󸀠 ⊆ Ω.

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1.5 Systems of elliptic equations | 53

Theorem 1.60 (Sufficient condition for weak maximum principle). Under the hypothesis (1.107), if λ1(s) (Ω󸀠 ) > 0, then the maximum principle holds for the operator −Δ + D in Ω󸀠 ⊆ Ω. Proof. By the variational characterization of the first eigenvalue, we have λ1(s) = min R(V) > 0 V∈H1 (Ω󸀠 ) V =0 ̸

so that Q(V) = Bs (V, V) > 0 for any V ≠ 0 in H10 (Ω󸀠 ). Assume that U ≤ 0 on 𝜕Ω󸀠 and −ΔU + D(x)U ≤ 0 in Ω󸀠 . Then, testing the equation with U + = (u+1 , . . . , u+m ), writing in the ith equation uj = u+j − u−j for i ≠ j, and recalling that −cij u+i u−j ≥ 0 if i ≠ j, we obtain that Bs (U + , U + ) ≤ 0, which implies U + ≡ 0 in Ω󸀠 . As an almost immediate consequence we get the following “classical” and “small measure” forms of the weak maximum principle (see [54, 97, 193, 201]). Theorem 1.61. The following statements hold: (i) If (1.84) holds and D is a. e. nonnegative definite in Ω󸀠 then the maximum principle holds for −Δ + D in Ω󸀠 . (ii) There exists δ > 0, depending on D, such that for any subdomain Ω󸀠 ⊆ Ω the maximum principle holds for −Δ + D in Ω󸀠 ⊆ Ω provided |Ω󸀠 | ≤ δ. Proof. (i) If the matrix D is nonnegative definite, then Q(Ψ) = Bs (Ψ, Ψ) ≥ ∫ |∇Ψ|2 > 0 Ω

for any Ψ ∈ H10 (Ω󸀠 ) \ {0}.

Hence the first symmetric eigenvalue is positive, and by Theorem 1.60 we get (i). (ii) It is a consequence of Theorem 1.60 and the property (vi) in Theorem 1.57. Remark 1.62. Obviously, the converse of Theorem 1.60 holds if D = C, i. e., if D is symmetric: if the maximum principle holds for −Δ + C in Ω󸀠 then λ1(s) (Ω󸀠 ) > 0. Indeed if λ1(s) (Ω󸀠 ) ≤ 0, since the system is cooperative (and symmetric), there exists a corresponding nontrivial nonnegative first eigenfunction Φ1 ≥ 0, Φ ≢ 0, and the maximum principle does not hold, since −ΔΦ1 + CΦ1 = λ1 Φ1 ≤ 0 in Ω󸀠 , Φ1 = 0 on 𝜕Ω󸀠 , while Φ1 ≥ 0 and Φ1 ≠ 0. However, the converse of Theorem 1.60 is not true for general nonsymmetric systems. Roughly speaking, the reason is that there is an equivalence between the validity of the maximum principle for the operator −Δ + D and the positivity of another eigenvalue, the principal eigenvalue λ1̃ , whose definition we recall below, and the inequality λ1̃ (Ω󸀠 ) ≥ λ1(s) (Ω󸀠 ), which can be strict, holds.

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54 | 1 Preliminaries Definition 1.63. The principal eigenvalue of the operator −Δ+D in an open set Ω󸀠 ⊆ Ω is defined as 2,N λ1̃ (Ω󸀠 ) = sup{λ ∈ ℝ : ∃Ψ ∈ Wloc (Ω󸀠 ; ℝm ) s. t.

Ψ > 0, −ΔΨ + D(x)Ψ − λΨ ≥ 0 in Ω󸀠 }

(1.108)

Let us recall some of the properties of the principal eigenvalue (see [54] and the references therein, and also [32] for the case of scalar nonlinear equations for a discussion and a more general framework). Theorem 1.64. Assume that the system (1.83) is fully coupled in an open set Ω󸀠 ⊆ Ω. Then: 2,N (i) there exists a positive eigenfunction Ψ1 ∈ Wloc (Ω󸀠 ; ℝm ) which satisfies −ΔΨ1 + D(x)Ψ1 = λ1̃ (Ω󸀠 )Ψ1 { { { Ψ1 > 0 { { { {Ψ1 = 0

in Ω󸀠

(1.109)

in Ω󸀠 on 𝜕Ω

󸀠

Moreover, the principal eigenvalue is simple, i. e., any function that satisfy (1.109) must be a multiple of Ψ1 (ii) the maximum principle holds for the operator −Δ + D in Ω󸀠 if and only if λ1̃ (Ω󸀠 ) > 0 2,N (Ω󸀠 ; ℝm ) such that Ψ > 0 and −ΔΨ + D(x)Ψ ≥ 0 in Ω󸀠 , then (iii) if there exists Ψ ∈ Wloc 󸀠 either λ1̃ (Ω ) > 0 or λ1̃ (Ω󸀠 ) = 0 and Ψ = cΨ1 for some c > 0 (iv) λ1̃ (Ω󸀠 ) ≥ λ1(s) (Ω󸀠 ), with equality if and only if Ψ1 is also the first eigenfunction of the symmetric operator −Δ + C in Ω󸀠 , C = 21 (D + Dt ). If this is the case, the equality C(x)Ψ1 = D(x)Ψ1 holds and, if m = 2, this implies that d12 = d21 . Proof. We refer to [54] for the proofs of (i)–(iii). For what concerns (iv), we observe that testing (1.109) with Ψ1 and recalling that the quadratic form associated to the operator −Δ+D coincides with the quadratic form associated to the symmetric operator −Δ + C, C = 21 (D + Dt ), we obtain that λ1̃ (Ω󸀠 ) =

Q(Ψ1 ) ≥ λ1(s) (Ω󸀠 ) (Ψ1 , Ψ1 )L2 (Ω)

with equality if and only if Ψ1 is the first symmetric eigenfunction, by Proposition 1.56(v). If this is the case, since Ψ1 satisfies the system (1.109) and the system (1.83), the equality D(x)Ψ1 = C(x)Ψ1 follows. Since Ψ1 is positive, if m = 2 we deduce that d12 = d21 .

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1.5 Systems of elliptic equations | 55

1.5.3 Comparison principles for semilinear elliptic systems We discuss now some weak and strong comparison principles for semilinear elliptic systems, analogous to those of Theorem 1.21 and Theorem 1.31 holding for equations. Let us consider a semilinear elliptic system of the type −ΔU = F(x, U) in Ω

{

U=0

on 𝜕Ω

(1.110)

i. e. −Δu1 = f1 (x, u1 , . . . , um ) in Ω { { { { { {. . . . . . ... { {−Δum = fm (x, u1 , . . . , um ) in Ω { { { { on 𝜕Ω {u1 = ⋅ ⋅ ⋅ = um = 0 for the unknown vector valued function U = (u1 , . . . , um ) : Ω → ℝm , where Ω is a bounded domain in ℝN and F = (f1 , . . . , fm ) : Ω × ℝm → ℝm is a C 1 function. A weak solution of (1.110) is a function U ∈ H10 (Ω) such that the function x 󳨃→ 2N 2N F(x, U(x)) belongs to Lq (Ω), with q > 1 if N = 2, q = N+2 if N ≥ 3 (note that N+2 is the 2N ∗ conjugate exponent of the critical Sobolev exponent 2 = N−2 ) and (see the notation about systems at the beginning of the section) ∫ ∇U ⋅ ∇Φ dx = ∫ F(x, U) ⋅ Φ dx Ω

(1.111)

Ω

for any Φ ∈ H10 (Ω). If U, V ∈ H1 (Ω), we write U ≤ V on 𝜕Ω, if the difference U − V weakly satisfies the inequality U − V ≤ 0 on 𝜕Ω, i. e., if (U − V)+ ∈ H10 (Ω). Moreover, we say that U satisfies in a weak sense the inequality − ΔU ≥ (≤) F(x, U) in Ω

(1.112)

if for any i = 1, . . . , m the component ui of U weakly satisfies −Δui ≥ (≤) fi (x, U)

in Ω,

i. e.,

∫ ∇ui ⋅ ∇φ dx ≥ (≤) ∫ fi (x, U)φ dx Ω

(1.113)

Ω

for any φ ∈ H01 (Ω) with φ ≥ 0 in Ω. This is equivalent to require that ∫ ∇U ⋅ ∇Φ dx ≥ (≤) ∫ F(x, U) ⋅ Φ dx Ω

(1.114)

Ω

for any Φ ∈ H10 (Ω) with Φ ≥ 0 in Ω.

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56 | 1 Preliminaries Definition 1.65. We say that the system (1.110) is – cooperative or weakly coupled in an open set Ω󸀠 ⊆ Ω if 𝜕fi (x, s1 , . . . , sm ) ≥ 0 𝜕sj –

for every (x, s1 , . . . , sm ) ∈ Ω󸀠 × ℝm

(1.115)

and every i, j = 1, . . . , m with i ≠ j. fully coupled in an open set Ω󸀠 ⊆ Ω along U ∈ H10 (Ω)∩C 0 (Ω; ℝm ) if it is cooperative in Ω󸀠 and in addition ∀I, J ⊂ {1, . . . , m} such that I ≠ 0, J ≠ 0, I ∩ J = 0, I ∪ J = {1, . . . , m} there exist i0 ∈ I, j0 ∈ J such that meas({x ∈ Ω󸀠 :

𝜕fi0 𝜕sj0

(x, U(x)) > 0}) > 0

(1.116)

Theorem 1.66 (Weak comparison principle in small domains for systems). Let Ω be a domain in ℝN , F : Ω × ℝm → ℝm a C 1 function and assume that (1.115) holds. Let A > 0 and U, V ∈ H1 (Ω) ∩ L∞ (Ω) such that ‖U‖L∞ (Ω) ≤ A, ‖V‖L∞ (Ω) ≤ A Then there exists δ > 0, depending on F and A such that the following holds: if Ω󸀠 ⊆ Ω is a bounded subdomain of Ω, measN ([u > v] ∩ Ω󸀠 ) < δ and −ΔU ≤ F(x, U), −ΔV ≥ F(x, V) in Ω󸀠 { U≤V on 𝜕Ω󸀠

(1.117)

then U ≤ V in Ω󸀠 . Proof. Assume that Ω󸀠 ⊆ Ω and U, V satisfy (1.117). Then, for any i = 1, . . . , m: fi (x, U(x)) − fi (x, V(x)) m 1

= ∑∫ j=1 0

𝜕fi (|x|, tU(x) + (1 − t)V(x)) dt(uj (x) − vj (x)) 𝜕sj

Let us define the matrix B(x) = (bij (x))m i,j=1 , where 1

bij (x) = − ∫ 0

𝜕fi [x, tU(x) + (1 − t)V(x)] dt 𝜕sj

(1.118)

Then we have that the function W = U − V = (w1 , . . . , wm ) satisfies −ΔW + B(x)W ≤ 0

{

W ≤0

in Ω󸀠 on 𝜕Ω󸀠

(1.119)

and if i ≠ j then bij (x) ≤ 0 by (1.115). Hence the result is a consequence of Theorem 1.61.

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1.5 Systems of elliptic equations | 57

Theorem 1.67 (Strong comparison principle for systems). Let Ω be a (bounded or unbounded) domain in ℝN , and let U, V ∈ C 1 (Ω) weakly satisfy −ΔU ≤ F(x, U); −Δv ≥ F(x, V) in Ω { U≤V in Ω

(1.120)

where F(x, U) : Ω × ℝm → ℝm is a C 1 function and (1.115) holds. 1. For every i ∈ {1, . . . , m} the following holds: either ui ≡ vi in Ω or ui < vi in Ω and, in the latter case, if ui , vi ∈ C 1 (Ω), ui (x0 ) = vi (x0 ) at a point x0 ∈ 𝜕Ω where the 𝜕v 𝜕u interior sphere condition is satisfied then 𝜕si (x0 ) < 𝜕si (x0 ) for any inward directional derivative. 2. If moreover U ∈ C 1 (Ω; ℝm ) is a solution of (1.110) and the system is fully coupled along U in Ω (i. e., also (1.116) with Ω󸀠 = Ω holds) then either U ≡ V in Ω or U < V in Ω (i. e., the same alternative holds for any component ui ). In the latter case, assume that U, V ∈ C 1 (Ω) and let x0 ∈ 𝜕Ω a point where U(x0 ) = (x ) < 𝜕V (x ) for any V(x0 ) and the interior sphere condition is satisfied. Then 𝜕U 𝜕s 0 𝜕s 0 inward directional derivative. Proof. The function W = U − V = (w1 , . . . , wm ) satisfies −ΔW + B(x)W ≤ 0

{

W ≤0

in Ω on 𝜕Ω

(1.121)

where 1

bij (x) = − ∫ 0

𝜕fi [x, tU(x) + (1 − t)V(x)] dt 𝜕sj

as in the previous theorem. Moreover, the linear system associated to the matrix B is cooperative because if i ≠ j then bij (x) ≤ 0 by (1.115). If U ∈ C 1 (Ω; ℝm ) is a solution of (1.110) and the system is fully coupled along U, then the linear system associated to the matrix B is fully coupled as well. Indeed, if x ∈ Ω, i0 ≠ j0 and that bi0 j0 (x) =

𝜕fi0 𝜕fi (x, U(x)) > 0 then since 𝜕s 0 (x, S) 𝜕sj0 j0 1 𝜕fi − ∫0 𝜕s 0 [x, tU(x) + (1 − t)V(x)] dt < 0. j0

≥ 0 for every S ∈ ℝm , we get

Thus it is enough to apply Theorem 1.53.

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2 Introduction to Morse theory In this chapter, we introduce Morse theory both in finite and infinite dimension. In the first section, we consider the case of functions defined on finite dimensional manifolds for which the main references are the books of M. Morse, J. Milnor and R. Bott [39, 177, 180]. We present in detail the Morse lemma, the deformation theorems and the Morse inequalities. The chief example of the height function on the torus will illustrate the meaning of the main results. In the second section, we will describe the basic Morse theory on infinite dimensional manifolds following the book of K. C. Chang [60]. We first indicate how to study the local behavior of a functional f near a critical point and then how to study its global behavior, obtaining some kind of Morse inequalities. Finally, we show that a mountain pass critical point of a functional f in a Banach space has Morse index less than or equal to one.

2.1 Morse theory on finite dimensional manifolds 2.1.1 Introduction Let M be a smooth, N-dimensional manifold and f : M 󳨀→ ℝ a C ∞ -function. A point x̄ ∈ M is called a critical point of f if, for a given local coordinate system (x1 , . . . , xN ) in a neighborhood U of x,̄ it is: 𝜕f (x)̄ = 0, 𝜕xi

∀i = 1, . . . , N.

In other words, this means that the induced derivative map f∗ : Tx̄ M 󳨀→ Tf (x)̄ ℝ on the tangent space at a point x̄ is zero. A number c ∈ ℝ is said to be a critical value of f if the set f −1 (c) contains at least one critical point; if this is not the case, we say that c is a regular value. Morse theory aims to establish a relationship between the critical points of f and the topology of the manifold M, as expressed, for example, by its homology and cohomology groups. One way of doing this is to study the sublevels of f to detect a change of their topology when a critical value is crossed. Let us illustrate this idea by a few examples. Example 2.1. Let f : ℝ 󳨀→ ℝ be a smooth function with only two critical values: a1 < a2 , corresponding to a local minimum and a local maximum, respectively, as in Figure 2.1. https://doi.org/10.1515/9783110538243-002

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60 | 2 Introduction to Morse theory

Figure 2.1: A function with two critical values.

Consider the sublevels: fa = { x ∈ ℝ : f (x) ≤ a } ,

a ∈ ℝ.

If a ≤ a1 , then fa is a half-line; if a ∈ (a1 , a2 ), then fa is the union of a segment and a half-line; if a > a2 , then fa is again a half-line. This shows that the topology of fa changes while crossing the critical values. Example 2.2. Let f : ℝ2 󳨀→ ℝ the function defined as f (x, y) = y2 − x 2 . Obviously, it has (0, 0) as only critical point whose corresponding critical value is zero. Let us consider the sublevels: f−ϵ = { (x, y) : y2 − x2 ≤ −ϵ } ,

fϵ = { (x, y) : y2 − x 2 ≤ ϵ } ,

ϵ > 0.

As we can see in Figure 2.2, they have different topological properties. Indeed, fϵ is a connected set, while f−ϵ is not. Example 2.3. Let T ⊂ ℝ3 be a torus laying on the plane (x, y), using a (x, y, z) system of coordinates as below. We consider the height function: f (x, y, z) = z, which has four critical points P1 , P2 , P3 , P4 with corresponding critical values 0 = z1 < z2 < z3 < z4 (see Figure 2.3). The sublevels: fi = { (x, y, z) ∈ T : z ≤ a < zi } ,

i = 1, . . . , 4

and f5 = { (x, y, z) ∈ T : z ≤ a, a > z4 } are described in Figure 2.4. It is clear that all these sets do not have the same homotopy type. Indeed f1 is the empty set, f2 is homeomorphic to a disk, f3 is homeomorphic to a cylinder, f4 is homeomorphic to a compact manifold of genus one having a circle as boundary and f5 is the full torus.

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2.1 Morse theory on finite dimensional manifolds | 61

Figure 2.2: Sublevels of the function f (x, y) = y 2 − x 2 .

Figure 2.3: Critical points of the function f (x, y, z) = z on the torus.

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62 | 2 Introduction to Morse theory

Figure 2.4: Sublevels of the function f (x, y, z) = z.

2.1.2 Morse lemma and deformation theorems A critical point x̄ of a function f on a manifold M, as in the previous section, is called nondegenerate if the Hessian matrix (

𝜕2 f ̄ (x)), 𝜕xi 𝜕xj

i, j = 1, . . . , N,

in local coordinates, is nonsingular. This definition does not depend on the coordinate system, as follows also from the intrinsic way of defining the Hessian of the function f as a symmetric bilinear form: f∗∗ (v, w) = v(w(f )) = ∑ ai bj i,j

𝜕2 f (x)̄ 𝜕xi 𝜕xj

̄ w = ∑ bj 𝜕x𝜕 (x). ̄ where v, w ∈ TMx̄ and, in a local coordinate system, v = ∑ ai 𝜕x𝜕 (x), i

j

Definition 2.4. Let x̄ be a critical point of the smooth function f : M 󳨀→ ℝ. The Morse index of x̄ is the maximal dimension of a subspace of Tx̄ M on which f∗∗ is negative

definite, i. e., it is the number of the negative eigenvalues of the hessian matrix

𝜕2 f . 𝜕xi 𝜕xj

We are going to show that the behavior of f at the critical point x̄ can be described by its Morse index.

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2.1 Morse theory on finite dimensional manifolds | 63

Lemma 2.5 (Morse lemma). Let M be a smooth, N-dimensional manifold and let x̄ be a nondegenerate critical point of a smooth function f : M 󳨀→ ℝ. If the Morse index of x̄ is k, then there exists a local coordinate system (y1 , . . . , yN ) in a neighborhood U of x,̄ i. e., a local chart φ: U 󳨀→ V, where V is a neighborhood of 0 ∈ ℝN , such that φ−1 (0) = x̄ { { { k N { { {f (φ−1 (y1 , . . . , yn )) = f (x)̄ − ∑ yi2 + ∑ yi2 i=1 i=k+1 {

(2.1)

Proof. Without loss of generality, we can assume that x̄ is the origin 0 ∈ ℝn and that f (x)̄ = f (0) = 0. We have 1 N

f (x)̄ = ∫ ∑ 0 i=1 N

𝜕f (tx , . . . , txn )xi dt 𝜕xi 1 1 1

= ∑ [∫ ∫ i,j=1 0 0

𝜕2 f (stx) dsdt]xi xj 𝜕xi 𝜕xj

N

= ∑ hij (x)xi xj . i,j=1

We can assume that hij = hji , otherwise we take hij = 21 (hij + hji ). Then, using the diago2

f nalization of quadratic forms and the fact that the matrix ( 𝜕x𝜕 𝜕x (0)) is nonsingular, we i

j

can construct a nonsingular change of coordinates which gives (2.1). Indeed, after a 2 f (0)) diagonal, by the nondegeneracy linear change of coordinates which makes ( 𝜕x𝜕 𝜕x i

j

hypothesis, we may assume that h11 (0) ≠ 0 so that, in a neighborhood of the origin f (x) = h11 [x12 +

∑

(i,j)=(1,1) ̸

hij (x) h11

xi xj ].

Then, introducing the coordinates: h1r (x) x ], h11 r r =1̸

y1 = √|h11 |[x1 + ∑

yr = xr for r ≠ 1

we have that, by the inverse function theorem, this is a regular change of coordinates and 󸀠 f = y12 + ∑ yi yj Hi,j (y1 , . . . , yn ). i,j>1

Repeating this procedure, or arguing by induction, we get (2.1).

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64 | 2 Introduction to Morse theory Remark 2.6. The proof can be done for functions f ∈ C 2 (M) (see [191]). Hence, as a consequence of (2.1) we get that the nondegenerate critical points of a C 2 -function f 2 f are isolated. This can also be deduced directly by the nonsingularity of the matrix 𝜕x𝜕 𝜕x i

j

𝜕f , . . . , 𝜕x𝜕f ) is locally invertible at x,̄ which implies that the vector-valued map ∇f = ( 𝜕x 1 N and hence vanishes only in x.̄ The functions f ∈ C 2 (M) which have only nondegenerate critical points are usually called Morse functions and it can be proved that they are an open dense set in the space C 2 (M) with respect to the uniform topology (see [40]).

Next, we introduce some deformation theorems suitable to study the sublevels of a function f : M 󳨀→ ℝ. A 1-parameter group of diffeomorphisms of a manifold M is a C 1 -map φ: ℝ × M 󳨀→ M such that: (i) ∀t ∈ M the map φt : M 󳨀→ M defined by φt (x) = φ(t, x) is a diffeomorphism of M onto itself; (ii) φt+s = φt ∘ φs ∀t, s ∈ ℝ. We recall that a vector field on M is a map v: M 󳨀→ TM such that π ∘ v: M 󳨀→ M is the identity where π: TM 󳨀→ M is the projection of the tangent bundle TM. For a given vector field v on M, we consider the differential equation

together with the initial condition:

𝜕η = v(η(t)) 𝜕t

η(0) = x,

x ∈ M.

(2.2)

(2.3)

Applying the theory of ordinary differential equation, we have the following. Proposition 2.7. A C 1 -vector field v on M, which vanishes outside of a compact set contained in M, generates a unique 1-parameter group of diffeomorphisms φt (x) of M, where φt (x) is the solution of (2.2), (2.3) at time t. This allows to prove the following. Theorem 2.8. Let f be a smooth real valued function on a manifold M and a, b ∈ ℝ, a < b. Assume that the set f −1 ([a, b]) is compact and does not contain critical points of f . Then the sublevel fa = { x ∈ M : f (x) ≤ a } is a deformation retract of the sublevel fb = { x ∈ M : f (x) ≤ b }, i. e., there exists a map (retraction) r: fb 󳨀→ fa such that i ∘ r is homotopic to the identity map of fb , where i: fa 󳨀→ fb is the inclusion map. Proof. It is well known (see, e. g., [208]) that using a partition of unity it is possible to construct on every smooth compact finite dimensional manifold a Riemannian metric, namely a smooth positive-definite symmetric bilinear form on tangent vectors. This clearly induces a scalar product on Tx M, for every x ∈ M for which we will use the notation ⟨v, w⟩ for two tangent vectors v and w.

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2.1 Morse theory on finite dimensional manifolds | 65

We consider the vector field grad f on M which is characterized by the identity ⟨v, grad f ⟩ = derivative of f in the direction of the vector v. Obviously, this vector field vanishes only at the critical points of f . Then it is welldefined the map ρ: M 󳨀→ ℝ, ρ = ⟨grad f 1,grad f ⟩ in f −1 ([a, b]) and ρ ≡ 0 outside of a

compact neighborhood of f −1 ([a, b]). The vector field v = ρ grad f satisfies the hypothesis of Proposition 2.7 and hence generates a 1-parameter group of diffeomorphisms φt : M 󳨀→ M. We have df (φt (x)) dφ (x) = ⟨ t , grad f ⟩ = ⟨x, grad f ⟩ = 1 dt dt ∀x ∈ M and ∀t ∈ ℝ such that φt (x) ∈ f −1 ([a, b]). Hence the map t 󳨃󳨀→ f (φt (x)) is linear as long as f (φt (x)) belongs to [a, b]. Moreover, the diffeomorphism φb−a : M 󳨀→ M carries fa onto fb . Finally, the family of maps rt : fb 󳨀→ fb defined by x rt (x) = { φt(a−f (x)) (x)

if f (x) ≤ a

if f (x) ∈ [a, b]

gives the desired retraction of fb onto fa since r0 is the identity, while r1 = fa . Before passing to the next theorem, we need some definitions. We denote by Bk a k-dimensional ball centered at the origin: Bk = {x ∈ ℝn : k ∑i=1 xi2 < 1 and xk+1 = ⋅ ⋅ ⋅ = xN = 0}, in the context of this section it will be called a k-cell. The operation of attaching a k-cell to a topological space Y can be defined as follows. We consider a continuous function g: 𝜕Bk 󳨀→ X which is a homeomorphism onto its image. The topological space obtained by taking the union of Y and the closed k-cell Bk with the equivalence relation which identifies any x ∈ 𝜕Bk with its image g(x) ∈ Y will be denoted by Y ∪g Bk (Y with a k-cell attached). The next theorem describes the change of topology of the sublevels of f while crossing a nondegenerate critical point. Theorem 2.9. Let f : M 󳨀→ ℝ be a smooth function and x̄ a nondegenerate critical point ̄ If there exists ϵ > 0 of f with Morse index k and corresponding critical value c = f (x). −1 such that f ([c − ϵ, c + ϵ]) is compact and has no critical points other than x,̄ then fc+ϵ has the homotopy type of fc−ϵ with a k-cell attached. Proof. By the Morse lemma (Lemma 2.5), there exists a neighborhood U of x̄ and a local system of coordinates (y1 , . . . , yn ) such that k

N

i=1

i=k+1

f (x) = f (x)̄ − ∑ yi2 + ∑ yi2 , ∀x ∈ U and y1 (x)̄ = ⋅ ⋅ ⋅ = yN (x)̄ = 0.

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66 | 2 Introduction to Morse theory By assumption, we can choose ϵ > 0 such that f −1 ([c − ϵ, c + ϵ]) is compact and contains only one critical point, namely x.̄ Moreover, ϵ can be taken so small that U is diffeomorphic to a neighborhood of the origin in ℝN and contains the closed ball B(ϵ) = {(y1 , . . . , yn ) : ∑ yi2 ≤ 2ϵ}. Let Bk be the cell Bk = {(y1 , . . . , yn ) : ∑ki=1 yi2 ≤ ϵ and yk+1 = ⋅ ⋅ ⋅ = yN = 0}. The resulting situation in the neighborhood of x̄ can be illustrated as in Figure 2.5.

Figure 2.5: Attaching a k-cell to a sublevel.

The intersection Bk ∩ fc−ϵ is 𝜕Bk so that fc−ϵ has a k-cell attached. We would like to prove that fc−ϵ ∪g Bk (g is the gluing map) is a deformation retract of fc+ϵ . To this aim, we construct a new function F: M 󳨀→ ℝ which coincides with f outside the neighborhood U and in U takes the form k

N

i=1

i=k+1

F = f − μ(∑ yi2 + 2 ∑ yi2 ) where μ: ℝ 󳨀→ [0, +∞) is a smooth function such that μ(0) > ϵ,

μ(r) = 0

for r ≥ 2ϵ

and

− 1 < μ󸀠 (r) ≤ 0

∀r ∈ ℝ.

Observe that the sublevel fc+ϵ is the same as the sublevel Fc+ϵ . Indeed outside the ellipsoid E = {∑ki=1 yi2 + 2 ∑ni=k+1 yi2 ≤ 2ϵ} the functions f and F coincide, since μ = 0. In the interior of E, we have k

N

i=1

i=k+1

F ≤ f = c − ∑ yi2 + 2 ∑ yi2 ≤ c +

N 1 k 2 ∑ yi + 2 ∑ yi2 ≤ c + ϵ. 2 i=1 i=k+1

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2.1 Morse theory on finite dimensional manifolds | 67

Moreover, the functions F and f have the same critical points; indeed, setting ξ = ∑ki=1 yi2 and η = ∑ni=k+1 yi2 we have that f = c − ξ + η. Consequently, 𝜕F = −1 − μ󸀠 (ξ + 2η) < 0, 𝜕ξ

𝜕F = 1 − 2μ󸀠 (ξ + 2η) ≥ 1 𝜕η

so that dF =

𝜕F 𝜕F dξ + dη. 𝜕ξ 𝜕η

Since dξ and dη are simultaneously zero only at the origin, it follows that F has no critical points in U other than the origin. We know that fc+ϵ = Fc+ϵ and F ≤ f ; therefore F −1 ([c − ϵ, c + ϵ]) ⊂ f −1 ([c − ϵ, c + ϵ]). Moreover, F(x)̄ = c − μ(0) < c − ϵ. As a consequence, F −1 ([c − ϵ, c + ϵ]) is compact and does not contain any critical point of F. Thus Theorem 2.9 implies that Fc−ϵ is a deformation retract of Fc+ϵ = fc+ϵ , and hence the assertion will be proved if we show that Fc−ϵ has the same homotopy type of fc−ϵ ∪g Bk . Note that Fc−ϵ = fc−ϵ ∪ H, with H = Fc−ϵ \ fc−ϵ as illustrated in Figure 2.6.

Figure 2.6: Attaching a k-cell to a sublevel.

We define a retraction rt of fc−ϵ ∪ H into fc−ϵ ∪g Bk in the following way: (i) rt (y1 , . . . , yN ) = (y1 , . . . , yk , tyk+1 , . . . , tyN )

if ξ ≤ ϵ.

Hence r1 is the identity map, r0 maps the set {ξ ≤ ϵ} into Bk and rt (Fc−ϵ ) ⊂ Fc−ϵ since 𝜕F > 0. 𝜕η

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68 | 2 Introduction to Morse theory (ii) rt (y1 , . . . , yN ) = (y1 , . . . , yk , st yk+1 , . . . , st yN ) if ϵ ≤ ξ ≤ η + ϵ, 1

ξ −ϵ 2 where st = t + (1 − t)( ) . η Hence r1 is the identity map, r0 maps the set {ϵ ≤ ξ ≤ η + ϵ} into the set f −1 (c − ϵ) and rt is defined as in (i) for ξ = ϵ. (iii) rt (y1 , . . . , yN ) = (y1 , . . . , yN ) if ξ ≥ η + ϵ, i. e. in fc−ϵ . It is easy to see that rt is the same as in (ii) if ξ = η + ϵ. By definition, rt maps fc−ϵ ∪ H into fc−ϵ ∪g Bk and the assertion is proved. In the same way, it could be proved that if the function f has q nondegenerate critical points with c as corresponding critical value and with Morse indices k1 , . . . , kq then the sublevel fc+ϵ has the same homotopy type as fc−ϵ ∪g1 Bk1 ∪ ⋅ ⋅ ⋅ ∪gq Bkq . This indicates that, starting with the sublevel fa , for a < minM f , it is possible somehow “to construct” the manifold M attaching cells as the value a growths crossing critical values of f , until when a reaches the maximum of f . This is the case of the torus and the height function considered in the Example 2.3, as shown in Figure 2.7.

2.1.3 Morse inequalities In this section, we will establish a relation between the number of critical points of a Morse function (i. e., a function which has only nondegenerate critical points) on a compact manifold M and the Betti numbers of M. We recall that the Betti numbers of M are the ranks of the homology (or relative homology) groups. We refer to [132] and [207] for definitions and an introduction to homology theory. Let us start with some preliminaries. We consider a triple of compact topological spaces X ⊃ Y ⊃ Z. Then we can define an exact homology sequence: 𝜕k

󳨀→ Hk+1 (X, Y) 󳨀→ Hk (Y, Z) 󳨀→ Hk (X, Z) 󳨀→ 𝜕k−1

󳨀→ Hk (X, Y) 󳨀󳨀󳨀→ ⋅ ⋅ ⋅ 󳨀→ H0 (X, Y) 󳨀→ 0

(2.4)

where Hi (⋅, ⋅) are the relative homology groups (see [207], Chapter IV) over a field G. Assuming that the ranks of the homology groups in (2.4) are all finite we denote by Pt (A, B) the Poincaré polynomial (or series) of the topological pair (A, B), i. e., Pt (A, B) = ∑ βi (A, B)t i i≥0

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2.1 Morse theory on finite dimensional manifolds | 69

Figure 2.7: Attaching cells as crossing critical values.

where βi (A, B) are the relative Betti numbers of the pair. Next, we consider the formal polynomial (or series) qt (X, Y, Z) = ∑ di (X, Y, Z)t i i≥0

(2.5)

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70 | 2 Introduction to Morse theory where di (X, Y, Z) is the rank of the image of the map 𝜕i in (2.4). We have the following. Lemma 2.10. Let X0 ⊂ X1 ⊂ ⋅ ⋅ ⋅ ⊂ Xn , n + 1 compact topological spaces. Then n

∑ Pt (Xj , Xj−1 ) = Pt (Xn , X0 ) + (1 + t)Qt j=1

(2.6)

where Qt = ∑nj=2 qt (Xj , Xj−1 , X0 ) is a polynomial with nonnegative integer coefficients. Proof. For any triple of topological spaces X ⊃ Y ⊃ Z and for any m ∈ ℕ, from the exact sequence (2.4) we get: β0 (X, Y) − β0 (X, Z) + β0 (Y, Z) − β1 (X, Y) + β1 (X, Z)

− β1 (Y, Z) + ⋅ ⋅ ⋅ + (−1)m βm (X, Y) − (−1)m βm (X, Z)

+ (−1)m βm (Y, Z) − (−1)m dm (X, Y, Z) = 0. Thus

(−1)m dm (X, Y, Z) = (−1)m−1 dm−1 (X, Y, Z) + (−1)m βm (X, Y) − (−1)m βm (X, Z) + (−1)m βm (Y, Z).

Multiplying both sides by (−1)m t m and summing on the index m, by (2.5) we obtain qt (X, Y, Z) = −tqt (X, Y, Z) + Pt (X, Y) − Pt (X, Z) + Pt (Y, Z) or, equivalently Pt (X, Y) + Pt (Y, Z) = Pt (X, Z) + (1 + t)qt (X, Y, Z). If we take as triple (X, Y, Z) the triple (Xj , Xj−1 , X0 ), j ≥ 2, we get Pt (Xj , Xj−1 ) = P(Xj , X0 ) − Pt (Xj−1 , X0 ) + (1 + t)qt (Xj , Xj−1 , X0 ). Then (2.6) follows, summing on the index j ≥ 2. For the next result, we need the property of the homology of a pair (X, Y) which relates it to the homology of the quotient space X/Y obtained identifying Y with only one selected point y0 ∈ Y. More precisely, it holds (see [132, 207]). Proposition 2.11. Assume that the topological pair (X, Y) satisfies: (i) X is an Hausdorff space; (ii) Y is closed in X; (iii) ∀x ∈ X/Y there exist two disjoints open sets U and V such that x ∈ U and Y ⊂ V;

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2.1 Morse theory on finite dimensional manifolds | 71

(iv) there exists an open neighborhood I of Y such that Y is a deformation retract of I and I ≠ Y. Then Hq (X, Y) ≃ Hq (X/Y), ∀q > 0 and H0 (X, Y) = H0# (X/Y) where H0# (X/Y) = {0} if X/Y is path-connected and H0# (X/Y) is a free group with (r − 1) generators if X/Y has r connected components. An example of a topological pair satisfying the conditions (i)–(iv) is the pair (Bk , 𝜕Bk ), where Bk is the unit ball in ℝk . Now we consider a compact manifold M as in the previous sections and a smooth Morse function f : M 󳨀→ ℝ. Let a0 < a1 < ⋅ ⋅ ⋅ < an be real numbers such that fa0 = 0, fan = M and each sublevel fai contains exactly i critical points, each one being the only one in the set fai \ fai−1 . By Theorem 2.9, we have H∗ (fai , fai−1 ) = H∗ (fai−1 ∪g Bλi , fai−1 ) = H∗ (Bλi , 𝜕Bλi )

(2.7)

where λi is the Morse index of the only critical point in fai \ fai−1 . Moreover, by Proposition 2.11 ℤ

Hq (Bλi , 𝜕Bλi ) = {

if q = λi

{0} if q ≠ λi ,

here ℤ is taken as the unitary commutative ring of the coefficients for the homology groups Hq . Hence the Poincaré polynomial of the pair (fai , fai−1 ) is t λi . Applying (2.6) to the spaces 0 = fa0 ⊂ fa1 ⊂ ⋅ ⋅ ⋅ ⊂ fan = M, by (2.7) we get n

n

i=1

i=1

∑ t λi = ∑ Pt (fai , fai−1 ) = Pt (M, 0) + (1 + t)Qt (f ). This procedure can be generalized to the case of more than one critical points corresponding to the same critical value obtaining: m

m

i=0

i=0

Pt (f ) = ∑ αi t i = ∑ βi (M)t i + (1 + t)Qt (f )

(2.8)

where m is the dimension of M and αi is the number of critical points of f with Morse i index i. The polynomial Pt (f ) = ∑m i=0 αi t is also called the Morse polynomial of f . In particular, from (2.8) we deduce that αi ≥ βi (M), i. e., any Morse function f on a compact manifold M has at least βi (M) critical points with index i. This, for example, allows to claim that any Morse function on the torus has at least four critical points: a maximum point, a minimum point and two saddle points. Now let us give a geometric explanation of (2.8), independently of Lemma 2.10. Let c be a critical value of f to which corresponds only one critical point with Morse index λ. Obviously, passing from the sublevel fc−ϵ to the sublevel fc+ϵ a power t λ should

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72 | 2 Introduction to Morse theory be added to the Morse polynomial of f . Let us try to understand how the Poincaré polynomial of the manifold changes at crossing the critical level c. By Theorem 2.9, we know that the sublevel fc+ϵ is obtained by fc−ϵ adding a λ-cell. An important observation is that the operation of attaching a λ-cell to a space can either “increase” the homology in dimension λ or “decrease” the homology in dimension (λ − 1). In Figure 2.8, there are some examples for the case λ = 1, 2.

Figure 2.8: Changing the homology of sets.

As a consequence, we have λ

Pt (fc+ϵ ) + t { { { or Pt (fc+ϵ ) = { { { λ−1 {Pt (fc−ϵ ) − t

(2.9)

Then, denoting by Pt (f |fa ) the Morse polynomial of the restriction of f to the sublevel fa and assuming that Pt (f |fc−ϵ ) = Pt (fc−ϵ ) (which holds, for example, if c is the

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2.2 Infinite dimensional Morse theory | 73

minimum of f ) from (2.9) we get Pt (f |fc+ϵ ) = Pt (f |fc−ϵ ) + t λ = Pt (fc−ϵ ) + t λ = Pt (fc+ϵ ) { { { = {or { { λ−1 {Pt (fc−ϵ ) + (1 + t)t

(2.10)

This explains why the term (1+t) appears and how we get (2.8) as the value c increases. The presence of this term (1 + t) in (2.8) is important and allows to deduce more information on the critical points of f , other than the inequalities αi ≥ βi (M). For example, if M = S2 and f is a function which has two minimum points, then by (2.8) we have 2 + α1 t + α2 t 2 = 1 + t 2 + (1 + t)(a0 + a1 t + a2 t 2 ) with ai ≥ 0. This implies that a0 ≥ 1, and hence α1 ≥ 1 which means that the function f must have at least one saddle point. Obviously, it could happen that Qt (f ) = 0, i. e., Pt (f ) = Pt (M) in which case f is called a perfect Morse function. It is worth observing that all this depends on the choice of the coefficients in the homology group. A class of perfect Morse functions is given by those ones which do not have critical points with consecutive Morse index. Finally, to find a perfect Morse function on a manifold M allows to compute the homology of M.

2.2 Infinite dimensional Morse theory 2.2.1 Critical groups and Morse lemma Let H be an Hilbert space and M a C 2 -Hilbert manifold (see, e. g., [14], Section 6.1 for the definition). The basic Morse theory for a C 2 -function f : M 󳨀→ ℝ is set up in two steps: (i) a local study of the behavior of the function f near its critical points, through the definition of the critical groups; (ii) a global study which focuses on the relationship between some numbers (Morse numbers) related to the number of critical points of f and the topological properties of the underlying manifold. In analogy with the finite dimensional case, we define the Morse index of a critical point u of f (i. e., a point where f 󸀠 (u) = 0) as the maximal dimension of a subspace of H on which the bilinear form 󸀠󸀠

B(v, w) = (f ∘ x−1 ) (x(u))(v, w)

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74 | 2 Introduction to Morse theory is negative definite, where x is a chart at u. If M is the whole Hilbert space, the Morse index of u is just the maximal dimension of a subspace of H on which f 󸀠󸀠 (u) is negative definite. The nullity of u is the supremum of the dimensions of subspaces of H on which the bilinear form B is zero. Finally, u is said to be a nondegenerate critical point if the linear operator L: H 󳨀→ H ∗ ≃ H defined by (Lv, w) = B(v, w),

∀v, w ∈ H

is invertible, where (⋅, ⋅) is the inner product on H. Note that, by the chain rule, the above definitions are independent of the choice of the chart x. Moreover, by the implicit function theorem, a nondegenerate critical point is isolated. We now define the critical groups of an isolated critical point and show their relation with the Morse index (see also [22]). Definition 2.12. Let u be an isolated critical point of f and let c = f (u). We define Cq (f , u) = Hq (fc ∩ U, fc ∩ U \ {u}),

q = 0, 1, . . .

the qth critical group of f at u, where U is a neighborhood of u which does not contain another critical point and Hq (⋅, ⋅) denotes the qth relative homology group over a field G as in the previous section. According to the excision property of the singular homology theory, the critical groups are independent of the choice of the neighborhood U. We show that for a nondegenerate critical point, the critical groups depend only on the Morse index. Theorem 2.13. Let u be an isolated critical point of f ∈ C 2 (M). Then (i) if u is a local minimum point of f G

Cq (f , u) = {

0

if q = 0 if q ≠ 0

(ii) if u is a nondegenerate critical point with Morse index m(u) = k ≥ 1, then G

Cq (f , u) = {

0

if q = k

if q ≠ k

Proof. (i) Let B be a closed neighborhood of u so small that f (v) > c = f (u), for every v ∈ B \ {u}. Then, by definition, Cq (f , u) = Hq (fc ∩ B, fc ∩ B \ {u}) = Hq ({u}, 0) = δq,0 G and the assertion is proved.

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2.2 Infinite dimensional Morse theory | 75

(ii) Let x be a chart at u and let B be a closed neighborhood of u contained in the domain of x. Then, for c = f (u), Cq (f , u) = Hq (fc ∩ B, fc ∩ B \ {u}) and it is enough to consider the case when M is an open subset of the space H. For simplicity, we assume u = 0 and c = f (u) = 0. Let L: H 󳨀→ H be the self-adjoint operator defined by (Lv, w) = f 󸀠󸀠 (0)(v, w),

∀v, w ∈ H.

Since, by assumption, L is invertible then the space H is the orthogonal sum of H + and H − which are the subspaces where L is positive definite and negative definite, respectively. Then we take a closed ball B with center at 0 so small that B ∩ H − ⊂ f0

(2.11)

+

B ∩ H ∩ f0 = {0}

f 󸀠󸀠 (v)(w, w) ≥ 0

(2.12)

∀v ∈ B, ∀w ∈ H + .

(2.13)

Let us define η: [0, 1] × B 󳨀→ B,

η(t, v) = (1 − t)v + tP(v)

where P is the orthogonal projector onto H − . For v ∈ F0 ∩ B, let us write g(t) = f (η(t, v)). It follows from (2.13) that, for all t ∈ [0, 1], g 󸀠󸀠 (t) = f 󸀠󸀠 ((1 − t)v + tP(v))((I − P)v) ≥ 0 so that g is convex on [0, 1]. But g(0) = f (v) ≤ 0, since v ∈ f0 and g(1) = f (P(v)) ≤ 0, by (2.11). Thus f (η(t, v)) = g(t) ≤ 0, for all t ∈ [0, 1]. Moreover, if η(t, v) = 0, for some t ∈ [0, 1] and some v ∈ f0 ∩ B, then v = 0, because η(t, v) = 0 implies P(v) = 0 so that, by (2.12), v = 0. Finally, H − ∩B\{0} is a deformation retract of f0 ∩B\{0} and H − ∩B is a deformation retract of f0 ∩ B. Since, by assumption, the Morse index of u is k, dim H − = k and we get Hq (f0 ∩ B, f0 ∩ B \ {0}) ≃ Hq (f0 ∩ B, H − ∩ B \ {0})

≃ Hq (H − ∩ B, H − ∩ B \ {0}) ≃ Hq (Bk , Sk−1 ) ≃ δq,k G.

Remark 2.14. The statement of Theorem 2.13 holds also in Banach spaces. We refer to [60, 154] for the proof (see also [6]).

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76 | 2 Introduction to Morse theory To study the local behavior of a function f ∈ C 2 (M) at a nondegenerate critical point, it is crucial to prove a Morse lemma, analogous to Lemma 2.5 for the finite dimensional case. We state it here and refer to [60] for the proof. Theorem 2.15 (Infinite dimensional Morse lemma). Let u be a nondegenerate critical point of a function f ∈ C 2 (M). Then there exist a neighborhood U of u and a local diffeomorphism Φ: U 󳨀→ Tu (M) (Tu (M) is the tangent space at u) with Φ(u) = 0, such that 1 f ∘ Φ−1 (ξ ) = f (u) + (d2 f (u)ξ , ξ ), 2

∀ξ ∈ Φ(U).

2.2.2 Morse inequalities In order to prove some kind of Morse identity, like (2.10) in the infinite dimensional case and a result analogous to that of Theorem 2.9, we need to define the Morse numbers of a function. Though everything could be done for a function f defined on a C 2 -Finsler manifold modelled on a Banach space, we will only consider the case of a C 2 -Hilbert manifold M and refer to [60] for the general case. Let f be a real valued function on M of class C 1 and assume that f has only isolated critical values and that each of them corresponds to a finite number of critical points. We denote by ci , i ∈ ℤ, the critical values of f and by Kci = {zji }m j=1 , i ∈ ℤ, the corresponding finite sets of critical points. Choosing ϵi ∈ (0, ηi ), ηi = min {ci+1 − ci , ci − ci−1 }, i ∈ ℤ, we define the following. Definition 2.16. For a pair of regular values a < b, the number Mq (a, b) = ∑ rank Hq (fci +ϵi , fci − ϵi ) a 0 such that: (i) η(0, u) = u, for all u ∈ M; ̄ (ii) η(t, u) = u, for all t ∈ [0, 1] and f (u) ∉ [c − ϵ,̄ c + ϵ]; (iii) η(t, ⋅) is a homeomorphism of M onto M for each t ∈ [0, 1]; (iv) η(1, fc+ϵ \ N) ⊂ fc−ϵ ; (v) f ∘ η(t, u) is nonincreasing in t, for all (t, u) ∈ [0, 1] × M. By the deformation theorems, we have that for functions satisfying the (PS) condition, the Morse numbers are well-defined, i. e., they are independent of the choice of {ϵi }. As a consequence of the deformation theorem and the homotopy invariance of the homology groups, we have a connection between the critical groups and the Morse numbers. Theorem 2.20. Assume that c is an isolated critical value of f ∈ C 1 (M) and Kc = {zj }j=1,...,m . Then, for ϵ > 0 sufficiently small: m

H∗ (fc+ϵ , fc−ϵ ) ≅ H∗ (fc , fc \ Kc ) ≅ ⨁ C∗ (f , zj ) j=1

Hence, for regular values a < b of f we have mi

Mq (a, b) = ∑ ∑ rank Cq (f , zji ), a r such that F(e) ≤ 0.

(2.15)

c = inf max F(γ(t))

(2.16)

Let γ∈Γ t∈[0,1]

where Γ is the set of all paths joining u = 0 and u = e, i. e., Γ = { γ ∈ C([0, 1], E) : γ(0) = 0, γ(1) = e } Then, if F satisfies the (PS)c condition, the number c is a positive critical level of F, namely there exists u ∈ E such that F(u) = c and F 󸀠 (u) = 0, in particular u ≠ 0 and u ≠ e. The original proof of this theorem is contained in [15] and it is also included in every book about variational methods; we refer, for example, to [14, 196, 225]. The aim of this section is to show that at a mountain pass level c (i. e., c given by (2.16)) of a C 2 -functional F there exists a critical point with Morse index at most one. As defined in Section 2.2.1, the Morse index m(u) of a critical point u of F is the maximal dimension of a subspace on which F 󸀠󸀠 (u) is negative definite. Thus we define

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2.2 Infinite dimensional Morse theory | 79

the subspaces: E 0 (u) = ker F 󸀠󸀠 (u)

(2.17)

−

󸀠󸀠

(2.18)

+

󸀠󸀠

(2.19)

E (u) = { v ∈ E : ⟨F (u)v, v⟩ < 0 }

E (u) = { v ∈ E : ⟨F (u)v, v⟩ > 0 } and u will be a nondegenerate critical point of F if E 0 (u) = {0}.

Theorem 2.23. Let F be a C 2 -functional satisfying the hypothesis of Theorem 2.21 and let Kc = {z ∈ E : F 󸀠 (z) = 0} ∩ F −1 (c), c given by (2.16). Then, if Kc is discrete, there exists u ∈ Kc with Morse index m(u) ≤ 1. Moreover, if Kc = {u} and u is nondegenerate, then m(u) = 1. To prove this theorem, we need some preliminary result. Lemma 2.24. Let F: E 󳨀→ ℝ be a C 2 -functional satisfying the hypotheses of Theorem 2.22 and let c be the critical mountain pass level defined by (2.16). Then, if Kc is discrete, there exists a critical point u ∈ Kc , such that C1 (F, u) ≠ 0, where C1 (F, u) denotes the first critical group of F at u, as in Definition 2.12. Proof. We refer to Theorem 1.5 of [60]. Lemma 2.25. Let F: E 󳨀→ ℝ a C 2 -functional and assume that u is a critical point of F with finite Morse index m(u). Then Cq (f , u) = 0,

for every q ≤ m(u) − 1.

Proof. See [154]. We can now prove 2.23. Proof of Theorem 2.23. If u is the only critical point in Kc and is nondegenerate, then applying Theorem 2.13 and Remark 2.14 as well as Lemma 2.24, we get that m(u) = 1. In the general case, by Lemma 2.24, we know that there exists a critical point u ∈ Kc such that C1 (F, u) ≠ 0. On the other side, by Lemma 2.25 we have that Cq (F, u) = 0, for every q ≤ m(u) − 1. Combining this information, we obtain m(u) ≤ 1. Another way of getting Theorem 2.23 without using explicitly the critical groups is shown in ([14], Theorem 12.31) in the case when E is an Hilbert space and Kc reduces to only one nondegenerate critical point. It combines different ingredients already used to prove Theorem 2.13, Lemma 2.24 and Lemma 2.25 and shows more clearly the connection between the geometric construction of the mountain pass critical level and the Morse index of a corresponding critical point. Therefore, we outline it below.

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80 | 2 Introduction to Morse theory Alternative proof of Theorem 2.23. We assume that E is a Hilbert space and Kc = {u} and u is nondegenerate. Moreover, for simplicity, we take u = 0 while the point 0 where F(0) = 0 in the statement of Theorem 2.22 will be denoted by u0 and write E + and E − instead of E ± (u) as defined in (2.18) and (2.19). Hence E = E + ⊕ E − and, arguing by contradiction, we assume that dim E − ≥ 2. By the Morse lemma in infinite dimension (see [60], Lemma 4.1), up to a smooth change of coordinates, we have that 󵄩 󵄩2 󵄩 󵄩2 F(u) = c − 󵄩󵄩󵄩u− 󵄩󵄩󵄩 + 󵄩󵄩󵄩u+ 󵄩󵄩󵄩 where u = u+ + u− , u− ∈ E − and u+ ∈ E + . Given β > α > 0, we consider a neighborhood Uα,β of u = 0, defined by 󵄩 󵄩 󵄩 󵄩 Uα,β = { u = u− + u+ : 󵄩󵄩󵄩u− 󵄩󵄩󵄩 < α, 󵄩󵄩󵄩u+ 󵄩󵄩󵄩 < β } . Then, for every u ∈ U α,β with ‖u+ ‖ = β we have F(u) ≥ c − α2 + β2 . Thus, given d > c, there exists ϵ > 0 such that, if α < β ≤ ϵ, then 󵄩 󵄩 inf{F(u) : u ∈ U α,β , 󵄩󵄩󵄩u+ 󵄩󵄩󵄩 = β} ≥ d. Taking α and β sufficiently small, we can also assume that inf{F(u) : u ∈ U α,β } > 0. Note that, since F(u0 ) = 0 and F(e) ≤ 0, neither u0 nor e belong to U α,β . By the definition of the mountain pass critical level c, taken δ ∈ ]0, d − c[, we can find γ ∈ Γ such that sup{F(γ(t)) : t ∈ [0, 1]} ≤ c + δ. By the deformation lemma, Theorem 2.19, we can find a deformation η such that: a) η ∘ γ ∈ Γ (Γ defined in the statement of Theorem 2.22); b) η(Fc+δ \ Uα,β ) ⊂ Fc−δ . As a consequence of b), we have that the path γ must necessarily intersect Uα,β , otherwise we get a contradiction with the definition of the critical level c. Since γ connects the point u0 and e which do not belong to Uα,β , we deduce that there exist t1 < t2 ∈ (0, 1) such that the points vi = γ(ti ), i = 1, 2, belong to 𝜕Uα,β , while γ(t) ∉ U α,β for all t < t1 or t > t2 . Since 󵄩 󵄩 F(vi ) ≤ c + δ < d ≤ inf{F(u) : u ∈ U α,β , 󵄩󵄩󵄩u+ 󵄩󵄩󵄩 = β}

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2.2 Infinite dimensional Morse theory | 81

it follows that ‖vi− ‖ = α and ‖vi+ ‖ < β, i = 1, 2, for vi = vi− + vi+ , i. e., vi∓ are the projections of vi on the spaces E − and E + . Let πi ⊂ 𝜕Uα,β denote the segments connecting vi with vi− . Since we are assuming that dim E − ≥ 2, we can connect v1− and v2− with an arc σ contained in 𝜕Uα,β ∩ E − . This allows to construct a new path γ̃ defined as follows: γ(t)

̃ ={ γ(t)

π1 ∪ σ ∪ π2

if t ∈ [0, t1 ] ∪ [t2 , 1] if t ∈ [t1 , t2 ]

and γ̃ belongs to Γ. Since for v ∈ 𝜕Uα,β ∩ E − one has that ‖v− ‖ = α and ‖v+ ‖ = 0, then F(v) = c − α2 ,

∀v ∈ 𝜕Uα,β ∩ E − .

This implies that sup F(u) < c. σ

On the other side supπi F(u) ≤ c + δ, i = 1, 2, so that sup F(u) ≤ c + δ, γ̃

and the path γ̃ does not intersect Uα,β . Then, as before, applying the deformation ̃ lemma we get, as in a) and b), a new path η ∘ γ̃ ∈ Γ such that sup{F(γ(t)), t ∈ [0, 1]} ≤ c − δ, which is a contradiction with the definition of the mountain pass level c. Hence for the critical point u ∈ Kc it must be m(u) ≤ 1. To rule out that m(u) = 0, we observe that in such case, since u is nondegenerate, then u is a local minimum for F and E = E + . Taking Uα,β such that inf{F(𝜐) : 𝜐 ∈ 𝜕Uα,β } ≥ d > c, the previous arguments can be repeated, reaching again a contradiction. We conclude by mentioning that similar arguments can be used to compute the Morse index of critical points obtained by other mini-max procedure. We refer to [60, 121] and the references therein for the statements and details.

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3 Morse theory for semilinear elliptic equations In this chapter, we introduce the Morse index of solutions of semilinear elliptic equations and show some first properties and applications. In particular, we will consider the case of least energy solutions, both positive or sign changing, for which the Morse index can be explicitly computed. As an application, we will show the uniqueness of the positive solution of Morse index one in planar convex bounded domains as obtained in [158]. Finally, we will prove estimates for the Morse index of symmetric sign changing solutions, both in the autonomous and nonautonomous case, following the papers [7] and [179].

3.1 Introduction Let Ω be a Lipschitz domain in ℝN , N ≥ 2, and Γ0 , Γ relatively open disjoint subsets of 𝜕Ω satisfying the conditions (1.28) and (1.29) of Chapter 1. Though in this chapter we will focus mostly on Dirichlet boundary conditions, we give the main definitions and some general results in the case of mixed Dirichlet– Neumann problems since they will be studied in Chapter 6. Let us consider the following semilinear elliptic boundary value problem: −Δu = f (x, u) in Ω { { { {u = 0 on Γ0 { { { { 𝜕u = g(x, u) on Γ { 𝜕𝜐

(3.1)

where 𝜐 denotes the outer normal to 𝜕Ω. We assume that f = f (x, s): Ω × ℝ 󳨀→ ℝ and g = g(x, s): Γ × ℝ 󳨀→ ℝ are locally Hölder continuous functions in Ω × ℝ and they are differentiable with respect to the second variable with f,

𝜕f ∈ C 0 (Ω × ℝ); 𝜕s

g,

𝜕g ∈ C 0 (Γ × ℝ) 𝜕s

(3.2)

We consider bounded weak solutions of (3.1), i. e., functions u ∈ H01 (Ω∪Γ)∩L∞ (Ω), where H01 (Ω ∪ Γ) is the Sobolev space defined in Chapter 1, such that ∫ ∇u ⋅ ∇φ dx = ∫ f (x, u)φ dx + ∫ g(x 󸀠 , u)φ dx󸀠 , Ω

Ω

Γ

∀φ ∈ H01 (Ω ∪ Γ).

(3.3)

Let us observe that since f is locally Hölder continuous, by standard elliptic regularity results u ∈ C 2 (Ω). Corresponding to such a solution of (3.1), we consider the quadratic https://doi.org/10.1515/9783110538243-003

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84 | 3 Morse theory for semilinear elliptic equations form Qu (ψ) = ∫|∇ψ|2 dx − ∫ Ω

Ω

𝜕f 𝜕g (x, u)|ψ|2 dx − ∫ (x󸀠 , u)|ψ|2 dx 󸀠 𝜕s 𝜕s

(3.4)

Γ

for any ψ ∈ Cc1 (Ω ∪ Γ). The Morse index of a solution is defined as follows. Definition 3.1. Let u ∈ H01 (Ω ∪ Γ) ∩ L∞ (Ω) be a weak solution of (3.1). We say that: (i) u is stable (or has zero Morse index) if Qu (ψ) ≥ 0 ∀ψ ∈ Cc1 (Ω ∪ Γ); (ii) u has Morse index equal to the integer m(u) ≥ 1 if m(u) is the maximal dimension of a subspace of Cc1 (Ω ∪ Γ) where the quadratic form Qu (ψ) is negative definite; (iii) u has infinite Morse index if, for any integer k ≥ 1 there exists a k-dimensional subspace of Cc1 (Ω ∪ Γ) where Qu is negative definite. The above definition is the same as the one given in Chapter 2 if we observe that a weak solution of (3.1) is a critical point of the functional J(v) =

1 ∫|∇v|2 dx − ∫ F(x, v) dx − ∫ G(x 󸀠 , v) dx 󸀠 2 Ω

(3.5)

Γ

Ω

v

v

in the space H01 (Ω ∪ Γ) ∩ L∞ , where F(x, v) = ∫0 f (x, s) ds and G(x, v) = ∫0 g(x, s) ds. Thus the quadratic form (3.4) is the one corresponding to the second derivative of the functional J, i. e., Qu (ψ) = J 󸀠󸀠 (u)(ψ, ψ). In general, to use the Morse index in applications it is convenient to relate it to the number of negative eigenvalues of a suitable linear operator. Therefore, we consider the following eigenvalue problem: 𝜕f { −Δw − (x, u)w = λw { { { 𝜕s { { w = 0 { { { { { { 𝜕w − 𝜕g (x, u)w = λw { 𝜕𝜐 𝜕s

in Ω (3.6)

on Γ0 on Γ

where u is a given solution of (3.1). The linear problem (3.6) is the one corresponding to the linearization of (3.1) at a solution u. The linear operator Lu : H01 (Ω∪Γ) → (H01 (Ω∪ Γ))∗ defined, for any v ∈ H01 (Ω ∪ Γ) by Lu (v)(z) = J 󸀠󸀠 (u)(v, z) = ∫ ∇v ⋅ ∇z dx − ∫ Ω

Ω

𝜕f 𝜕g vz dx − ∫ vz dx󸀠 , 𝜕s 𝜕s Γ

z ∈ H01 (Ω ∪ Γ)

is the so-called linearized operator at the solution u and will be denoted by Lu = (−Δ −

𝜕f 𝜕 𝜕g ; − ) 𝜕s 𝜕𝜐 𝜕s

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(3.7)

3.1 Introduction | 85

according to the notations of Chapter 1 (see (1.74)). In the case of Dirichlet problems, i. e., when Γ = 0, we will use the simple notation Lu = −Δ −

𝜕f 𝜕s

Using the eigenvalue theory developed in Chapter 1, we obtain, in bounded domains, the following characterization of the Morse index. Theorem 3.2. Let Ω be as in (3.1) and assume further that it is bounded. Then the Morse index of a solution u to (3.1) equals the number of negative eigenvalues of problem (3.6). Proof. Let us denote by μ(u) the number of negative eigenvalues of (3.6). If the quadratic form Qu defined in (3.4) is negative definite on a k-dimensional supspace of Cc1 (Ω ∪ Γ), then, by (iii) of Theorem 1.42 in Chapter 1 we have that the kth eigenvalue λk of (3.6) is negative. Hence μ(u) ≥ m(u). On the other hand, if there are k negative eigenvalues of (3.6), by (i) of Theorem 1.44 there is a k-dimensional supspace of Cc1 (Ω ∪ Γ) where the quadratic form Qu is negative definite, hence m(u) ≥ μ(u). We stress that Theorem 3.2 holds in bounded domains because of the spectral theory developed in Chapter 1. Remark 3.3. If one of the derivative 𝜕f , 𝜕g is nonpositive (or, more generally, if the 𝜕s 𝜕s 1 linearized operator Lu is coercive in H0 (Ω ∪ Γ)), then other choices of eigenvalue problems could be possible to evaluate the Morse index of a solution u. For example, if 𝜕f ≤ 0, a modification of the construction done in Section 1.4 of Chapter 1 yields a 𝜕s compact operator in the space L2 (Γ) and a corresponding sequence of eigenvalues of the problem −Δw + c(x)w = 0 { { { {w = 0 { { { { 𝜕w + d(x)w = λw { 𝜕𝜐

in Ω on Γ0

(3.8)

on Γ

where c = − 𝜕f and d = − 𝜕g . 𝜕s 𝜕s This case occurs, in particular, in the study of harmonic functions subjected to nonlinear boundary conditions (see, e. g., [1], [27] and the references therein). In this case f (x, u) ≡ 0, so that the previous eigenvalue problem becomes −Δw = 0 { { { {w = 0 { { { { 𝜕w + d(x)w = λw { 𝜕𝜐

in Ω on Γ0 on Γ

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86 | 3 Morse theory for semilinear elliptic equations ≤ 0, modifying the construction of Section 1.4 of Chapter 1, we get a If instead 𝜕g 𝜕s compact linear operator in L2 (Ω) and a corresponding sequence of eigenvalues of the problem: −Δw + c(x)w = λw { { { {w = 0 { { { { 𝜕w + d(x)w = 0 { 𝜕𝜐

in Ω on Γ0

(3.9)

on Γ

In both cases, the eigenvalues share the same variational characterization as the one given in Chapter 1, so that the Morse index of a solution can be characterized by the number of negative eigenvalues of (3.8) or (3.9). Some other eigenvalue problems with weights have been considered in the literature (see [18, 120, 173] and the references therein). In particular, when Γ = 𝜕Ω in [173] the eigenvalue problem −Δwj + c(x)wj = λj m(x)wj { { { { 𝜕wj + d(x)w = λ n(x)w j j j { 𝜕𝜐

in Ω on 𝜕Ω

with positive weights m, n is considered and the eigenvalues sequence constructed by constrained minimization. In that paper, the weights can also vanish in part of the domain, but the nonnegativity of the coefficients c, d is assumed (or more generally coercivity of the corresponding linear operator). The general linear eigenvalue problem (3.6) that we consider, whose corresponding spectral theory has been developed in Chapter 1, with the same eigenvalue parameter in the equation and in the nonlinear boundary condition, has the advantage of not requiring the nonnegativity of the coefficients c and d. This is important while studying the Morse index of solutions of nonlinear problems as (3.1), since many choices of the nonlinearities f and g lead to negative or sign changing coefficients c = − 𝜕f , 𝜕s d = − 𝜕g in the linearization and to noncoercive linear operators. 𝜕s

3.2 Positive solutions of Dirichlet problems Let us consider the Dirichlet semilinear elliptic problem: −Δu = f (x, u) in Ω { u=0 on 𝜕Ω

(3.10)

where Ω is a smooth bounded domain in ℝN , N ≥ 2. Obviously, (3.10) is a special case of (3.1) when Γ = 0.

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3.2 Positive solutions of Dirichlet problems | 87

Under some hypotheses on the function f (x, s), we will show the existence of positive solutions of (3.10) with Morse index one. Finally, as an application of Morse theory, we will prove a uniqueness property of Morse index one solutions. 3.2.1 Existence of a positive solution by the mountain pass theorem The standard regularity assumptions on f usually required are that f is a Carathéodory function (see [14], Chapter 1, or [196] for the definition) and that it is locally Hölder continuous. However, since our aim is to study the Morse index of solutions, we will always assume that f : Ω × ℝ 󳨀→ ℝ is differentiable in the second variable and (3.2) holds. Let us start observing that if f satisfies the following superlinear condition: f (x, s) 𝜕f (x, s) > 𝜕s s

∀x ∈ Ω, ∀s ∈ ℝ \ {0}

(3.11)

then every solution u of (3.10) has Morse index m(u) greater than or equal to one. Indeed, multiplying the equation (3.10) by u and integrating on Ω we have ∫|∇u|2 dx = ∫ f (x, u)u dx. Ω

Ω

Hence, testing the quadratic form Qu defined in (3.4) on the function u itself we get Qu (u) = ∫[f (x, u) − Ω

𝜕f (x, u)u]u dx = 𝜕s

∫ Ω∩[u=0] ̸

[

f (x, u) 𝜕f − (x, u)]u2 dx. u 𝜕s

Thus if (3.11) holds and u does not vanish in Ω we have Qu (u) < 0 so that m(u) ≥ 1. If u changes sign, we can apply the same procedure in each nodal region of u, i. e., in each connected component of the set 𝒵 (u) = { x ∈ Ω : u(x) ≠ 0 } obtaining that the restriction of u to each nodal domain, extended to zero to the whole Ω, gives a direction in the space H01 (Ω) where the quadratic form Q(u) is negative. This implies that, if (3.11) holds, then m(u) ≥ n(u)

(3.12)

where n(u) denotes the number of nodal regions of the solution u. Now we will apply the mountain pass theorem (Theorem 2.22 of Chapter 2) to find a positive solution of (3.10).

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88 | 3 Morse theory for semilinear elliptic equations

(i)

We assume that f (x, s) satisfies the following additional hypotheses: 2N

there exists a1 ∈ L N+2 (Ω) and a2 > 0 such that 󵄨 󵄨󵄨 p 󵄨󵄨f (x, s)󵄨󵄨󵄨 ≤ a1 (x) + a2 |s| ∀(x, s) ∈ Ω × ℝ N +2 for 1 < p < = 2∗ − 1 if N ≥ 3, p > 1 if N = 2 N −2 2N 2N 󸀠 (2∗ = is the critical Sobolev exponent and (2∗ ) = ); N −2 N +2

(3.13)

(ii) f (x, s) = o(|s|) as s → 0, uniformly in x ∈ Ω;

(3.14)

(iii) there exist α > 2 and r ≥ 0 such that for |s| ≥ r 0 < αF(x, s) ≤ sf (x, s) ∀x ∈ Ω

(3.15)

where F is the primitive of f which appears in (3.5). The assumption (3.13) which states that f has subcritical growth is needed to prove the Palais–Smale compactness condition for the functional J in (3.5) (with Γ = 0). The hypotheses (3.14) and (3.15) imply that the functional J satisfies the geometrical conditions of the mountain pass theorem. We have the following. Theorem 3.4. If f satisfies (3.2), (3.13)–(3.15) then the Dirichlet problem (3.10) has a nontrivial classical positive solution u. Moreover, the Morse index m(u) is less than or equal to one and if (3.11) holds m(u) = 1. Proof. The existence of a nontrivial weak solution u of (3.10) can be obtained by a standard application of the mountain pass theorem to the functional J(u) =

1 ∫|∇u|2 dx − ∫ F(x, u) dx 2 Ω

(3.16)

Ω

in the space H01 (Ω). The proof can be found in many books on variational methods (e. g., in [196], Theorem 2.15 or in [212], Theorem 6.2). Then, by standard regularity theorems, it is easy to show that u belongs to C 2 (Ω)∩C 0 (Ω) and it is a classical solution of (3.10). In order to get a positive solution, we consider the function f (x, s) if s ≥ 0

f + (x, s) = {

0

if s < 0

,

x∈Ω

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3.2 Positive solutions of Dirichlet problems | 89

and observe that the functional J + (u) =

1 ∫|∇u|2 dx − ∫ F + (x, u) dx, 2 Ω

Ω

where F is the primitive of f such that F (0) = 0, satisfies the same hypotheses as the functional J. Thus, applying the mountain pass theorem, we get a critical point ũ of J + in H01 (Ω). It is easy to see that ũ ≥ 0, because using ũ − as a test function in the equation: (J + )󸀠 (u)̃ = 0 we get +

+

+

󵄨 󵄨2 ∫󵄨󵄨󵄨∇ũ − 󵄨󵄨󵄨 dx = ∫ f + (x, u)̃ ũ − dx = 0.

Ω

Ω

Then, again by regularity theorems, ũ is a classical solution of (3.10) and, by the strong maximum principle, ũ > 0 in Ω. By Theorem 2.23 of Chapter 2, we know that at the mountain pass critical level there exists a critical point u whose Morse index is less than or equal to one. Finally, if (3.11) holds we know the Morse index m(u)̃ must be greater than or equal to one. Hence m(u)̃ = 1. An important example of a nonlinearity f (x, s) which satisfies all hypotheses of Theorem 3.4, as well as (3.11) is f (x, s) = |s|p−1 s, with 1 < p < N+2 if N ≥ 3, 1 < p if N = 2. N−2 In this case, (3.10) becomes the famous Lane–Emden problem: −Δu = |u|p−1 u in Ω { u=0 on 𝜕Ω

(3.17)

3.2.2 Existence of a positive solution by constrained minimization Another way of getting a positive solution of (3.10) is by looking for critical points of the functional J in (3.16) on a suitable Hilbert manifold in the space H01 (Ω). To this aim, let us observe that any critical point of J belongs to the Nehari manifold 𝒩 defined by {

1

}

2

󸀠

𝒩 = { v ∈ H0 (Ω) \ {0} : ⟨J (v), v⟩ = ∫|∇v| dx − ∫ f (x, v)v dx = 0 } .

{

Ω

Ω

}

We will show that, under some hypotheses on the function f , the set 𝒩 is a C 1 -Hilbert manifold of codimension one in H01 (Ω) and is a natural constraint for the functional J, in the sense that any critical point of the restriction of J on 𝒩 is a critical point of J in the whole space H01 (Ω). Proposition 3.5. Let us assume that f satisfies (3.11), (3.13)–(3.15). Then 𝒩 is a C 1 -Hilbert manifold of codimension one and:

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90 | 3 Morse theory for semilinear elliptic equations (i) ∃r > 0 such that Br ∩ 𝒩 = 0; (ii) any critical point of J|𝒩 is a critical point of J on H01 (Ω); (iii) for any u ∈ H01 (Ω) \ {0} there exists t(u) > 0 such that t(u)u ∈ 𝒩 . Proof. Let us set G(v) = ⟨J 󸀠 (v), v⟩ so that 𝒩 is the zero level set of G. Clearly, G is of class C 1 on H01 (Ω) and, for any u ∈ 𝒩 , we have ⟨G󸀠 (u), u⟩ = J 󸀠󸀠 (u)(u, u) + ⟨J 󸀠 (u), u⟩ = J 󸀠󸀠 (u)(u, u) = Qu (u) < 0 by (3.11), as we have shown before. Hence G󸀠 (u) ≠ 0, for any u ∈ 𝒩 which means that 𝒩 is a C 1 -Hilbert manifold of codimension one. The manifold 𝒩 separates H01 (Ω) into two components and, by (3.13), (3.14) the one which contains the origin also contains a small ball around the origin, so that (i) holds. Let z be a critical point of J restricted to 𝒩 , then by the Lagrange multiplier theorem (see [150]) there exists λ ∈ ℝ such that J 󸀠 (z) = λG󸀠 (z). Taking the scalar product with z we get (J 󸀠 (z), z) = λ⟨G󸀠 (z), z⟩ which is impossible unless λ = 0, since z ∈ 𝒩 . Thus 𝒩 is a natural constraint for the functional J. To prove (iii), let us observe that if we fix u ∈ H01 (Ω), with ‖u‖ = 1 then, by the hypotheses on f , G(tu) > 0 for t > 0 and t small, while G(tu) < 0 for t large. Therefore, there exists t(u) > 0 such that t(u) ⋅ u ∈ 𝒩 . Remark 3.6. It could be proved that, for any u ∈ H01 (Ω) \ {0}, the number t(u) in (iii) is unique and t(u)u corresponds to the point where the functional Φ(t) = J(tu) achieves its maximum (see [225]). The properties of the Nehari manifold allow to prove the following existence theorem. Theorem 3.7. Let f satisfy (3.11), (3.13)–(3.15) and 1 f (x, s)s − F(x, s) ≥ c, 2

∀x ∈ Ω, s ∈ ℝ

(3.18)

for some constant c ∈ ℝ. Then the Dirichlet problem (3.10) has a nontrivial classical positive solution with Morse index equal to one. Proof. Let us start by proving that the functional J is bounded from below on the Nehari manifold 𝒩 . Indeed for any u ∈ 𝒩 , by (3.18), we have 1 J(u) = ∫[ uf (x, u) − F(x, u)] dx ≥ c|Ω|. 2 Ω

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3.2 Positive solutions of Dirichlet problems | 91

As in Theorem 3.4, we have that J satisfies the Palais–Smale compactness condition (see e. g. [196]), consequently the infimum of J on 𝒩 is achieved at a point u ∈ 𝒩 (see e. g. [14], Theorem 7.12). Then u is a critical point of J|𝒩 , and hence by (ii) of Proposition 3.5 it is also a critical point of J in H01 (Ω) and so a weak solution of (3.10). Proceeding as in the proof of Theorem 3.4, i. e.,using the modified functional J + , we get a positive classical solution. Since u is a minimizer of a C 2 -functional on a Hilbert manifold of codimension one, we have that the Morse index m(u) is less or equal than one. Indeed, denoting by T(u) the tangent space of the manifold 𝒩 at u we have that J 󸀠󸀠 (u)(v, v) ≥ 0

∀v ∈ T(u).

(3.19)

This holds because for every v ∈ T(u) we consider a C 1 -curve γ: [−1, 1] 󳨀→ 𝒩 such that γ(0) = u and γ 󸀠 (0) = v. Since J 󸀠 (u)(v) = 0, we get that J ∘ γ: [−1, 1] 󳨀→ ℝ is twice differentiable at t = 0 with derivative 󵄨󵄨 𝜕2 󵄨 (J ∘ γ)󵄨󵄨󵄨 = ⟨J 󸀠󸀠 (u)v, v⟩. 2 󵄨󵄨t=0 𝜕t 2

Recalling that J(u) is the minimum of J on 𝒩 , we infer that 𝜕t𝜕 2 (J ∘ γ)|t=0 ≥ 0. Then m(u) ≤ 1, since T(u) has codimension one and using (3.11) we get m(u) = 1, as in Theorem 3.4. Definition 3.8. A solution u of (3.10) which minimizes the functional J on the Nehari manifold 𝒩 is called least-energy solution. Since all solutions of (3.10) belong to 𝒩 , this is equivalent to say that u has the least energy among all solutions of (3.10). that

Let us denote by α the minimum of J on the Nehari manifold 𝒩 . It could be proved α = inf max J(γ(t)) γ∈Γ t∈[0,1]

where Γ = { γ ∈ C 0 ([0, 1], H01 (Ω)) : γ(0) = 0, J(γ(1)) < 0 } (see [225], Theorem 4.2). This interesting min-max characterization of α also would allow to prove that a minimizer u of J on 𝒩 has Morse index m(u) ≤ 1 (and hence m(u) = 1 if (3.11) holds), with the same arguments used to prove that a mountain pass critical point has Morse index less than or equal to one (see Section 2.2.3 of Chapter 2).

3.2.3 Uniqueness of solutions of Morse index one of Lane-Emden problems Let us consider the Lane–Emden Dirichlet problem (3.17) for positive solutions and its more general version obtained by adding a linear perturbation:

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92 | 3 Morse theory for semilinear elliptic equations −Δu = up + λu in Ω { { { u>0 in Ω { { { on 𝜕Ω {u = 0

(3.20)

where Ω is a bounded smooth domain in ℝN , N ≥ 2, λ ∈ ℝ and p > 1. An important question related to the study of (3.20) is to know whether the solution is unique, whenever it exists. It is not difficult to provide cases when (3.20) admits more than one solution, as in the case of the annulus or more general annular domains (see, e. g., [58, 129, 157]) or in some nonconvex domains as the ones dumb-bell shaped (see [86]). In the case of convex domains, a conjecture has been formulated, with its roots in the papers [86] and [122]. Conjecture 1. If Ω is a bounded and convex domain, then there exists only one solution of (3.20) for the full range of λ and p > 1 for which solutions exist. Note that a necessary condition to have a solution of (3.20) is λ < λ1 (Ω) where λ1 (Ω) is the first eigenvalue of the operator −Δ in H01 (Ω). This is easily proved by multiplying (3.20) by the first eigenfunction of −Δ and integrating. Moreover, no solutions of (3.20) exist if N ≥ 3, Ω is star-shaped, p ≥ N+2 and λ ≤ 0 N−2 as a consequence of the famous Pohozaev’s identity [192]. N+2 On the other side, if λ < λ1 (Ω), if N ≥ 2 or if N ≥ 3 and 1 < p < N−2 , a solution of (3.20) always exists, using the mountain pass theorem or the constrained minimization method described in the previous section. Let us also observe that, if 0 < p < 1, then uniqueness holds in any smooth bounded domain [44, 46]. When Ω is a ball the conjecture has been proved for the full range of the values of λ and p for which existence holds, mainly exploiting ODE techniques. Indeed, the famous symmetry result of Gidas, Ni, Nirenberg [122] asserts that every positive solution of (3.20) is radial and radially decreasing (see Chapter 6) so that (3.20) can be rewritten as an ordinary differential equation. When λ = 0, the proof is quite easy and can be obtained by rescaling arguments and the uniqueness for O. D. E. initial value problem (see also [122]). Instead, when λ ≠ 0 the uniqueness in the ball is much more difficult to obtain and the complete result is spread in several papers [4, 5, 182, 209, 226]. When Ω is not a ball, only few results are available. In the case λ = 0, some are of perturbative type like that of [228] for domains close to a ball or that of [133] where the exponent p is close to N+2 in dimension N ≥ 3 and the domain Ω is assumed to N−2 be symmetric and convex in N orthogonal directions. We also quote [67] for a partial result in star-shaped domains. As regards general results, the only ones to our knowledge are those contained in [74, 86, 158], again for the case λ = 0. In [74] and [86] the case of domains in ℝ2 , symmetric and convex in two orthogonal directions are considered and, by different methods, it is proved that only one positive solution exists for any exponent p > 1.

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3.2 Positive solutions of Dirichlet problems | 93

Instead in the paper [158] only the uniqueness of the least energy solution in convex domains in the plane is proved. However, the proof is very interesting and applies more generally to solutions of Morse index one. Since it is a nice application of Morse theory, we will detail it below. Finally, let us quote two recent results for the case when Ω is a square in ℝ2 obtained by a “computer-assisted” proof; see [175, 176]. They apply also when λ ∈ [0, λ1 (Ω)] and seem to be the only ones available for λ ≠ 0. Now, following [158], we will show that in a convex domain in ℝ2 problem (3.20) with λ = 0 has only one positive solution with Morse index one. We start by proving some preliminary results. The first one holds in any bounded domain in any dimension and for every λ < λ1 (Ω). Lemma 3.9. Let Ω be any smooth bounded convex domain in ℝN , N ≥ 2, then there exists p0 > 1 such that (3.20) has only one positive solution for any p ∈ (1, p0 ] and for any λ < λ1 (Ω). Proof. If u1 and u2 are two distinct solutions, then w = u1 − u2 must change sign, otherwise the identity p−1 0 = ∫ u1 (−Δu2 − λu2 ) − u2 (Δu1 − λu1 ) dx = ∫ u1 u2 (up−1 2 − u1 ) dx Ω

Ω

deduced from (3.20) would imply u1 ≡ u2 . Now let un be a solution of (3.20) with p = pn , pn ↘ 1 and set Mn = max un = un (Qn ), for some Qn ∈ Ω. By using the method of moving plane (see Chapter 6), we can show that, since Ω is convex, there exists a neighborhood V of 𝜕Ω such that Qn ∉ V, ∀n. p −1 We claim that Mn n 󳨀→ λ1 (Ω) − λ as n → +∞. p −1 p −1 First of all, we show that Mn n is bounded. Suppose that Mn n 󳨀→ +∞ and consider the function ũ n (x) = p

x+Q 1 un ( pn −1n ). Mn Mn 2

which satisfies the equation −Δũ n = ũ nn +

λ

pn −1

p −1 Mn n

ũ n in Mn 2 Ω − Qn .

By standard elliptic estimates, ũ n converges uniformly to a function ũ ∈ C 2 (K), for any compact set K in ℝN and ũ satisfies −Δũ = ũ { { { ũ > 0 { { { ̃ =1 {u(0)

in ℝN

in ℝN

Let λR and ΦR be respectively the first eigenvalue and the corresponding positive eigenfunction of the operator −Δ in the ball BR (0) with zero Dirichlet boundary condition.

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94 | 3 Morse theory for semilinear elliptic equations Using the Hopf’s boundary lemma and the fact that λR 󳨀→ 0 as R → +∞, for R large we have 0>

∫ ũ 𝜕BR (0)

𝜕ΦR ̃ R dx > 0 dσ = (1 − λR ) ∫ uΦ 𝜕𝜐 BR (0)

p −1

which is a contradiction and shows that Mn n is bounded. Thus, up to a subsequence, p −1 Mn n 󳨀→ μ. u Let ū n = Mn , which is a solution of the problem n

−Δū n = Mnpn −1 ū pnn + λū n { ū n = 0

in Ω on 𝜕Ω

By elliptic estimates, ū n converges in C 2 (Ω) ∩ C 0 (Ω) to a function ū which satisfies −Δū = μū + λū in Ω { { { ū > 0 in Ω { { { on 𝜕Ω {ū = 0

Hence μ + λ = λ1 (Ω) and ū = φ1 , the first eigenfunction of −Δ in Ω, so that μ = λ1 (Ω) − λ and the claim is proved. Now suppose that the assertion of the theorem is false, i. e., let us assume that un and vn are two distinct solutions of (3.20) with p = pn ↘ 1. Since ū n 󳨀→ φ1 uniformly, p −1 then ū pn −1 󳨀→ 1, and hence the convergence of Mn n to λ1 − λ implies that upnn −1 󳨀→ λ1 − λ uniformly in any compact set of Ω. p −1 Obviously, the same happens to the sequence v̄n n where v̄n =

wn =

un −vn ‖un −vn ‖ ∞

satisfy

−Δwn = gn wn + λwn { wn = 0 pn

vn . ‖vn ‖∞

The functions

in Ω on 𝜕Ω

pn

n where gn = uun −v 󳨀→ λ1 −λ for what we have proved before. Again by standard elliptic n −vn estimates we deduce that wn 󳨀→ φ1 uniformly. This is not possible since φ1 does not change sign while we have shown, at the beginning of the proof, that wn must change sign.

The next result applies to Morse index one solutions of (3.20) with λ = 0, in convex domains in the plane. It shows that this type of solutions are nondegenerate, i. e., zero is not an eigenvalue of the linearized operator (3.7) which, in the case of the Lane– Emden problem becomes Lu = −Δ − p|u|p−1 for a solution u of (3.20).

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(3.21)

3.2 Positive solutions of Dirichlet problems | 95

Theorem 3.10. Assume that Ω is a smooth bounded convex domain in ℝ2 and u is a classical solution of (3.20), for λ = 0, with Morse index m(u) = 1. Then u is nondegenerate. Proof. Since m(u) = 1, only the first eigenvalue of Lu is negative. Therefore, arguing by contradiction, we assume that the second eigenvalue λ2 = λ2 (Lu ) is zero. Then a corresponding eigenfunction φ2 solves −Δφ2 − pup−1 φ2 = 0

{

φ2 = 0

in Ω on 𝜕Ω

(3.22)

Consider a point Q = (x0 , y0 ) ∈ ℝ2 which will be chosen later and the function: w(x, y) = (x − x0 )

𝜕u 𝜕u + (y − y0 ) 𝜕x 𝜕y

which satisfies − Δw − pup−1 w = 2up

in Ω

(3.23)

Multiplying (3.20) by φ2 and (3.22) by u, integrating and subtracting, we get (p − 1) ∫ up φ2 dx = 0.

(3.24)

Ω

Instead, multiplying (3.22) by w and (3.23) by φ2 , integrating and subtracting we have −∫w 𝜕Ω

𝜕φ2 ds = −2 ∫ up φ2 dx = 0 𝜕𝜐

(3.25)

Ω

by (3.24). Now we observe that by Courant’s theorem, the nodal line l(φ2 ) = { x ∈ Ω : φ2 (x) = 0 } , of the eigenfunction φ2 divides Ω into two regions. Then either the closure of l(φ2 ) does not touch 𝜕Ω or it does at two points. If the first case happens, then we choose the point Q as any point inside Ω and we deduce that w(x, y) < 0

for any point (x, y) ∈ 𝜕Ω. 𝜕φ

Then by (3.25), we get a contradiction since 𝜕𝜐2 has only one sign on 𝜕Ω, and by Hopf’s boundary lemma cannot be identically zero. In the case when the closure of l(φ2 ) touches 𝜕Ω, then we denote by Pi ∈ 𝜕Ω, i = 1, 2, the two points in l(φ2 ) ∩ 𝜕Ω. Note that if they coincide we are back in the previous case. Again we distinguish two possibilities: either the tangent lines to 𝜕Ω

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96 | 3 Morse theory for semilinear elliptic equations at the points Pi are parallel or not. In the second case, we choose the point Q in the definition of the function w as the intersection of the two tangent lines and observe 𝜕φ that w and 𝜕𝜐2 both simultaneously change sign at Pi , i = 1, 2. This implies that the 𝜕φ

product w 𝜕𝜐2 has only one sign on 𝜕Ω, and hence (3.25) leads again to a contradiction. If instead the two tangent lines are parallel, we assume that the x-axis is the common direction and set w = 𝜕u . Then, repeating the same argument as before, we obtain 𝜕x ∫w 𝜕Ω

𝜕φ2 ds = 0 𝜕𝜐

which is again a contradiction to the Hopf’s boundary lemma. We can now prove the uniqueness result. Theorem 3.11. Let Ω be a smooth bounded convex domain in ℝ2 . Then for any p > 1 and λ = 0 there exists only one solution of (3.20) with Morse index one. Proof. By Lemma 3.9, we know that there exists p0 > 1 such that (3.20) has a unique solution for p ∈ (1, p0 ). Obviously, this solution is the least energy solution which has Morse index one, and hence is nondegenerate by Theorem 3.10. Let (1, p)̄ be the maximal interval with this uniqueness property. If p̄ = +∞, the assertion is proved; otherwise, since all solutions are nondegenerate, we have that there is only one solution of Morse index one also for p = p.̄ This is a consequence of the implicit function theorem and of the fact that the Morse index of a solution does not change, as p varies, as long as it stays nondegenerate (since by Theorem 1.44 the eigenvalues depend continuously on the coefficients). Arguing by contradiction, let us assume that there exists a sequence pn ↘ p̄ and two distinct solutions un , vn of (3.20) with Morse index one for p = pn . By standard elliptic estimates, we have that un , vn both converge in C 2 (Ω) to the unique solution ū corresponding to p = p.̄ Set wn = un − vn

and

w̄ n =

wn . ‖wn ‖ H 1 (Ω)

(3.26)

0

Then w̄ n satisfies −Δw̄ n = αn (x)w̄ n

{

w̄ n = 0

in Ω on 𝜕Ω

(3.27)

1

where αn (x) = ∫0 pn (tun (x) + (1 − t)vn (x))pn −1 dt. Moreover, w̄ n 󳨀→ w̄ weakly in H01 (Ω) and w̄ ≠ 0. In fact by (3.27), we have 1 − ∫|∇w̄ n |2 dx = ∫ αn w̄ n2 dx = p̄ ∫ ū p−1 w̄ 2 dx + o(1) ̄

Ω

Ω

Ω

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(3.28)

3.3 Sign changing solutions of Dirichlet problems | 97

which implies w̄ ≠ 0. Passing to the limit in (3.27), we get −Δw̄ = p̄ ū p−1 w̄ { { { w̄ ≠ 0 { { { {w̄ = 0 ̄

in Ω (3.29)

in Ω on 𝜕Ω

which is a contradiction since we assumed ū to be nondegenerate. Remark 3.12. In the recent paper [99], using some uniform estimates obtained in [146], it is proved that when the exponent p is large all solutions of (3.20) in convex planar domain have Morse index one. Thus combining this result with Theorem 3.11, we get that the uniqueness conjecture is true for p > p1 , for some exponent p1 > 1. The proof of [99] relies on a careful analysis of the asymptotic behavior of solutions of (3.20) performed in [103, 104] and [98]. Remark 3.13. The proof of Theorem 3.11 relies on the nondegeneracy of the solutions and on the fact that the uniqueness holds when the exponent p is close to one. This last fact is true regardless of the Morse index and also in higher dimension, while the proof of the nondegeneracy strongly uses the fact that the Morse index of the solution is one and that the domain is planar.

3.3 Sign changing solutions of Dirichlet problems In this section, we consider again the general semilinear elliptic problem (3.10) and study the existence and the Morse index of sign changing solutions. In the case when the nonlinear term is superlinear, but not necessarily odd, the existence of sign changing solutions in general bounded domains is not obvious. The first result is due to Castro, Cassio and Neuberger in [57], then other results have been obtained in [20] and in [23]. In the next section, we will use the approach of [23] to prove the existence of a least-energy sign changing solution and show that its Morse index is two. In the other sections, we will study the case of symmetric solutions to give an estimate of their Morse index. Often we will refer to a sign changing solution as a nodal solution since it must have at least two nodal domains.

3.3.1 Existence of a solution with Morse index two by constrained minimization We study the Dirichlet problem (3.10) under some hypotheses on the function f (x, s). Some of them are the same as for the case of positive solutions but for the reader’s convenience we prefer to state precisely what is needed for the existence of sign changing solutions.

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98 | 3 Morse theory for semilinear elliptic equations So let us consider the problem (3.10) where Ω is a smooth bounded domain in ℝN , N ≥ 2. We assume that the function f (x, s) satisfies the following assumptions: (i) f ∈ C 1 (Ω × ℝ, ℝ),

f (x, 0) = 0

∀x ∈ Ω;

(3.30)

(ii) there exists p ∈ (1,

(iii)

N +2 ) if N ≥ 2 or p > 1 if N = 2, N −2

󵄨󵄨 𝜕f 󵄨󵄨 󵄨 󵄨 such that 󵄨󵄨󵄨 (x, s)󵄨󵄨󵄨 ≤ c(1 + |s|p−1 ) ∀x ∈ Ω, s ∈ ℝ; 󵄨󵄨 𝜕s 󵄨󵄨 f (x, s) 𝜕f (x, s) > 𝜕s s

(iv)

∀ x ∈ Ω, s ≠ 0;

(3.31)

(3.32)

there exist R > 0 and α > 2 such that 0 < αF(x, s) ≤ sf (x, s), s

∀x ∈ Ω, |s| ≥ R

(3.33)

where F(x, s) = ∫ f (x, s) ds is a primitive of f; 0

(v) the second Dirichlet eigenvalue μ2 of the operator −Δ−

𝜕f (x, 0) 𝜕s

on Ω is positive.

(3.34)

Note that (3.32) is the same as (3.11) which is important to estimate the Morse index of a solution of (3.10). We consider again the functional J associated to (3.10) J(v) =

1 ∫|∇v|2 dx − ∫ F(x, v) dx, 2 Ω

Ω

v ∈ H01 (Ω).

(3.35)

Then we define the set: M = { u ∈ H01 (Ω) : u+ ≠ 0, u− ≠ 0, ⟨J 󸀠 (u), u+ ⟩ = ⟨J 󸀠 (u), u− ⟩ = 0 } where u+ and u− are the positive and negative part of u, i. e., u+ = max{u, 0}, u− = max{−u, 0}. The set M is not a manifold in H01 (Ω) but we will show that M ∩ H is a manifold where H = H01 (Ω) ∩ H 2 (Ω), endowed with the scalar product from H 2 (Ω). Obviously, M contains all sign changing solutions of (3.10) and is contained in the Nehari manifold 𝒩 previously defined. It is often called the nodal Nehari set.

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3.3 Sign changing solutions of Dirichlet problems | 99

Proposition 3.14. The set M ∩ H is a C 1 -manifold of codimension two in H. For the proof of the above result, we need the following lemma whose proof is contained in [23].1 Lemma 3.15. Let us define the functionals: 󵄨 󵄨2 Q± (u) = ∫ ∇u ⋅ ∇u± dx = ± ∫󵄨󵄨󵄨∇u± 󵄨󵄨󵄨 dx Ω

Ω

Ψ± (u) = ∫ f (x, u)u± dx, Ω

u ∈ H01 (Ω). Then, setting f 󸀠 (x, s) = 𝜕f (x, s), we have: 𝜕s a) Q± is differentiable at u ∈ H, with derivative Q󸀠± ∈ H −1 (Ω) given by Q󸀠± (u)v = ± ∫ ((−Δu)v + ∇u ⋅ ∇v) dx. [±u>0]

b) Q± |H ∈ C 1 (H). c) Ψ± ∈ C 1 (H01 (Ω)) with derivative given by Ψ󸀠± (u)v = ± ∫ f 󸀠 (x, u)uv dx ±

∫ f (x, u)v dx [±u>0]

[±u>0]

In particular, Q󸀠+ (u)u− = Q󸀠− (u)u+ = Ψ󸀠+ (u)u− = Ψ󸀠− (u)u+ = 0 󵄨 󵄨2 Q󸀠± (u)u± = ± 2 ∫ ∇u∇u± dx = 2 ∫󵄨󵄨󵄨∇u± 󵄨󵄨󵄨 dx Ω

Ψ󸀠± (u)u±

󸀠

Ω

± 2

= ∫ f (x, u)(u ) dx ± ∫ f (x, u)u± dx. Ω

Ω

Proof of Proposition 3.14. Let us observe that M ∩ H is the intersection of the zero level sets of the functionals g± defined by g± (u) = ⟨J 󸀠 (u), u± ⟩ = Q± (u) − Ψ± (u),

u ∈ H,

(3.36)

i. e. M ∩ H = { u ∈ H : u+ ≠ 0, u− ≠ 0, g+ (u) = 0 = g− (u) }. 1 Note that in the paper [23] the authors make the convention that u− = min{u, 0} and, therefore, the lemma appears slightly different.

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100 | 3 Morse theory for semilinear elliptic equations The previous lemma implies that g± |H ∈ C 1 (H) and, since ∫Ω ∇u∇u± dx ∫Ω f (x, u)u± dx for u ∈ M ∩ H, we have 2 󵄨 󵄨2 g+󸀠 (u)u+ = ∫(󵄨󵄨󵄨∇u+ 󵄨󵄨󵄨 − f 󸀠 (x, u)(u+ ) ) dx,

g+󸀠 (u)u− = 0

2 󵄨 󵄨2 = ∫(󵄨󵄨󵄨∇u− 󵄨󵄨󵄨 − f 󸀠 (x, u)(u− ) ) dx,

g−󸀠 (u)u+ = 0

=

Ω

g−󸀠 (u)u−

Ω

for u ∈ M ∩ H. Hence the assumption (3.32) yields 2

g+󸀠 (u)u+ = ∫[∇u∇u+ − f 󸀠 (x, u)(u+ ) ] dx = ∫ [ Ω

[u>0]

f (x, u) 2 − f 󸀠 (x, u)](u+ ) dx < 0 u

and 2

2

g−󸀠 (u)u− = ∫[−∇u∇u− − f 󸀠 (x, u)(u− ) ] dx = ∫[−f (x, u)u− − f 󸀠 (x, u)(u− ) ] dx Ω

Ω

f (x, u) 2 − f 󸀠 (x, u)](u− ) dx < 0 = ∫ [ u [u 0. Then there exists ϵ > 0 and a homotopy h: (JXc+ϵ ∪ A) × [0, 1] 󳨀→ JXc+ϵ ∪ A such that: (i) ht (JXd ∪ A) ⊂ JXd ∪ A, ∀d ≤ c + ϵ, t ∈ [0, 1]; (ii) h1 (JXc+ϵ ∪ A) ⊂ JXc−ϵ ∪ A. The existence of a least-energy nodal solution is given by the following theorem. Theorem 3.17. Under the assumptions (3.30)–(3.34), the infimum β (see (3.37)) is achieved by a function ū which is a sign changing solution of (3.10) with Morse index two and exactly two nodal domains. Proof. We have assumed (3.34) but, in order to simplify the proof, we suppose the stronger hypothesis that the first eigenvalue μ1 of the operator −Δ − 𝜕f (x, 0) is positive. 𝜕s This is obviously the case when 𝜕f (x, 0) = 0, so that μ1 is just the first eigenvalue of 𝜕s −Δ. For the more general case, we refer to [23]. Let e1 be the first normalized Dirichlet eigenfunction of the operator −Δ − 𝜕f (x, 0) 𝜕s on Ω and consider the set Sr = { w ∈ E1⊥ , ‖w‖H 1 (Ω) = r } ⊂ H01 (Ω) 0

where E1 = ℝe1 is the space spanned by e1 . Observe that α = inf J(Sr ) > 0 and Sr ∩ A = 0 (see [20]). Setting γ = α2 , we consider the inclusion γ

jc : (JXc ∪ A, JX ∪ A) 󳨅→ (H01 (Ω), H01 (Ω) \ Sr )

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102 | 3 Morse theory for semilinear elliptic equations for any c ≥ γ, which is well-defined since Sr ∩ A = 0. It induces a homomorphism: γ

jc∗ : H 2 (H01 (Ω), H01 (Ω) \ Sr ) 󳨀→ H 2 (JXc ∪ A, JX ∪ A) where H ∗ (C, D) stands for the Alexander–Spanier cohomology of the pair with D ⊂ C with integer coefficients. Next, we prove that H 2 (H01 (Ω), H01 (Ω) \ Sr ) ≅ ℤ.

(3.38)

Indeed, the pair (H01 (Ω), H01 (Ω) \ Sr ) is the same as the product pair (E1 , E1 \ {0}) × (E1⊥ , E1⊥ \ Sr ). The Künneth theorem (see [207]) shows that H 2 (H01 (Ω), H01 (Ω) \ Sr ) ≅ H 1 (E1⊥ , E1⊥ \ Sr ) ≅ H̃ 0 (E1⊥ \ Sr ) ≅ ℤ which proves (3.38). Now we can define c̄ = inf{c ≥ γ : jc∗ is injective}. Then c̄ ≥ α, since JXc ∪ A ⊂ H01 (Ω) \ Sr , hence jc∗ = 0 for c < α. Next, we show c̄ ≤ β

(3.39)

with β given by (3.37). To prove this, let ϵ > 0 and choose u ∈ M such that J(u) < β + ϵ2 . Since M is contained in the Nehari manifold 𝒩 , by Remark 3.6 we have J(λu+ − μu− ) ≤ J(u) for every λ, μ ≥ 0. Now we recall that, by the assumption (3.33) on f (x, s), lim J(tu) = −∞.

t→0

Therefore, there exists some number R > 0 such that J(λu+ − μu− ) ≤ 0

whenever max{λ, μ} ≥ R.

Approximating u+ and u− with suitable functions v1 , v2 ∈ P, we get J(λv1 − μv2 ) ≤ β + ϵ J(λv1 − μv2 ) ≤ γ

for 0 ≤ λ, μ ≤ R,

if max{λ, μ} ≥ R.

Now we consider the sets 𝒞 = { λv1 − μv2 : 0 ≤ λ, μ ≤ R } ⊂ [v1 , v2 ] ⊂ X,

𝜕𝒞 = { λv1 − μv2 ∈ 𝒞 : min{λ, μ} = 0 or max{λ, μ} = R } where [v1 , v2 ] is the span of {v1 , v2 }.

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3.3 Sign changing solutions of Dirichlet problems | 103

We have the following inclusions: i

β+ϵ

(𝒞 , 𝜕𝒞 ) → 󳨅󳨀 (JX

γ

jβ+ϵ

∪ A, JX ∪ A) 󳨅󳨀󳨀→ (H01 (Ω), H01 (Ω) \ Sr ).

We claim that the induced map ∗ i∗ ∘ jβ+ϵ : H 2 (H01 (Ω), H01 (Ω) \ Sr ) 󳨀→ H 2 (𝒞 , 𝜕𝒞 )

is an isomorphism. We choose e2 ∈ E1⊥ with ‖e2 ‖H 1 (Ω) = 1 and consider the sets 0

𝒞1 = { λe1 + μe2 : |λ| ≤ R, 0 ≤ μ ≤ R } = (BR ∩ E1 ) × [0, R] ⋅ e2 ,

𝜕𝒞1 = { λe1 + μe2 : |λ| = R or μ ∈ {0, R} } .

Clearly, (𝒞 , 𝜕𝒞 ) can be deformed to (𝒞1 , 𝜕𝒞1 ) within (H01 (Ω), H01 (Ω) \ Sr ). This shows that ∗ i∗ ∘ jβ+ϵ is an isomorphism if, and only if, the inclusion i1 : (𝒞1 , 𝜕𝒞1 ) 󳨅→ (H01 (Ω), H01 (Ω) \ Sr ) induces an isomorphism. Now (𝒞1 , 𝜕𝒞1 ) ≅ (E1 ∩ BR , E1 ∩ SR ) × ([0, R]e2 , {0, R}e2 )

(H01 (Ω), H01 (Ω) \ Sr ) ≅ (E1 , E1 \ {0}) × (E1⊥ , E1⊥ \ Sr ) Since the inclusions (E1 ∩ BR , E1 ∩ SR ) 󳨅→ (E1 , E1 \ {0}),

([0, R] ⋅ e2 , {0, R ⋅ e2 }) 󳨅→ (E1⊥ , E1⊥ \ Sr ) induce isomorphisms on cohomology levels, the chain follows by the naturality of the Künneth maps. ∗ ∗ is injective, and thus c̄ ≤ β + ϵ, for ϵ > 0 Now, since i∗ ∘ jβ+ϵ is an isomorphism, jβ+ϵ arbitrary. Hence (3.39) holds. Next, we show that Sc̄ ⊄ A

(3.40)

where Sc̄ was defined as the set of the critical points of J at c̄ level. Indeed, if this was ̄ not true then by Lemma 3.16 we can take ϵ > 0 and a homotopy h: (JXc+ϵ ∪ A) × [0, 1] 󳨀→ ̄ c+ϵ ∗ ∗ ∗ JX ∪ A such that h1 ∘ jc−ϵ = j , where ̄ ̄ c+ϵ γ

γ

h∗1 : H 2 (JXc−ϵ ∪ A, JX ∪ A) 󳨀→ H 2 (JXc+ϵ ∪ A, JX ∪ A) ̄

̄

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104 | 3 Morse theory for semilinear elliptic equations γ

γ

̄ ̄ ∗ is induced by h1 : (JXc+ϵ ∪ A, JX ∪ A) 󳨀→ (JXc−ϵ ∪ A, JX ∪ A). Hence, since jc+ϵ ̄ ̄ is injective, jc−ϵ has to be injective as well. This however contradicts the definition of c,̄ so that (3.40) holds. Now let ū ∈ Sc̄ \ A, then ū is a sign changing solution of (3.10). In particular, ū belong to the nodal Nehari set M and, therefore, c̄ = J(u)̄ ≥ β, which together with (3.39) implies that ū is a minimum for (3.37). Let us show that the Morse index m(u)̄ is exactly 2. By (3.32), since ū changes sign, m(u)̄ ≥ 2 (see (3.12)). To show that m(u)̄ ≤ 2, let us observe that ū ∈ H, by elliptic regularity. Then, since by Proposition 3.14, we have that M ∩ H is a C 1 -manifold of codimension two, arguing exactly as in the proof of Theorem 3.7, we get that m(u)̄ ≤ 2. Finally, since ū changes sign and has Morse index two, by (3.12) ū must have exactly two nodal domains.

3.3.2 Estimates of Morse index for symmetric sign changing solutions: the autonomous case In this section, we consider smooth bounded domains Ω ⊂ ℝN , N ≥ 2, symmetric with respect to an hyperplane which, without loss of generality, can be assumed to pass through the origin. More precisely, we denote by Ti = { x = (x1 , . . . , xN ) ∈ ℝN , xi = 0 } ,

i ∈ {1, . . . , N}

(3.41)

the hyperplane orthogonal to the unit vector, ei = {0, . . . , 0, 1, 0, . . . , 0} and denote by Ω−i = { x ∈ Ω, xi < 0 }

and

Ω+i = { x ∈ Ω, xi > 0 }

(3.42)

the two subdomains in which Ω is divided by Ti . We assume that Ω is symmetric with respect to Ti , for some i = 1, . . . , N, so that Ω−i is just the reflection of Ω+i with respect to Ti . Then we consider the autonomous version of problem (3.10) which we rewrite as −Δu = f (u) in Ω

{

u=0

on 𝜕Ω

(3.43)

where f = f (x, s) = f (s) satisfies the hypothesis (3.30)–(3.34). As it will be clear from the proof, all results of this section strongly depend on the fact that the nonlinear term f does not depend on the x-variable. We consider a sign changing solution u which has the same symmetry as Ω, i. e., u is even in the xi -variable, i ∈ {1, . . . , N}. Note that such a symmetric solution of (3.43) always exists under the above assumptions on f , applying the variational methods of Section 3.3.1 in the subspace of H01 (Ω) given by the functions even in the xi -variable, i ∈ {1, . . . , N}.

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3.3 Sign changing solutions of Dirichlet problems | 105

In particular, we will consider radial solutions of (3.43) when Ω is either a ball or an annulus centered at the origin. The aim of this section is to prove estimates from below on the Morse index of symmetric sign changing solutions whose nodal sets does not touch 𝜕Ω, following the ideas of [7]. This will be based on the general fact that each symmetry direction for a nodal solution u and for the domain, as defined above, produces a negative eigenvalue for the linearized operator at u which we recall to be defined, in the case of the Dirichlet problem, as Lu = −Δ − f 󸀠 (u).

(3.44)

Moreover, from the estimate on the Morse index we will deduce information on the nodal set of the solution. Let us denote by λk the eigenvalues of the operator Lu in Ω with homogeneous Dirichlet boundary conditions and by μi the first eigenvalue of Lu in the domain Ω−i , i ∈ {1, . . . , N}. Thus, there exists a function ψi solution of −Δψi − f 󸀠 (u)ψi = μi ψi { { { ψi = 0 { { { {ψi > 0

in Ω−i

(3.45)

on 𝜕Ω−i in

Ω−i

Our proofs will be based on the study of the sign of μi when u is symmetric with respect to the xi -direction. Let us start pointing out the following properties (earlier considered in [70] for the case of the eigenvalues of the Laplacian). Proposition 3.18. If u is a solution of (3.43) even in the xi -variable, for some i ∈ {1, . . . , N} then the odd extension of ψi to Ω, defined by ψi (x) ψ̃ i (x) = { −ψi (x1 , . . . , xi−1 , −xi , xi+1 , . . . , xN )

if x ∈ Ω−i

if x ∈ Ω+i

is an eigenfunction for the linearized operator Lu with corresponding eigenvalue μi . Hence μi = λβ(i) for some β(i) ≥ 2. If u is even in k variables, say x1 , . . . , xk , 1 ≤ k ≤ N, then the eigenfunctions ψ̃ 1 , . . . , ψ̃ k are linearly independent so they produce k eigenvalues λβ(1) , . . . , λβ(k) of Lu in Ω. Proof. Since ψi vanishes on the hyperplane Ti , we have 𝜕ψi = 0, 𝜕xj

𝜕2 ψi =0 𝜕xj2

on Ti ∩ Ω for j ≠ i

and, by the equation in (3.45), 𝜕2 ψi =0 𝜕xi2

on Ti ∩ Ω.

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106 | 3 Morse theory for semilinear elliptic equations Hence, by the symmetry of u in the xi -variable, it follows easily that the functions ψ̃ i are Dirichlet eigenfunctions of Lu in Ω with corresponding eigenvalue μi = λβ(i) . Since ψ̃ i changes sign (it has precisely two nodal domains, namely Ω−i and Ω+i ), then β(i) ≥ 2. The second part of the assertion is a consequence of the oddness of ψ̃ i in the xi variable. The general statement on the Morse index of a symmetric solution is the following (see [7]). Theorem 3.19. Let Ω be symmetric with respect to the hyperplane Ti and convex in the xi -direction for some i ∈ {1, . . . , N} (i. e. if x and y belong to Ω and lay on the same line parallel to the xi -axis then the segment joining x and y is contained in Ω). If u is a sign changing solution of (3.43) even in the xi -variable and such that the closure of its nodal set N(u) = { x ∈ Ω : u(x) = 0 } does not intersect 𝜕Ω then the eigenvalue μi (see (3.45)) is negative. Moreover, for the Morse index m(u), it holds m(u) ≥ k + 2

(3.46)

where k ∈ {1, . . . , N} is the number of variables xi such that u is even in xi . Proof. Let us consider the function

𝜕u 𝜕xi

in Ω−i . By the symmetry of u in the xi variable,

𝜕u = 0 on 𝜕Ω−i ∩ Ti . On the other hand, since N(u) does not touch 𝜕Ω we we have that 𝜕x i have that u does not change sign near 𝜕Ω, so we can assume u > 0 near the boundary. Then the smoothness of 𝜕Ω and the assumption that Ω is convex in the xi direction imply that

𝜕u ≥0 𝜕xi In addition,

𝜕u 𝜕xi

satisfies Lu (

𝜕u 𝜕u 𝜕u ) − Δ( ) − f 󸀠 (u) = 0. 𝜕xi 𝜕xi 𝜕xi

Since u changes sign in Ω−i , necessarily

where in

Ω−i .

on 𝜕Ω−i .

𝜕u 𝜕xi

(3.47)

must change sign and be negative some-

Hence, there exists a domain D strictly contained in Ω−i such that 𝜕u < 0 in D 𝜕xi

and

𝜕u = 0 on 𝜕D. 𝜕xi

(3.48)

We deduce from (3.47)–(3.48) that the first eigenvalue of Lu in D is zero and as a consequence, the first eigenvalue μi of Lu in Ω−i is negative as we wanted to prove. Then (see Proposition 3.18) the corresponding eigenvalue λβ(i) = μi of Lu in Ω is negative, and to it there corresponds an eigenfunction ψ̃ i odd with respect to xi . On the other side, under the assumption (3.32) the first eigenvalue λ1 of Lu in Ω is negative. Moreover, since u is even in xi and changes sign also the second eigenvalue

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3.3 Sign changing solutions of Dirichlet problems | 107

of Lu in the space of even functions must be negative, again by (3.32). Summing up all the symmetric directions, we get that the Morse index m(u) satisfies (3.46). The assumption that Ω is convex in the xi -direction is not essential, but simplifies the proof otherwise other parts of 𝜕Ω−i , neither on 𝜕Ω nor on Ti , should be considered. Below we will consider the case of an annulus and it will be clear how to deal with domains which have holes. In particular, Theorem 3.19 applies to nodal radial solutions in the ball which are even in all variables x1 , . . . , xN and whose nodal sets do not touch 𝜕Ω so that, using also Proposition 3.18, we deduce that their Morse index is at least N + 2. However, in the radial case, we can have a better estimate which also takes into account the number of nodal regions of u and the number of the radial negative eigenvalue of Lu . This has been derived in [106] in the case when Ω is a ball and also works when the domain is an annulus. Let Ω be a ball or an annulus centered at the origin and u a nodal radial solution of (3.43). We denote by mrad (u) the radial Morse index of u, i. e., the number of negative radial eigenvalues of the linearized operator Lu . Theorem 3.20. If Ω is a ball or an annulus in ℝN , N ≥ 2 and u is a radial solution of (3.43) with n = n(u) ≥ 2 nodal domains then m(u) ≥ mrad + N(n − 1).

(3.49)

Since the assumption (3.32) implies that mrad (u) ≥ n (as for (3.11), (3.12)), then from (3.49) we deduce m(u) ≥ n + N(n − 1).

(3.50)

As for Theorem 3.19, the proof is based on using the partial derivatives of u to produce negative eigenvalues whose corresponding eigenfunctions are odd with respect to a hyperplane passing through the origin. However, since we now consider also the case of an annulus and want to get a more precise estimate we need some extra work. Proof of Theorem 3.20. For any i = 1, . . . , N, we consider the hyperplanes Ti and the domains Ω−i as defined in (3.41), (3.42). Note that now Ω−i is a half-ball or a half-annulus determinated by Ti , for each i = 1, . . . , N. Let us fix n ∈ ℕ, n ≥ 2 and let us denote by un a radial solution of (3.43) having n nodal domains. We denote by A1 , . . . , An the nodal regions of un counting them starting from the outer boundary in such a way that 𝜕A1 contains 𝜕Ω, if Ω is a ball, or the outer boundary of Ω, if Ω is an annulus. Since un is radial we have that Aj are annuli for j ∈ {1, . . . , n − 1} while An is a ball if Ω is a ball or another annulus if so is Ω.

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108 | 3 Morse theory for semilinear elliptic equations Let us first consider the case of the ball so that Aj = { x ∈ Ω : Rj+1 < |x| < Rj } ,

j = 1, . . . , n − 1

An = { x ∈ Ω : |x| < Rn }

where Rj , j = 2, . . . , n are the nodal radii and R1 is the radius of the ball Ω. tion

We consider the derivatives

𝜕un , 𝜕xi

Lun (

i = 1, . . . , N, which, as in (3.47), satisfy the equa𝜕un )=0 𝜕xi

in Ω.

(3.51)

on Ω ∩ Ti .

(3.52)

Using the symmetry of un , we have 𝜕un =0 𝜕xi Then we consider the half-nodal regions A−i,j = Aj ∩ Ω−i ,

j = ⋅ ⋅ ⋅ , n and i = 1, . . . , N.

To simplify the notation, let us fix i = 1 and focus on the function

A−1,j ,

that we simply denote by

A−j ,

whatever we prove for

𝜕un , 𝜕xi

𝜕un 𝜕x1

(3.53) 𝜕un 𝜕x1

in the sets

will hold, with obvious

changes, for the other derivatives i = 2, . . . , N. Let us observe that for each nodal region Aj , writing un (r) = un (|x|) there exists at least one value rj ∈ (Rj , Rj+1 ), j = 1, . . . , n − 1, such that dun (r ) = 0. dr j

(3.54)

Notice that if the nonlinearity f = f (s) satisfies the condition sf (s) ≥ 0 then rj is the unique radius in (Rj , Rj+1 ) such that (3.54) holds in Aj , j = 1, . . . , n − 1. Then, since un is radial we have that

𝜕un 𝜕x1

≡ 0 on the spheres

Sj = { x ∈ ℝN : |x| = rj } ,

j = 1, . . . , n − 1.

(3.55)

Let us fix one rj ∈ (Rj , Rj+1 ) for each j = 1, . . . , n − 1 (i. e., just one value of the radius in the interval (Rj , Rj+1 ) such that (3.54) holds) and consider the sets Nj− = { x ∈ ℝN : rj > |x| > rj−1 } ∩ Ω−1 ,

j = 1, . . . , n − 2

We observe that for j = 1, . . . , n − 2, by (3.51) and (3.54) we have 𝜕u { { L ( n ) = 0 in Nj− { { un 𝜕x1 { { { { 𝜕un = 0 on 𝜕Nj− { 𝜕x1 Brought to you by | University of Bath Authenticated Download Date | 8/4/19 9:36 AM

(3.56)

3.3 Sign changing solutions of Dirichlet problems | 109

Thus

𝜕un 𝜕x1

is an eigenfunction of the linearized operator Lun in Nj− corresponding to the

zero eigenvalue which is the first one or a higher one according to the fact that Nj− .

changes sign or not in Moreover, also in the set

𝜕un 𝜕x1

− Nn−1 = { x ∈ ℝN : rn−1 > |x| ≥ 0 } ∩ Ω−1

the function

𝜕un 𝜕x1

− satisfies (3.56) (for j = n − 1). Hence also in the set Nn−1 zero is an 𝜕u

eigenvalue for Lun with corresponding eigenfunction 𝜕xn . 1 In conclusion, we have obtained (n − 1) adjacent regions where an eigenvalue of − Lun is zero. This implies that in the domain N − = ⋃n−1 j=1 Nj the hth eigenvalue λh of Lun is zero for some h ≥ n − 1. Since N − is strictly contained in Ω−1 , by construction we have that the hth eigenvalue λh of Lun in Ω−1 is negative for some h ≥ n − 1. In particular, λn−1 = λn−1 (Lun ) < 0 in Ω−1 and so are all λk = λk (Lun ) in Ω−1 for k ≤ n − 1. Reflecting by oddness with respect to T1 , the corresponding eigenfunctions we get eigenfunctions of Lun in the whole Ω corresponding to the same (n − 1) negative eigenvalues λk , k = 1, . . . , n − 1. Repeating the same arguments for all i = 1, . . . , N, we get at least (n − 1) negative eigenvalues λk (un ) in the domains Ω−i , for each i = 1, . . . , N, which give eigenvalues of Lun in the whole Ω whose corresponding eigenfunctions are odd with respect to Ti , i = 1, . . . , N. Note that, by symmetry, λk (Lun , Ω−i ) = λk (Lun , Ω−s ) for i ≠ s, i, s = 1, . . . , N, k = 1, . . . , n − 1 but the corresponding eigenfunctions are linearly independent, because they are odd with respect to orthogonal axes. Hence the multiplicity of each eigenvalue λk of Lun in Ω is at least N and so we have got at least N(n − 1) negative eigenvalues. Since the eigenfunctions we have found are not radial, adding mrad (un ), we get the estimate (3.49) and then (3.50), by (3.32). The case when Ω is an annulus follows in a similar, slightly easier, way, since the only difference is that the last nodal domain An is an annulus, so that it does not need to be treated in a different way with respect to the other regions Aj , j = 1 . . . , n − 1. N+2 Remark 3.21. In the case when the nonlinearity in (3.43) is f (u) = |u|p−2 u, p ∈ (1, N−2 ) if N ≥ 3 and p ∈ (1, +∞) if N = 2, and the domain Ω is a ball, a result of [143] (see also [23] for N = 2) shows that for a radial solution u of (3.43) with n nodal domains the radial Morse index mrad (u) is exactly n. Hence, in this case, (3.49) and (3.50) are equivalent.

An interesting “symmetry breaking” result can be immediately deduced from Theorem 3.20.

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110 | 3 Morse theory for semilinear elliptic equations Corollary 3.22. If Ω is a ball or an annulus in ℝN , N ≥ 2, then a least energy nodal solution of (3.43) cannot be radial. Proof. By Theorem 3.17, we know that a least energy nodal solution has Morse index two, and hence, by (3.49), cannot be radial. More generally, the results of Theorem 3.19 allow to give information on the nodal set N(u) of a solution u of (3.43) which is symmetric in some directions and has a low Morse index. Corollary 3.23. Let Ω be symmetric and convex in k directions, say x1 , . . . , xk , k ∈ {1, . . . , N} and u be a solution of (3.43) with the nonlinearity f which is either convex or has its first derivative f 󸀠 convex. Assume further that u is even in the k variables x1 , . . . , xk and m(u) < k + 2.

(3.57)

Then the closure of the nodal set N(u) intersects 𝜕Ω. In particular, if Ω is a ball or an annulus, a least energy solution of (3.43) has this property. Proof. By Theorem 3.19, we have the estimate (3.46) which contradicts (3.57) unless N(u) intersects 𝜕Ω. If u is a least energy solution then, by Theorem 3.17 its Morse index m(u) is equal to 2. On the other side, by the symmetry results of Chapter 6 (see Definition 6.5, Theorem 6.20 and Theorem 6.22) under the convexity assumptions on the nonlinearity f (s), we know that any solution with Morse index less or equal to N is foliated Schwarz symmetric, which in particular, means that it is axially symmetric. Therefore, u is symmetric in (N − 1) directions, and hence N(u) ∩ 𝜕Ω ≠ 0; otherwise, by Theorem 3.19 we would have m(u) ≥ (N − 1) + 2 > 2. 3.3.3 Estimates of Morse index for symmetric sign changing solutions: the nonautonomous case The results of the previous section strongly rely on the fact that the equation in (3.43) is autonomous. Indeed, in the proofs of Theorem 3.19 and Theorem 3.20 it is used that 𝜕u the derivatives 𝜕x are solutions of the equation (3.47), which is obtained by (3.43) difi ferentiating with respect to xi . Of course, this is not true if the nonlinearity f depends on the x-variable. For this reason, the proofs of the above theorems do not extend to the general problem (3.10). In the special case when f (x, s) = |x|α f (s), α ≥ 0 and Ω is a bounded radially symmetric domain, some Morse index estimates have been obtained. First, in [179] the 2-dimensional case is considered and with a simple change of coordinates some bounds from below have been derived, showing also that the Morse

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3.3 Sign changing solutions of Dirichlet problems | 111

index goes to infinity as α → +∞. Later some partial results in all dimensions, including the case of some Schrödinger–Hénon systems have been obtained in [168]. Finally, in [12] more complete results are proved by studying a related singular eigenvalue problem. However, it is still an open question to understand whether Morse index bounds can be obtained for nodal solutions of (3.10) in the presence of general nonlinearities f (x, s). Here, we describe the results of [179] which are obtained in a simple way using the estimates for the autonomous case of Section 3.3.2. Let us consider the following problem: −Δu = |x|α f (u) in Ω

{

u=0

(3.58)

on 𝜕Ω

where α > 0, Ω ⊂ ℝ2 is either a ball or an annulus centered at the origin and f : ℝ 󳨀→ ℝ is C 1,β on bounded sets of ℝ. We also assume the following condition on f : f 󸀠 (s) >

f (s) s

∀s ∈ ℝ \ {0}

(3.59)

which is equivalent to (3.32) for our type of nonlinearities. In the case when f (u) = |u|p−1 u, with p > 1, (3.58) is the well-known Hénon equation studied in [144] −Δu = |x|α |u|p−1 u

{

u=0

x∈Ω

on 𝜕Ω

(3.60)

which has been extensively analyzed since the work of Ni [181]. A part from its mathematical interest, it appears in several applications, in particular in astrophysics [144, 178]. The existence of a positive radial solution of (3.60) is obtained in [181] when Ω is a ball centered at zero in ℝN , N ≥ 3 and 1 < p < N+2+2α or N = 2 and p = 1, by using N−2 the mountain pass theorem. For the same range of exponents, using the arguments of [23], it is possible to prove the existence of a nodal radial solution, which has the least energy among all nodal radial solutions. In the case when Ω is an annulus, the same existence results hold for any p > 1, since no lack of compactness occurs in the setting of radial functions. For the more general problem (3.58) with Ω ⊂ ℝ2 , applying the results of [23] we have that a least energy nodal solution exists, also in general bounded domains, and has Morse index 2, if f satisfies (3.59) and the conditions: 󵄨 󵄨 f (0) = 0 and ∃p > 1 : 󵄨󵄨󵄨f 󸀠 (s)󵄨󵄨󵄨 ≤ c(1 + |s|p−1 ), ∀s ∈ ℝ,

(3.61)

∃R > 0, θ > 2 : 0 < θ ∫ f (t) dt ≤ sf (s), ∀|s| ≥ R.

(3.62)

s

0

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112 | 3 Morse theory for semilinear elliptic equations Therefore, in radial domains the question whether the least energy nodal solution is radial or not arises. As for the autonomous case, the answer will be deduced by an estimate of the Morse index for nodal radial solutions of (3.58). Indeed, we have the following result. Theorem 3.24. Let u be a radial sign changing solution of (3.58). Then the Morse index m(u) is greater than or equal to 3. Moreover, if (3.59) holds, then the Morse index of u is at least n(u) + 2, where n(u) denotes the number of nodal regions of u. As a consequence of this theorem, we get the following symmetry breaking result. Corollary 3.25. Assume (3.59), (3.61) and (3.62). Then any least energy nodal solution of (3.58) is not radially symmetric. Let us point out that Corollary 3.25 was already shown for the Hénon problem (3.60) for every dimension N ≥ 2 but only for particular values of α: for α large in [24] by a comparison of energy argument and for α small in [37, 38] by an asymptotic analysis as α → 0, of the least energy nodal solutions. In contrast with the symmetry breaking result of Corollary 3.25, we observe that least energy nodal solutions of (3.58) are foliated Schwartz symmetric by the results described in Chapter 6. The proof of Theorem 3.24 is obviously different from that of Theorem 3.19 and Theorem 3.20. It relies on a suitable change of variable which works well in ℝ2 and has already been used in [64] (see also [65, 127]). Thus, before proving Theorem 3.24, let us introduce the change of variable and describe its properties. For a point x = (x1 , x2 ) ∈ ℝ2 , let us denote by (r, θ) its polar coordinates, namely: x1 = r cos θ,

x2 = r sin θ,

r = √x12 + x22 .

So, for a function u defined in a domain of ℝ2 we can write u(x1 , x2 ) = u(r cos θ, r sin θ) = u(r, θ). We recall the following formulae: ∇x = (

𝜕 1 𝜕 𝜕 1 𝜕 𝜕 𝜕 , ) = (cos θ − sin θ , sin θ + cos θ ), 𝜕x1 𝜕x2 𝜕r r 𝜕θ 𝜕r r 𝜕θ |∇x |2 = (

2

2

2

2

𝜕 𝜕 𝜕 1 𝜕 ) +( ) = ( ) + 2( ) 𝜕x1 𝜕x2 𝜕r r 𝜕θ

(3.63)

𝜕2 𝜕2 𝜕2 1 𝜕 1 𝜕2 + = + + . 𝜕x12 𝜕x22 𝜕r 2 r 𝜕r r 2 𝜕θ2

(3.64)

and Δx =

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3.3 Sign changing solutions of Dirichlet problems | 113

In order to define a change of variable x 󳨃→ y in ℝ2 , we set y = (y1 , y2 ) ∈ ℝ2 with associate polar coordinates (s, σ), i. e., y1 = s cos σ,

y2 = s sin σ,

s = √y12 + y22

so that for a function v we write v(y1 , y2 ) = v(s cos σ, s sin σ) = v(s, σ). Then for a number k > 0, we consider the following transformation: Tk : ℝ2 󳨀→ ℝ2 ,

Tk (y) = y|y|k−1 ,

(3.65)

setting Tk (0, 0) = (0, 0) and x = Tk (y). In polar coordinates, the transformation Tk reads Tk (s, σ) = (sk , σ),

i. e. r = sk , θ = σ.

(3.66)

Some properties of this map are summarized in the following lemma. Lemma 3.26. We have: (i) Tk is a homeomorphism whose inverse is 1

Tk−1 x = x|x| k −1 ,

i. e. Tk−1 = T 1 .

(3.67)

k

(ii) In Cartesian coordinate, the Jacobian matrix of Tk is, ∀y ≠ 0, |y|2 + (k − 1)y12 𝜕(x1 , x2 ) k−3 (y) = |y| [ JTk (y) = 𝜕(y1 , y2 ) (k − 1)y y 1 2

(k − 1)y1 y2 |y|2 + (k − 1)y22

],

and 󵄨󵄨 󵄨 2k−2 . 󵄨󵄨det JTn (y)󵄨󵄨󵄨 = k|y|

(3.68)

(iii) Given a function ψ defined on a subset of ℝ2 , we set φ = ψ ∘ Tk−1 and x = Tk (y) for y ≠ 0. Then ψ is differentiable at y if and only if φ is differentiable at x. (iv) Let ψ, φ, x as before and r, s, σ and θ as in (3.66). Then (ψ2s +

1 1 2 2−2k ψ )s = k 2 φ2r + 2 φ2θ , s2 σ r

∀s ≠ 0

(3.69)

which implies that 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 min{1, k 2 }󵄨󵄨󵄨∇φ(x)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨∇ψ(y)󵄨󵄨󵄨 |y|2−2k ≤ max{1, k 2 }󵄨󵄨󵄨∇φ(x)󵄨󵄨󵄨

(3.70)

∀y ≠ 0. Moreover, if ψ is radially symmetric, then 󵄨 󵄨2 󵄨 󵄨2 k 2 󵄨󵄨󵄨∇φ(x)󵄨󵄨󵄨 = 󵄨󵄨󵄨∇φ(y)󵄨󵄨󵄨 |y|2−2k ,

∀y ≠ 0.

(3.71)

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114 | 3 Morse theory for semilinear elliptic equations Proof. The statements (i), (ii) and (iii) are just matter of computation. Regarding (iv), the identity (3.69) follows from (3.65). From (3.69), we infer that min{1, k 2 }(φ2r +

1 2 1 1 φ ) ≤ (φ2s + 2 φ2σ ) ≤ max{1, k 2 }(φ2r + 2 φ2θ ) r2 θ s r

which combined with (3.63) implies (3.70). If ψ is radially symmetric, it is also clear that (3.71) follows from (3.69) since ψσ ≡ 0 and φθ ≡ 0. Recalling that Ω ⊂ ℝ2 is either a ball or an annulus centered at the origin we set Ωk = Tk−1 (Ω). Lemma 3.27. Let 1 ≤ r < ∞. Then Sk : Lr (Ωk ) 󳨀→ Lr (Ω, |x|

2−2k k

defined by Sk ψ := ψ ∘ Tk−1 ,

),

is a continuous linear isomorphism such that 󵄨 󵄨r 󵄨 󵄨r 2−2k ∫ 󵄨󵄨󵄨ψ(y)󵄨󵄨󵄨 dy = k −1 ∫󵄨󵄨󵄨φ(x)󵄨󵄨󵄨 |x| k dx,

Ωk

Ω

with φ = ψ ∘ Tk−1 .

(3.72)

Proof. In the case when Ω is an annulus centered at the origin, (3.72) derives by the standard change of variable theorem, using (3.67) and (3.68). In the case when Ω = B(0, R) is a ball centered at the origin and radius R > 0, the singularity at zero of Tk or Tk−1 causes no problem, since we can reduce the arguments to the previous case by approximation with annuli. Indeed, 󵄨 󵄨 ∫ 󵄨󵄨󵄨h(z)󵄨󵄨󵄨 dz = lim+ δ→0

B(0,R)

∫ B(0,R)\B(0,δ)

󵄨󵄨 󵄨 󵄨󵄨h(z)󵄨󵄨󵄨 dz,

∀h ∈ L1 (B(0, R)).

Then the monotone convergence theorem, passing to the limit, gives the result for the ball. In the same way, we can prove the following lemma. Lemma 3.28. Let F: ℝ 󳨀→ ℝ be a continuous function. Then F ∘ ψ ∈ L1 (Ωk ) if, and only if, F ∘ φ ∈ L1 (Ω, |x|

2−2k k

) with φ = ψ ∘ Tk−1 . Moreover,

∫ F(ψ(y)) dy = k −1 ∫ F(φ(x))|x|

dx.

(3.73)

Ω

Ωk

We point out that if k = |x|α at (3.58) coincide.

2−2k k

2 , α+2

then

2−2k k

= α and so the weights |x|

2−2k k

at (3.73) and

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3.3 Sign changing solutions of Dirichlet problems | 115

Lemma 3.29. The map Sk : H01 (Ωk ) 󳨀→ H01 (Ω),

defined by Sk ψ := ψ ∘ Tk−1 ,

is a continuous linear isomorphism. Moreover, setting φ = ψ ∘ Tk−1 , we have 1 1 󵄨2 󵄨 󵄨2 󵄨2 󵄨 󵄨 min{k, } ∫󵄨󵄨󵄨∇φ(x)󵄨󵄨󵄨 dx ≤ ∫ 󵄨󵄨󵄨∇ψ(y)󵄨󵄨󵄨 dy ≤ max{k, } ∫󵄨󵄨󵄨∇φ(x)󵄨󵄨󵄨 dx k k Ωk

Ω

Ω

for all ψ ∈ H01 (Ωk ) and 󵄨 󵄨2 󵄨 󵄨2 k ∫󵄨󵄨󵄨∇φ(x)󵄨󵄨󵄨 dx = ∫ 󵄨󵄨󵄨∇ψ(y)󵄨󵄨󵄨 dy, Ω

Ωk

1 ∀ψ ∈ H0,rad (Ωk ).

Proof. Here, we use (3.67)–(3.71) and proceed as in the proof of Lemma 3.27. Now we consider the change of variable (3.65) restricted to radial functions. For a radial function u: Ω ⊂ ℝ2 󳨀→ ℝ, we define the radial function v: Ωk 󳨀→ ℝ by setting v(y) = u(Tk (y)), i. e., v(s) = u(sk ) = u(r),

r = sk , r = |x|, s = |y|.

(3.74)

Then an easy computation yields 1 1 vss (s) + vs (s) = k 2 s2k−2 (urr (sk ) + k yr (sk )), s s

s > 0.

So, using the previous notation in polar coordinates, since r = |x|, s = |y|, r = sk , we infer that 2

Δv(y) = k 2 |y|2k−2 Δu(Tk (y)) = k 2 |x|2− k Δu(x).

(3.75)

Hence, if u is a radial solution of the Hénon type equation (3.58), then v: Ωk 󳨀→ ℝ is a radial function that satisfies −Δv(y) = k 2 |y|2k−2+kα f (v(y)) y ∈ Ωk

{

v=0

on 𝜕Ωk

Thus if we choose k such that 2k − 2 + kα = 0,

i. e., k =

2 , α+2

(3.76)

then we infer that − Δv(y) = (

2

2 ) f (v(y)), α+2

y ∈ Ωk , v = 0 on 𝜕Ωk .

(3.77)

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116 | 3 Morse theory for semilinear elliptic equations 2 transforms radial solutions of the nonauThis means that the map Tk , for k = α+2 tonomous problem (3.64) into radial solutions of the autonomous problem (3.77). This will be very useful to estimate the Morse index of nodal solutions of (3.58), so to prove Theorem 3.24. To this aim, we consider the quadratic form associated to a solution u of (3.58), i. e.,

󵄨2 󵄨 Qu (φ) = ∫󵄨󵄨󵄨∇φ(x)󵄨󵄨󵄨 − ∫|x|α f 󸀠 (u)φ(x)2 dx, Ω

Ω

2 , α+2

and the one relative to v = u ∘ Tk , for k =

φ ∈ H01 (Ω)

i. e.,

2

2 󵄨 󵄨2 ) ∫ f 󸀠 (v)ψ(y)2 dy, Qv (ψ) = ∫ 󵄨󵄨󵄨∇ψ(y)󵄨󵄨󵄨 dy − ( α+2 Ωk

Ωk

ψ ∈ H01 (Ωk ).

The crucial point for the proof of Theorem 3.24 is the following result. Proposition 3.30. Let v, ψ ∈ H01 (Ωk ) and set u = v ∘ Tk−1 , φ = ψ ∘ Tk−1 . Then Qv (ψ) ≥

2 Q (φ), α+2 u

∀ψ ∈ H01 (Ωk )

(3.78)

and Qv (ψ) =

2 Q (φ), α+2 u

∀ψ radial in H01 (Ωk ).

Proof. It is a direct consequence of Lemma 3.28 and Lemma 3.29. Proof of Theorem 3.24. Let u be a radial nodal solution of (3.58) and v(y) the transα formed function defined in (3.74) for k = α+2 which is a nodal radial solution of (3.77). Observe that the eigenvalue problem for the linearized operator associated to (3.77) is 2

2 { {−Δψ − ( ) f 󸀠 (v)ψ = λψ in Ωk α + 2 { { on 𝜕Ωk {ψ = 0

(3.79)

2

Hence, if ψ is a radial eigenfunction for (3.79), the function φ defined by φ(s α+2 ) = ψ(s) is a radial eigenfunction for the eigenvalue problem 2

α+2 { {−Δφ − |x|α f 󸀠 (u)φ = λ( ) |x|α φ in Ω 2 { { on 𝜕Ω {φ = 0 by (3.75), (3.76).

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(3.80)

3.3 Sign changing solutions of Dirichlet problems | 117

We know, from [7] (see also Theorem 3.20) that the Morse index of v is at least 3 and greater than or equal to n(u) + 2, if (3.59) holds. More precisely, the problem (3.79) has a negative first eigenvalue λ1,rad to which there corresponds a radial eigenfunction ψ1,rad and two other negatives eigenvalues λ2 = λ3 with corresponding eigenfunction ψ2 and ψ3 . By the proof of Theorem 3.20 (see also Theorem 3.19), we have that ψ2 (y1 , y2 ) is even in y2 and odd in y1 , ψ3 (y1 , y2 ) is even in t1 and odd in y2 .

Hence, in particular, Qv (ψ1,rad ) < 0

and

Qv (ψi ) < 0, i = 2, 3.

Moreover, if (3.59) holds then the radial eigenvalues for (3.79) λi,rad are also negative, for i = 2, . . . , n(u). Let us denote by ψi,rad , i = 2, . . . , n(u), the associated radial eigenfunctions. As we have observed, the change of variable Tk guarantees that φi,rad defined by ψi,rad = φi (Tk (y)) with i = 1, 2, . . . , n(u) are radial eigenfunctions of (3.80) for λ = λi,rad . Even though φ2 and φ3 defined by φi (x) = ψ(Tk−1 (x)), i = 2, 3, are not eigenfunctions of (3.80), they correspond to directions in which the quadratic form induced by Qu is negative definite, which follows from (3.78). Using the symmetries of φ1,rad , . . . , φn(u),rad , φ2 , φ3 , it is easy to see that they are all mutually orthogonal with respect to both the bilinear forms (u, w) 󳨀→ ∫|x|α uw dx

and

Ω

(u, w) 󳨀→ ∫(∇u∇w − |x|α f 󸀠 (u)uw) dx. Ω

Therefore we deduce that Qu (w) < 0 for every nonzero w in the span [φ1,rad , φ2 , φ3 ] or for every nonzero w in the span [φ1,rad , . . . , φn(u),rad , φ2 , φ3 ] if (3.59) holds. This proves the assertion. Theorem 3.24 just proved gives an estimate on the Morse index of a nodal radial solution of (3.58), which is independent of the exponent α of the nonlinearity. It is an interesting question to see how the weight |x|α , and hence its exponent α influences the Morse index of a solution. In this direction, we describe the following result, also proved in [179]. Theorem 3.31. Let α > 0 be even and let u be a radial nodal solution of (3.58). Then u has Morse index greater than or equal to α + 3. If in addition (3.59) holds, then the Morse index of u is at least n(u) + α + 2. The proof of this theorem relies on a modification of the previous change of variable that works fine for the case when α is even. This change of variable is the key

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118 | 3 Morse theory for semilinear elliptic equations argument to prove the existence of many negative eigenvalues of the problem (3.80). A variant of it was used in [189] in higher dimensions to pass from doubly symmetric solutions of a supercritical problem in dimension 2m, m ≥ 2, to axially symmetric solutions of a subcritical problem in dimension m + 1. To prove Theorem 3.31, there is not a change of dimension but a somehow similar idea is applied to create a correspondence between eigenfunctions of linearized operator of two different problems. To the aim of proving Theorem 3.31, let us define this other change of variable in 2 ℝ which involves changing both polar coordinates r and θ. Given k > 0 and m ∈ ℕ, we set 2π ], m σ θ= . m

Tk,m : [0, ∞) × [0, 2π] 󳨀→ [0, ∞) × [0, Tk,m (s, σ) := (sk ,

σ ), m

r = sk ,

(3.81)

Obviously, Tk,1 is just Tk of (3.66). Consider any continuous function ψ defined on a radially symmetric domain Ω in ℝ2 in the Cartesian coordinates (y1 , y2 ). Then using the polar coordinates we can write ψ(y1 , y2 ) = ψ(s cos σ, s sin σ) = ψ(s, σ) and we set −1 φ(x1 , x2 ) = φ(r, θ) = ψ(Tk,m (r, θ)).

Hence φ is a function defined for θ ∈ [0, 2π ] which, since ψ(s, 0) = ψ(s, 2π), can be m extended 2π -periodically and continuously for all θ ∈ [0, 2π]. We still denote this exm tension by φ and we observe that, if it is smooth, by direct computation, we have 2 1 1 m2 k 2 1 k 2 r 2− k [φrr + φr + 2 φθθ ] = ψss + φs + 2 ψσσ . r s r s

Hence if we choose k =

1 , m

for the Laplacian in cartesian coordinates we have m−2 |x|2(1−m) Δφ(x) = Δψ(y).

(3.82)

In view of the relation (3.82) involving the Laplacian of φ and ψ, we will apply the above procedure to work with the Hénon type equations (3.58) in the case when α = 2(m − 1), with m ≥ 2, that is for every α even. Indeed α = 2(m − 1) ⇐⇒ k =

1 2 = m α+2

which coincides with the relation (3.76) between k and α. Note that, in the complex plane, the above transformation T 1 ,m is just the one 1

which sends z into z m , z ∈ C.

k

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3.3 Sign changing solutions of Dirichlet problems | 119

With the above choice of α, we consider a radial nodal solution u of (3.58). By Theorem 3.24, we know that u has Morse index greater than or equal to 3 and at least n(u) + 2 if (3.59) is also satisfied. We will use the change of variable (3.81) with k = m1 to construct α + 2 = 2m convenient nonradial directions on which the quadratic form Qu (w) is negative. Proof of Theorem 3.31. Let α = 2(m − 1), with m ≥ 2, k = m1 , and let u be a radial nodal solution of (3.58). Then, by (3.77) the radial function v = u ∘ Tk solves 1 {−Δv = 2 f (v) in Ωk m { v = 0 on 𝜕Ωk { Therefore, by results of [7], see Theorem 3.20, there exists two eigenfunctions ψ2 and ψ3 for the eigenvalue problem 1 󸀠 {−Δψ − 2 f (v)ψ = λψ in Ωk m { on 𝜕Ωk {ψ = 0

(3.83)

with the following properties: (i) the corresponding eigenvalues λ2 = λ3 are negative; (ii) ψ2 is even with respect to y2 and odd with respect to y1 , while ψ3 is even with respect to y1 and odd with respect to y2 ; (iii) ψ2 (y1 , y2 ) > 0 if y1 > 0, while ψ3 (y1 , y2 ) > 0 if y2 > 0. Next, applying the change of variables (3.81), we consider the functions φm,i (r, θ) = ψi ∘ T −1 (r, θ), i = 2, 3, extended by periodicity as before for all θ ∈ [0, 2π], so to have 1 ,m m

𝜕ψ

them defined on the whole Ω. Then, by the conditions 𝜕σ2 = 0 and ψ3 = 0 at σ = 0, we have that φm,i , i = 2, 3, are C 2 (Ω)-functions and by (3.82) they satisfy −Δφ − |x|α f 󸀠 (u)φ = λ|x|α φ in Ω

{

φ=0

(3.84)

on 𝜕Ω

with λ = λi m2 . Moreover, it is easy to see that both φm,i , i = 2, 3, have 2m nodal sets, π . This means that each one is a first each one being an angular sector of amplitude m eigenfunction of (3.84) in that sector with corresponding eigenvalue λi m2 < 0. In particular, φm,2 is the first eigenfunction in the sector Ωm,2 = { (x1 , x2 ) = (r cos θ, r sin θ) ∈ Ω, θ ∈ [−

π π , ]} 2m 2m

while φm,3 is the first eigenfunction in the sector Ωm,3 = { (x1 , x2 ) = (r cos θ, r sin θ) ∈ Ω, θ ∈ [0,

π ] }. m

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120 | 3 Morse theory for semilinear elliptic equations Then, by the monotonicity of the first eigenvalue with respect to the domain, we have that the first eigenvalue in Ωn,2 or Ωn,3 are also negative for every integer 1 ≤ n < m, Ωn,i defined as before, replacing m by n, for i = 2, 3. The corresponding eigenfunctions, say φn,i , extended by oddness with respect to the anticlockwise part of the boundary of Ωn,i and periodically, with angular period 2π , give rise to other two eigenfunction for (3.84), for every n ∈ {1, . . . , m}. By construcn tion, their symmetry or antisymmetry, all these pairs of eigenfunctions are mutually orthogonal with respect to both the bilinear forms (u, w) 󳨃󳨀→ ∫|x|α uw dx

and

Ω

(u, w) 󳨃󳨀→ ∫[∇u∇w − |x|α f 󸀠 (u)uw] dx Ω

so that we get 2m negative eigenvalues for (3.84) corresponding to nonradial directions. Counting also the first radial eigenvalue, which is negative, and the second, up to the n(u)-th radial eigenvalue which are also negative if (3.59) holds, we get the assertion, since α = 2(m − 1). In the particular case of the Hénon equation (3.60), the same change of variable can be used to prove the uniqueness of radial solutions, up to multiplication by −1, having n nodal sets. Moreover, it can be shown that the least energy nodal radial solution of (3.60) is nondegenerate in the space of radial function (see [179], Theorem 1.5). This is achieved by passing again to the autonomous problem and arguing as in Proposition 4 of [188]. This nondegeneracy result, together with the property, by Theorem 3.31, that the Morse index of nodal radial solutions tends to +∞ along sequences of even exponents α → +∞, indicates that there should be infinitely many branches of nonradial nodal solutions of (3.60) bifurcating from the nodal radial solutions. This should derive by arguments similar to those described in Chapter 5, where some applications of Morse theory to bifurcation are outlined.

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4 Morse index of radial solutions of Lane–Emden problems In this chapter, we will compute exactly the Morse index of radial solutions of the classical Lane–Emden problem −Δu = |u|p−1 u in B { u=0 on 𝜕B

(4.1)

where B is the unit ball of ℝN , N ≥ 2, centered at the origin 0 ∈ ℝN and 1 < p < ps with N+2 ps = N−2 if N ≥ 3 and ps = +∞ if N = 2. The results we present in the next sections hold either for p sufficiently close to ps = N+2 , if N ≥ 3 or for p sufficiently large if N = 2. N−2 As we will see, there is a substantial difference between the case N ≥ 3 and N = 2 which derives from the different asymptotic behavior of the solutions of (4.1) when p tends to the limit value ps . In order to compute the Morse index, we will use a spectral decomposition, for the linearized operator at a radial solution, which will be described in Section 4.1. The same approach can be used to study the Morse index of radial solutions in annuli, in particular to the aim of proving existence of nonradial solutions bifurcating from the radial ones. This will be shown in Chapter 5. We point out that, in the case of the annuli, Morse index estimates can be obtained by some nonexistence Liouville-type results for finite Morse index solutions (see [25]).

4.1 Spectral decomposition for the linearized operator at a radial solution In this section, we describe a way of decomposing the spectrum of the linearized operator at a radial solution which has been used in several papers to study similar problems in an annulus [21, 129, 160, 185]. Since in our case we are dealing with problem (4.1) in a ball, we use an approximation procedure by annuli with a small hole in order to avoid singularities at the origin [105, 106]. Most of the proofs of this section are taken from [21, 105, 106] and [129]. From now on, we denote by u a radial solution of (4.1). The interesting case will be when u changes sign. Indeed problem (4.1) admits only one positive solution, as recalled in Section 3.2.3, which is radial by the symmetry result of [122]. Since it is unique it must be the least energy solution (see Section 3.2.2) and, therefore, its Morse index is equal to one. This means that the linearized operator Lu : H 2 (B) ∩ H01 (B) → L2 (Ω) 󵄨 󵄨p−1 Lu (v) = −Δv − p󵄨󵄨󵄨u(x)󵄨󵄨󵄨 v

(4.2)

https://doi.org/10.1515/9783110538243-004

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122 | 4 Morse index of radial solutions of Lane–Emden problems has only one negative eigenvalue to which there corresponds a radial eigenfunction, since u is also the least energy solution of (4.1) in the space of radial functions. Thus we assume that u is a radial nodal solution of (4.1). Since the nonlinearity which defines the equation in (4.1) is f (s) = |s|p−1 s, p > 1, it satisfies all conditions stated in Section 3.3.1 and Section 3.3.2, so we can apply all results there described. In particular, if n = n(u) denotes the number of nodal domains of u, by Theorem 3.20 we have m(u) ≥ mrad (u) + N(n − 1)

(4.3)

where, as usual, m(u) is the Morse index of u and mrad (u) is its radial Morse index, i. e., the number of negative radial eigenvalues of Lu (i. e., eigenvalues which are associated to a radial eigenfunction). In the special case of the Lane–Emden problem, the radial Morse index can be explicitly computed. Theorem 4.1. Let u be a radial solution of (4.1) with n = n(u) nodal regions. Then mrad (u) = n.

(4.4)

This has been proved in [23] when u has only two nodal regions (see also Section 3.3.1) and in ([143], Proposition 2.9) for any number of nodal domains. Note also that for any fixed number n ∈ ℕ+ , there exists only one, up to the sign at the origin, radial solution of (4.1) with n nodal regions. This is a well-known result which is contained, for instance, in [145]. Let us denote by μ1 < μ2 ≤ ⋅ ⋅ ⋅ ≤ μi ≤ . . . ,

μi 󳨀→ +∞

as i → +∞

the eigenvalues of the linearized operator (4.2) counted according to their multiplicity. It is useful to recall their min-max characterization (see Chapter 1) μi =

inf

max R(v),

W⊂H01 (B) v∈W v=0 ̸ dim W=i

i ∈ ℕ+ ,

(4.5)

where R(v) is the Rayleigh quotient R(v) =

Qu (v)

∫B v(x)2 dx

(4.6)

and Qu is the quadratic form associated to Lu , namely 󵄨 󵄨2 󵄨 󵄨p−1 Qu (v) = ∫[󵄨󵄨󵄨∇v(x)󵄨󵄨󵄨 − p󵄨󵄨󵄨u(x)󵄨󵄨󵄨 v(x)2 ] dx. B

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(4.7)

4.1 Spectral decomposition for the linearized operator at a radial solution

| 123

Since u is a radial solution of (4.1), we can also consider the sequence of the radial eigenvalues of Lu that we denote by βi ,

i ∈ ℕ+

again counted according to their multiplicity. For the βi ’s, an analogous min-max characterization holds: βi =

inf

max R(v),

1 W⊂H0,rad (B) v∈W v=0 ̸ dim W=i

i ∈ ℕ+

(4.8)

1 where R(v) is as in (4.6) and H0,rad (B) denotes the subspace of H01 (B) made by radial functions. To study the spectrum of the linearized operator Lu , a suitable procedure is to decompose it as a sum of the spectrum of a radial weighted operator and the spectrum of the Laplace–Beltrami operator on the unit sphere. This leads to a weighted eigenvalue problem with a singularity at the origin. To bypass this difficulty, we first approximate the ball B by annuli with a small hole, showing that the number of the negative eigenvalues of the operator Lu stabilizes when the size of the hole is sufficiently small. Therefore, we consider the annuli:

Ah = { x ∈ ℝN :

1 < |x| < 1 } , h

h ∈ ℕ+

(4.9)

and denote by μhi ,

i ∈ ℕ+

the Dirichlet eigenvalues of Lu in Ah counted according to their multiplicity. Again they can be characterized as μhi =

max Rh (v)

inf

V⊂H01 (Ah ) v∈V v=0 ̸ dim V=i

(4.10)

where Rh is the corresponding Rayleigh quotient Rh (v) =

Qh (v) ∫A v(x)2 dx

(4.11)

h

and Qh : H01 (Ah ) → ℝ is the associated quadratic form 󵄨 󵄨2 󵄨 󵄨p−1 Qh (v) = ∫ [󵄨󵄨󵄨∇v(x)󵄨󵄨󵄨 − p󵄨󵄨󵄨up (x)󵄨󵄨󵄨 v(x)2 ] dx. Ah

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124 | 4 Morse index of radial solutions of Lane–Emden problems Let us denote by k h the number of negative eigenvalues μhi of Lu in Ah . We also use the notation βih ,

i ∈ ℕ+

for the radial Dirichlet eigenvalues of Lu in Ah counted with their multiplicity. Again, we have βih =

inf

max Rh (v),

i ∈ ℕ+

1 V⊂H0,rad (Ah ) v∈V v=0 ̸ dim V=i

(4.12)

where Rh is as in (4.11). h Finally, let krad be the number of radial eigenvalues of Lu in Ah . It is easy to see, using the canonical embedding H01 (Ah ) ⊂ H01 (B) and the min-max characterizations (4.5), (4.10) and (4.8), (4.12), that the following inequalities hold: μhi ≥ μi

and βih ≥ βi ,

∀i, h ∈ ℕ+ .

(4.13)

Similarly, we have μhi ≥ μh+1 i

and βih ≥ βih+1 ,

∀i, h ∈ ℕ+ .

(4.14)

Lemma 4.2. Let p ∈ (1, ps ) be fixed. Then μhi ↘ μi

and

βih ↘ βi

as h → +∞,

∀i ∈ ℕ+

Proof. It is an immediate consequence of Corollary 1.47. By Lemma 4.2 and (4.13), it follows that the number of negative eigenvalues (resp., negative radial eigenvalues) of the linearized operator Lu in B coincides with the numh ber k h (resp., krad ) of the negative eigenvalues (resp., negative radial eigenvalues) of Lu in Ah , for h large. So we have the following. Lemma 4.3. Let p ∈ (1, ps ) and let u be a solution to (4.1). Then there exists h󸀠 ∈ ℕ+ such that: h a) m(u) = k h and mrad (u) = krad ∀h ≥ h󸀠 . b) In particular, if u is the least energy nodal radial solution of (4.1) then, by Theorem 3.17 applied in the space of radial functions, it follows that h krad = 2 ∀h ≥ h󸀠 .

In order to make a decomposition of the spectrum of the linearized operator Lu , we consider the auxiliary weighted linear operator L̃ hu : H 2 (Ah ) ∩ H01 (Ah ) 󳨀→ L2 (Ah ) defined by 󵄨 󵄨p−1 L̃ hu (v) := |x|2 (−Δv − p󵄨󵄨󵄨u(x)󵄨󵄨󵄨 v),

x ∈ Ah

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(4.15)

4.1 Spectral decomposition for the linearized operator at a radial solution

| 125

and denote by μ̃ hi , i ∈ ℕ+ , its eigenvalues counted with their multiplicity. Observe that the corresponding eigenfunctions ψ satisfy 󵄨p−1 󵄨 h ψ(x) { , x ∈ Ah {−Δψ(x) − p󵄨󵄨󵄨u(x)󵄨󵄨󵄨 ψ(x) = μ̃ i 2 |x| { { on 𝜕Ah {ψ = 0 Since u is radial, we also consider the following linear operator: L̃ hu,rad : H 2 (( h1 , 1)) ∩ H01 (( h1 , 1)) 󳨀→ L2 (( h1 , 1)) written in polar coordinates: N −1 󸀠 󵄨 󵄨p−1 L̃ hu,rad (v) := r 2 (−v󸀠󸀠 − v − p󵄨󵄨󵄨u(x)󵄨󵄨󵄨 v), r

1 r ∈ ( , 1) h

(4.16)

and denote by β̃ ih , i ∈ ℕ+ , its eigenvalues counted with their multiplicity. Obviously, β̃ ih are nothing else than the radial eigenvalues of L̃ hu . We also set k̃ h := # { i ∈ ℕ+ such that μ̃ hi < 0 } ,

h k̃rad := # { i ∈ ℕ+ such that β̃ ih < 0 } .

(4.17) (4.18)

Denoting by σ(⋅) the spectrum of a linear operator we have the following decomposition. Lemma 4.4. σ(L̃ hu ) = σ(L̃ hu,rad ) + σ(−ΔSN−1 )

∀h ∈ ℕ+

(4.19)

where ΔSN−1 denotes the Laplace–Beltrami operator on the unit sphere SN−1 , N ≥ 2. Proof. Let us denote by λk , k = 0, 1, . . . , the eigenvalues of −ΔSN−1 . It is well known (see [34]) that λk = k(k + N − 2) ∀k ∈ ℕ.

(4.20)

Given μ ∈ σ(L̃ hu ), we consider an associated eigenfunction ψ satisfying: ψ p−1 { {−Δψ − p|u| ψ = μ 2 |x| { { ψ = 0 {

in Ah on 𝜕Ah

Then we choose k ∈ ℕ and an eigenfunction φ of −ΔSN−1 associated to λk . The function w(r) = ∫ ψ(r, θ)φ(θ) dθ SN−1

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126 | 4 Morse index of radial solutions of Lane–Emden problems satisfies −w󸀠󸀠 −

N −1 N −1 󸀠 w = ∫ (−ψrr − ψr )φ dθ r r SN−1

= ∫ (−Δψ + SN−1

= p|u|p−1 w +

1 Δ N−1 ψ)φ dθ r2 S μ 1 w + 2 ∫ (ΔSN−1 ψ)φ dθ. r2 r SN−1

Integrating the last term by parts, we get −w󸀠󸀠 −

μ − λk N −1 󸀠 w − p|u|p−1 w = w, r r2

which implies that the numbers (μ − λk ) are eigenvalues of the operator L̃ hu,rad . Hence μ = (μ − λk ) + λk ∈ σ(L̃ hu,rad ) + σ(−ΔSN−1 ). Vice versa, we consider β ∈ σ(L̃ hu,rad ) and λk ∈ σ(−ΔSN−1 ) and choose corresponding eigenfunctions w and φ. Defining ψ(x) = w(|x|)φ(

x ), |x|

we have N −1 󸀠 w w )φ − 2 ΔSN−1 φ r r λ β + λk β ψ = [p|u|p−1 w + 2 w]φ + 2k wφ = p|u|p−1 ψ + r r r2

−Δψ = (−w󸀠󸀠 −

which implies that β + λk ∈ σ(L̃ hu ). Thus, by (4.19), we can write μ̃ hj = β̃ ih + λk ,

for i, j ∈ ℕ+ and k ∈ ℕ.

(4.21)

Note that in (4.21) only β̃ ih depend on the exponent p in (4.1) while, by (4.20), the eigenvalues λk depend only on the dimension N. Moreover, the multiplicity of λk is (see [34]): Nk − Nk−2

(4.22)

where for s ∈ ℤ N −1+s (N − 1 + s)! { )= , {Ns = ( N −1 (N − 1)!s! { { {Ns = 0,

if s ≥ 0 if s < 0

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(4.23)

4.1 Spectral decomposition for the linearized operator at a radial solution

| 127

The next result shows that the number of the negative eigenvalues of the linearized operator Lu in the annulus Ah is the same as the one of the corresponding weighted operator. Lemma 4.5. We have k h = k̃ h

and

h h krad = k̃rad

h h where k h , krad , k̃ h and k̃rad are as in Lemma 4.3 and in (4.17), (4.18).

Proof. We first show that k h ≥ k̃ h . Let ψ be an eigenfunction for the operator L̃ hu corresponding to a negative eigenvalue μ̃ h < 0. Hence p−1 h ψ { {−Δψ − p|u| ψ = μ̃ |x|2 { { {ψ = 0

in Ah

(4.24)

on 𝜕Ah

Multiplying (4.24) by ψ and integrating on Ah , we get ψ(x)2 󵄨 󵄨2 󵄨 󵄨p−1 Qhu (ψ) = ∫ [󵄨󵄨󵄨∇ψ(x)󵄨󵄨󵄨 − p󵄨󵄨󵄨u(x)󵄨󵄨󵄨 ψ(x)2 ] dx = μ̃ h ∫ dx < 0 |x|2 Ah

Ah

namely ψ makes the quadratic form Qhu negative. Then k h ≥ k̃ h since the set of such eigenfunctions ψ is a space of dimension k̃ h . To prove that k h ≤ k̃ h , let us assume, by contradiction, that k h > k̃ h and let W be the k h -dimensional subspace spanned by the orthogonal eigenfunctions φi associated to the negative Dirichlet eigenvalues of Lu in Ah : W = span{φ1 , φ2 , . . . , φkh } ⊂ H01 (Ah ). By the variational characterization of the eigenvalues of L̃ hu , we have μ̃ hkh

≤ max

∫A (|∇v(x)|2 − p|u(x)|p−1 v(x)2 ) dx h

v∈W v=0 ̸

∫A

h

v(x)2 |x|2

dx

0

(4.32)

indicating also the dependence on the exponent p in (4.1). We start by summarizing the basic properties of the solutions unp . Note that also the case n = 1, i. e., when the solution is positive, is included. Proposition 4.7. Let p ∈ (1, ps ) then: (i) unp (0) = ‖unp ‖∞ ;

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130 | 4 Morse index of radial solutions of Lane–Emden problems (ii) in each nodal region the map r 󳨃→ unp (r), r = |x| has exactly one critical point (which is either a local maximum or a local minimum point, and they alternate); (iii) ∫B |∇unp (y)|2 dy = ∫B |unp (y)|p+1 dy 󳨀→ nSN/2 , as p → ps . n n Now we denote by ri,p the ordered nodal radii of unp , i. e., unp (ri,p ) = 0, i = 1, . . . , n: n n n n 0 < r1,p < r2,p < ⋅ ⋅ ⋅ < rn−1,p < rn,p = 1.

(4.33)

Moreover, we denote by sni,p the unique maximum point of |unp | in each nodal region Bni,p , i = 0, . . . , n − 1: n Bn0,p = { x ∈ ℝN : |x| < r1,p },

n n Bni,p = { x ∈ ℝN : ri,p < |x| < ri+1,p },

i = 1, . . . , n − 1 if n ≥ 2.

(4.34)

Finally, we consider the restriction of |unp | to the ith nodal region: 󵄨 󵄨 uni,p = 󵄨󵄨󵄨unp 󵄨󵄨󵄨χBn , i,p

i = 0, . . . , n − 1

(4.35)

and define 󵄩 󵄩 󵄨 󵄨 n Mi,p = 󵄩󵄩󵄩uni,p 󵄩󵄩󵄩∞ = 󵄨󵄨󵄨unp (sni,p )󵄨󵄨󵄨,

i = 0, . . . , n − 1

(4.36)

The asymptotic behavior of these quantities is described in the following. Proposition 4.8. For any i = 0, . . . , n − 1 and p → ps , we have 󵄨 󵄨2 󵄨 󵄨p+1 ∫󵄨󵄨󵄨∇uni,p (y)󵄨󵄨󵄨 dy = ∫󵄨󵄨󵄨uni,p (y)󵄨󵄨󵄨 dy 󳨀→ SN/2 ;

(4.37)

󵄨 󵄨2∗ ∫󵄨󵄨󵄨uni,p (y)󵄨󵄨󵄨 dy 󳨀→ SN/2 ;

(4.38)

󵄨 󵄨 N (p−1) 󳨀→ SN/2 ; ∫󵄨󵄨󵄨uni,p (y)󵄨󵄨󵄨 2

(4.39)

unp ⇀ 0 in H01 (B);

(4.40)

B

B

B

B

n Mi,p

sni,p 󳨀→ 0 sni,p 󳨀→ 0, n ri+1,p

n n ri,p (Mi−1,p )

(4.41)

󳨀→ +∞;

(so that

n ri,p

󳨀→ 0);

for i ≠ n − 1, if n ≥ 2; p−1 2

n sni,p (Mi,p )

p−1 2

󳨀→ +∞ 󳨀→ 0

if n ≥ 2; if n ≥ 2.

(4.42) (4.43) (4.44) (4.45)

Some crucial estimates for |unp | in each nodal region are contained in the following.

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4.2 Asymptotic analysis of radial solutions | 131

Proposition 4.9. We have 󵄨󵄨 n 󵄨󵄨 󵄨󵄨up (x)󵄨󵄨 ≤

n M0,p

[1 +

n p−1 (M0,p )

N(N−2)

2

|x| ]

N−2 2

∀x ∈ Bn0,p .

(4.46)

) and n ≥ 2 there exists γ = γ(α, n) ∈ (0, 1), γ(α, n) → 1 as Moreover, for any α ∈ (0, N−2 2 4 α → 0 and δi = δi (α, n) ∈ (0, N−2 ), i = 1, . . . , n − 1 such that for p ≥ ps − δi it holds 󵄨󵄨 n 󵄨󵄨 󵄨󵄨up (x)󵄨󵄨 ≤

n Mi,p

[1 +

N−2 2α n )p−1 |x|2 ] 2 (Mi,p N(N−2)2

n ∀x ∈ Ci,p

(4.47)

where 1

n n Ci,p = { x ∈ ℝN : γ − N sni,p < |x| < ri+1,p } (⊂ Bni,p ) n with Bni,p and Mi,p defined as in (4.34) and (4.36).

We consider now, for n ∈ ℕ+ , the tail sets n−1

n Ti,p = ⋃ Bnj,p , j=1

i = 0, . . . , n − 1.

(4.48)

n n n Hence T0,p = B, T1,p = B \ Bn0,p , . . . , Tn−1,p = Bnn−1,p . Then we consider the rescaled functions n zi,p (x) =

1 n |x| up ( ), p−1 n Mi,p (M n ) 2 i,p

n n x ∈ T̃ i,p = (Mi,p )

p−1 2

n Ti,p

(4.49)

for i = 0, . . . , n − 1, which are radial and solve 󵄨 n 󵄨󵄨p−1 n n −Δzi,p = 󵄨󵄨󵄨zi,p 󵄨󵄨 zi,p { { { { n zi,p = 0 { { { { n n n 󸀠 n {zi,p (si,p ) = 1 and (zi,p ) (si,p ) = 0

n in T̃ i,p

n on 𝜕(T̃ i,p )

(4.50)

Moreover, by (4.32), it holds n (−1)i zi,p >0

n in B̃ ni,p = (Mi,p )

p−1 2

Bni,p .

(4.51)

The asymptotic behavior of these functions, as p → ps is given by the following. Proposition 4.10. As p → ps , n ( − 1)i zi,p 󳨀→ U

n z0,p 󳨀→ U

2 in Cloc (ℝN ),

2 in Cloc (ℝN \ {0}) for i = 1, . . . , n − 1 (n ≥ 2)

(4.52) (4.53)

where U is defined as in (4.29).

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132 | 4 Morse index of radial solutions of Lane–Emden problems A part from the technical statements, the asymptotic behavior of the radial solutions of (4.1) as p → ps , which is deduced by the previous propositions, can be described as follows. All nodal regions Bni,p , i = 0, . . . , n − 1, of unp shrink to the origin, n as ri,p → 0. Moreover, also the maximum or minimum points sni,p converge to 0 but n faster than the nodal radius ri+1,p , so that they do not see each other in the limit. The n estimate (4.41) allows to use the local maxima Mi,p as rescaling parameters in defining n the rescaled functions zi,p , while (4.44) implies that the limit domains for these functions, and the problem (4.50) that they solve, are ℝN \ {0} or ℝN for the rescaling in the first region. Therefore, the limit of the problems (4.50) is the problem (4.28) which explains why (4.52) and (4.53) hold. The unique positive solution U of (4.28) satisfying (4.30) is explicitly given in (4.29) and usually is referred as the standard bubble. Thus, the limit profile of the solutions unp , as p → ps , looks like a superposition of n bubbles or, in other words, like a tower of n standard bubbles. Moreover, the restrictions of the solution unp to each nodal domain carry the same energy as stated in (4.37) and (4.38). This peculiar behavior of the nodal radial solutions also induces an interesting blow-up (in time) phenomenon in the associated parabolic problem with initial data close to the stationary radial solutions ([59], see also [171]).

4.2.2 The case N = 2 Now we consider the 2-dimensional case. We will only analyze the asymptotic behavior of the least energy nodal radial solution of (4.1) as the exponent p → ps = +∞. This solution has only two nodal domains and radial Morse index two as it can be seen by repeating the minimizing procedure of Theorem 3.17 on the nodal Nehari set 1 in H0,rad (B). The reason for analyzing only the least energy nodal solution relies on the fact that an accurate study of the asymptotic behavior of nodal solutions, as p → +∞, in dimension two is very difficult and has been done only for solutions with two nodal regions. Indeed, unlike the higher dimensional case, the sign-changing solutions can behave in a different way in each nodal domain, making so their analysis quite involved. We believe that a complete study of nodal radial solutions with any number of nodal regions should be possible though technically very complicated; however, it has not been done so far. Since the least energy nodal radial solution has exactly two nodal domains, we denote it simply by up , neglecting the dependence of n(up ) as compared with the case N ≥ 3. The asymptotic behavior of up , as p → +∞, has been accurately studied in [137]. For general bounded domains, the asymptotic analysis of both positive and sign changing solutions to (4.1) has been done in [103] and [102], but the estimates are not as precise as in the radial setting.

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4.2 Asymptotic analysis of radial solutions | 133

We summarize below the main results on the asymptotic behavior of the solutions up and we refer to [137] for the proofs. First, we recall two well-known properties of up , similar to (i) and (ii) of Proposition 4.7, for the case N ≥ 3: (i) up (0) = ‖u‖∞ ;

(4.54)

(ii) in each nodal region the solution up has only one critical point

(namely the maximum or the minimum point of up ).

(4.55)

From now on, we will assume that up (0) > 0 and denote by rp the unique nodal radius of up and by sp the unique minimum radius of up , i. e., rp ∈ (0, 1) is such that up (rp ) = 0

(4.56)

󵄩 󵄩 sp ∈ (0, 1) is such that 󵄩󵄩󵄩u−p 󵄩󵄩󵄩∞ = u−p (sp ) = −up (sp )

(4.57)

and

where, as usual, u−p is the negative part of up . Proposition 4.11. Let (up ) be a family of least energy radial nodal solutions to (4.1) with up (0) > 0. Let us define −2

:= pup (0)p−1 ,

−2

:= pup (sp )p−1

(ϵp+ ) (ep− )

(4.58)

and the rescaled functions zp+ (x) := p zp− (x) := p

up (ϵp+ x) − up (0) up (0)

,

up (sp )

,

up (ϵp− x) − up (sp )

x∈

B , ϵp+

(4.59)

x∈

B . ϵp−

(4.60)

Then ϵp± → 0,

1 in Cloc (ℝ2 ),

zp+ → U zp−

→ Zl

(4.61)

in

1 Cloc (ℝ1

\ {0})

(4.62) (4.63)

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134 | 4 Morse index of radial solutions of Lane–Emden problems as p → ∞, where U(x) := log(

2

1

1 + 81 |x|

(4.64)

) 2

is the regular solution of U

in ℝ2

−Δu = e { { { { { { ∫ eU dx = 8π { { { ℝ2 { { { {U(0) = 0

(4.65)

and Zl (x) := log(

2(γ + 2)2 δγ+2 |x|γ (δγ+2 + |x|γ+2 )2

(4.66)

),

with l = lim

p→+∞

sp

ϵp−

1

≈ 7.1979,

γ = √2l2 + 4 − 2,

γ + 4 γ+2 δ=( ) l γ

(4.67)

is a singular radial solution of −ΔZ = eZ + Hδ0 { { { Z { { { ∫ e dx < +∞

ℝ2 (4.68)

{ℝ2 l

where H = − ∫0 eZl (s) s ds and δ0 is the Dirac measure centered at 0. Moreover, rp

ϵp+

󳨀→ +∞

and

ϵp− rp

󳨀→ +∞.

(4.69)

Finally, there exists c > 0 such that 󵄨 󵄨p−1 p|y|2 󵄨󵄨󵄨up (y)󵄨󵄨󵄨 ≤ c

∀y ∈ B.

(4.70)

All previous statements have been proved in [137] with the exception of (4.70) which corresponds to the property P3k in ([103], Proposition 2.2). Let us explain and comment on the results of Proposition 4.11. As in the case of higher dimension, one would like to detect the limit profile of the solutions up , as p → +∞, by some limit problem in the whole ℝ2 . However, in dimension two there is not an obvious limit equation for two reasons:

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4.2 Asymptotic analysis of radial solutions | 135

(i) the nonlinearity f (u) = |u|p−1 u does not converge to the exponential function, as p → +∞, (ii) the L∞ norm of the solution does not blow-up, as p → +∞, so that it cannot be used as rescaling parameter for the solution. This motivates the choice of ϵp± in (4.58) as a parameter to define the rescaled functions zp± in (4.59), (4.60). Let us recall that the asymptotic analysis of solutions of (4.1) in bounded domains Ω ⊂ ℝ2 started in [198] and [197] where the authors considered the case of families of least energy (hence positive) solutions and, in some domains, proved concentration results as well as some asymptotic estimates. However, they did not identify a “limit problem.” This was done later in [3] (see also [114]) by showing that suitable scalings 1 of the least energy solutions converge in Cloc (ℝ2 ), as p → +∞, to the function U in (4.64) which is a regular solution of the famous Liouville problem (4.65) in the plane. They also showed that the L∞ -norm of the least energy solutions converge to √e, thus confirming a previous conjecture of [61]. Concerning sign changing solutions, the asymptotic analysis was started in [136] by considering a family of low-energy nodal solutions but adding the hypothesis that the minimum and the maximum of the solutions are “comparable” (see [136] for the precise definition). In this case, the positive and the negative part of the solutions separate, each one having the limit profile (after scaling) of the regular solution U of the Liouville problem (4.65). However, as shown in [137], this is not the case of nodal radial solutions in the ball. The newelty, and somehow surprising result, of [137] is that the limit profiles of the positive and negative parts of the least energy nodal radial solutions up are different. Indeed, assuming up (0) > 0, by (4.62) we have that the rescaling zp+ converge to the regular solution of (4.65), while, by (4.63), the rescaling zp− converge to a singular radial solution Zl (given by (4.66)) of the problem (4.68) in the plane. Hence, asymptotically, the solutions up look like a superposition of different bubbles, given by a regular and a singular solutions of (4.65) and (4.68). So again, we can say that the limit profile is a tower of two bubbles, but the bubbles are different, unlike the higher dimensional case. Moreover, it can be proved that each bubble carries a different energy which emphasizes the difference with the case N ≥ 3. The same kind of phenomenon is shown to happen in some symmetric domains by analysing the asymptotic behavior of low energy symmetric solutions (see [103]). In [102], similar results are shown for low Morse index symmetric solutions. However, as pointed out before, in this more general situations, precise estimates are not available. As in higher dimensions, the asymptotic behavior of these bubble-tower solutions induces a peculiar blow-up (in time) phenomenon in the associated parabolic problem with initial data close to this kind of stationary solutions [100, 101, 109].

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136 | 4 Morse index of radial solutions of Lane–Emden problems Finally, we observe that the Liouville problems (4.65) and (4.68) are important both in geometry and in physics. They arise in the study of surfaces with prescribed Gaussian curvature and in the study of vortices in the Chern–Simon theory [214].

4.3 Computation of the Morse index of radial solutions in dimension N ≥ 3 At the end of Section 4.1, as a result of the spectral decomposition we were left out with the estimates of the first n eigenvalues β̃ ih , i = 1, . . . , n, of the linear weighted operator L̃ hu,rad in the interval ( h1 , 1) (see (4.16)). Since, as in Section 4.2, the results will depend on the number of nodal regions of the radial solutions, as well as on the exponent p, we will indicate explicitly the dependence on them. Therefore, the radial solutions with n nodal regions of (4.1) will be denoted by unp and the operator in (4.16) and its eigenvalues will be denoted by L̃ h,n p,rad and by β̃ h (n, p). i

The first result is about an estimate for the nth eigenvalue.

1 󸀠󸀠 n Proposition 4.12. Let h ∈ ℕ+ , p ∈ (1, ps ) and h󸀠󸀠 p = hp (n) = [ r n ] + 1, where r1,p is the 1,p

first nodal radius of unp , as defined in (4.33). Then

β̃ nh (n, p) > −(N − 1) for any h ≥ h󸀠󸀠 p.

(4.71)

dun (r)

󸀠󸀠 p Proof. Let η(r) = dr , then, by the choice of h󸀠󸀠 p it follows that for any h ≥ hp it holds 1 n < r1,p , so that the function η satisfies h

{ { L̃ h,n { p,rad η = −(N − 1)η { { { { {η( 1 ) < 0 { { { h { { { {η(1) < 0 if n is odd or η(1) > 0 if n is even,

1 in ( , 1) h

the last alternative depending on the assumption uhp (0) > 0 and the sign are strict, by the Hopf lemma. Moreover, by the behavior of unp , we know that, for h ≥ h󸀠󸀠 p , the 1 function η has exactly (n − 1) zeros in the interval ( h , 1), given (if n ≥ 2) by the points sni,p , i = 1, . . . , n − 1, which satisfy (4.42). Let w be an eigenfunction of L̃ h,n associated to the eigenvalue β̃ h (n, p), i. e., n

p,rad

{ { L̃ h,n w = β̃ nh (n, p)w { { p,rad { { { {w( 1 ) = w(1) = 0 { n

1 in ( , 1) n

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4.3 Computation of the Morse index of radial solutions in dimension N ≥ 3

| 137

It is well known that w has exactly n nodal regions. Then, arguing by contradiction, let us assume that β̃ nh (n, p) ≤ −(N − 1). If β̃ nh (n, p) = −(N − 1) then η and w are two solutions of the same Sturm–Liouville equation β̃ h (n, p) 󸀠 󵄨p−1 󵄨 (r N−1 v󸀠 ) + [p󵄨󵄨󵄨up (r)󵄨󵄨󵄨 r N−1 + n 3−N ]v = 0, r

1 r ∈ ( , 1) n

and they are linearly independent, because η(1) ≠ 0 = w(1). As a consequence of the Sturm separation theorem, the zeros of η and w must alternate. Since η has (n − 1) zeros, then w must have (n−1) nodal regions and this gives a contradiction. If β̃ nh (n, p) < −(N − 1), then by the Sturm comparison theorem, η must have a zero between any two consecutive zeros of w. As a consequence, since we know that w has (n − 1) zeros in ( h1 , 1) and also w( h1 ) = w(1) = 0, then η must have n zeros in ( h1 , 1), which gives again a contradiction. Now we should study the eigenvalues β̃ ih (n, p), for i = 1, . . . , n − 1, of the linear operator L̃ h,n (see (4.16)) which is defined in the annulus (4.9). It is convenient to p,rad choose h depending on p (and n) as follows: p−1

n hnp = max{h󸀠p , h󸀠󸀠 p , (M0,p )

+ 1}

(4.72)

n where h󸀠p is as in Proposition (4.6), h󸀠󸀠 p as in Proposition (4.12) and M0,p is defined in (4.36). Then we consider the eigenvalues

β̃ i (n, p) = β̃ ih (n, p)

for h = hnp , i ∈ ℕ+

(4.73)

so that we can drop the dependence on h. By the choice of hnp and the previous results, we have that β̃ i (n, p) < 0 for i = 1, . . . , n − 1 and for every p ∈ (1, ps ). In the same way we shorten the notation for the linear operator, setting L̃ np,rad = L̃ h,n p,rad

taking h = hnp .

The crucial result to estimate the eigenvalues β̃ i (n, p), i = 1, . . . , n − 1, when p is close to ps is the following. Proposition 4.13. For any n ∈ ℕ+ , lim inf β̃ 1 (n, p) ≥ −(N − 1),

(4.74)

lim inf β̃ i (n, p) ≥ −(N − 1) for i = 1, . . . , n − 1.

(4.75)

p→ps

and hence p→ps

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138 | 4 Morse index of radial solutions of Lane–Emden problems Obviously, (4.75) follows from (4.74). It will be later proved that indeed equality holds in (4.74) and (4.75). The proof of Proposition 4.13 is very long and requires several nontrivial estimates, therefore, we will only explain its strategy, referring to [106] for all details. To get (4.74), it is natural to study the normalized eigenfunction associated to β̃ 1 (n, p), for any p ∈ (1, ps ), which is radial and positive. Let us denote it by φnp . Hence it satisfies φn N − 1 n󸀠 󵄨󵄨 n 󵄨󵄨p−1 n n 󸀠󸀠 { ̃ (n, p) p { −φ − φ − p u φ = β 󵄨 󵄨 { 1 p p p p 󵄨 󵄨 { r { r2 { { n 1 { { {φp ( n ) = φnp (1) = 0 hp {

in (

1 , 1) hnp

(4.76)

To obtain (4.74), one would like to pass to the limit into (4.75) and study the limit eigenvalue problem. Since the term p|unp |p−1 in (4.76) is not bounded, this cannot be done straightforwardly but a scaling procedure should be used in order to get something in the limit. Usually these scalings are done using as parameters the unbounded quantity given by p|unp |p−1 . However, since unp has several nodal regions and blows up in each of them at a different rate, is convenient to use various rescalings of the eigenfunction φnp with different parameters. More precisely, we define ̂n,i φ p = ̂ n,i = (M n ) where x ∈ A p i,p

1 n ) (Mi,p

p−1 2

(p−1)(N−2) 4

φnp (

|x| n ) (Mi,p

p−1 2

),

i = 0, . . . , n − 1

(4.77)

Anp and Anp = { y ∈ ℝN :

1 < |y| < 1 } hnp

n where Mi,p are the L∞ -norm of unp in the corresponding ith nodal region, defined in (4.36). ̂n,i The functions φ p satisfy the eigenvalue problem:

̂n,i φ { p n { ̃ ̂n,i ̂n,i {−Δφ p − Vi,p (x)φ p = β1 (n, p) |x|2 { { { n,i ̂p = 0 {φ

̂ n,i x∈A p ̂ n,i on 𝜕A p

(4.78)

where n Vi,p (x) = p

󵄨󵄨 |x| 󵄨󵄨 n 󵄨󵄨up ( p−1 n p−1 󵄨 (Mi,p ) 󵄨 (M n ) 2 1

i,p

󵄨󵄨p−1 󵄨 )󵄨󵄨󵄨 . 󵄨󵄨 Mn

Note that by (4.41) and the fact that (4.44) and (4.45) imply that M ni,p → +∞, as p → ps , i+1,p

̂ n,i is ℝN \ {0}. This suggests that ∀i = 0, . . . , n − 1, it follows that the limit domain of A p

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4.3 Computation of the Morse index of radial solutions in dimension N ≥ 3

| 139

n a limit eigenvalue problem, obtained by considering the limits of the potential Vi,p should play a role to evaluate the limit of the eigenvalues β̃ 1 (n, p) as p → ps . This limit eigenvalue problem is the one for the weighted linear operator:

L̃ ∗ v = |x|2 [−Δv − V(x)v],

x ∈ ℝN

(4.79)

where 2

N(N − 2) N +2 ( ) N − 2 N(N − 2) + |x|2

V(x) = ps U(x)ps −1 =

(4.80)

and we recall that U, defined in (4.29), is the solution of (4.28) satisfying (4.30). Then the corresponding eigenvalue problem is − Δη − V(x)η = λ

η

2

|x|

,

x ∈ ℝN \ {0}

(4.81)

One of the main steps in computing the Morse index of unp is to show that the first eigenvalue β̃ ∗ of (4.81) is exactly β̃ ∗ = −(N − 1).

(4.82)

This result has its own interest independent of the computation of the Morse index of unp . It also holds in dimension N = 2 with an appropriate definition of the potential V(x) and will play a crucial role in the computation of the Morse index of least energy nodal radial solution of (4.1) in the 2-dimensional ball that we describe in the next section. Therefore, we will prove (4.82) in Section 4.5. Going back to (4.78), let us observe that, passing to the limit as p → ps , it could ̂n,i happen that the rescaled eigenfunctions φ p all converge to zero, for i = 1, . . . , n − 1, in which case nothing is obtained in the limit. Since this cannot be excluded, the asymptotic analysis of the eigenvalues β̃ 1 (n, p) is done in [106] by evaluating with great accuracy the contribution to the limit of β̃ 1 (n, p) given by each nodal region of unp . Once Proposition 4.13 is obtained, we can prove the following exact formula for the Morse index of the solution unp . Theorem 4.14. Let N ≥ 3 and unp be a radial solution of (4.1) with n ∈ ℕ+ nodal domains. Then m(unp ) = n + N(n − 1)

(4.83)

for p sufficiently close to ps . Proof. We recall that we have approximated the ball B by an annulus Ah , choosing h = hnp as in (4.72). Then we have considered the radial weighted linear operators L̃ np,rad whose eigenvalue we have denoted by β̃ (n, p), i ∈ ℕ+ (see (4.73)). i

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140 | 4 Morse index of radial solutions of Lane–Emden problems To simplify the notation, we set L̃ np = L̃ hunp for h = hnp , where L̃ hunp is defined in (4.15) and we denote its eigenvalues by μ̃ i (n, p) = μ̃ hi (n, p),

for h = hnp ,

i ∈ ℕ+

emphasizing the dependence on n and p. The number of negative eigenvalues of L̃ hp is denoted by k̃ (n) = k̃ h (n). p

p

We have seen, by Proposition 4.6, that determining the Morse index m(unp ) is equivalent to counting the number k̃p (n) of negative eigenvalues μ̃ i (n, p) of the operator L̃ np . Hence we should show that k̃p (n) = n + N(n − 1)

for p close to ps .

(4.84)

for i, j ∈ ℕ+ , k ∈ ℕ

(4.85)

By (4.21), we have that μ̃ j (n, p) = β̃ i (n, p) + λk ,

where λk are the eigenvalues of the Laplace–Beltrami operator −ΔSN−1 on the unit sphere SN−1 , N ≥ 3. As we already mentioned in (4.20), λk = k(k + N − 2) (≥ 0),

k = 0, 1, . . .

with multiplicity (see [34]) Nk − Nk−2

(4.86)

k̃p (n) ≥ n + N(n − 1)

(4.87)

β̃ 1 (n, p) ≤ ⋅ ⋅ ⋅ ≤ β̃ n (n, p) < 0 ≤ β̃ n+1 (n, p) ≤ . . .

(4.88)

where Ns , s ∈ ℤ, is defined in (4.23). By (4.27), we already know that

and that

By (4.88), since λk ≥ 0, it immediately follows that β̃ i (n, p) + λk ≥ 0 ∀i ≥ n + 1, ∀k ≥ 0

(4.89)

so that all the eigenvalues β̃ i (n, p) with i ≥ n + 1 cannot produce any negative eigenvalue μ̃ j (n, p) by the formula (4.85). Next, we analyze the contribution given by the last negative eigenvalues β̃ n (n, p). Observe that λ1 = N − 1 and, by Proposition 4.12, β̃ n (n, p) > −(N − 1), hence we get β̃ n (n, p) + λk > 0,

∀k ≥ 1.

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(4.90)

4.3 Computation of the Morse index of radial solutions in dimension N ≥ 3

|

141

On the other side, from (4.88) and observing that λ0 = 0, we have that β̃ n (n, p) + λ0 = β̃ n (n, p) < 0.

(4.91)

Hence, by (4.85) and (4.91) we have that β̃ n (n, p) is a negative eigenvalue of L̃ np , which is radial and simple, since by (4.86) it follows that λ0 has multiplicity one. In fact, the eigenfunctions corresponding to λ0 are just constants. Furthermore, because of (4.90), this eigenvalue is the only negative eigenvalue obtained by summing β̃ n (n, p) with the eigenvalues of −ΔSN−1 . Then (4.83) is obviously proven in the case n = 1. In the case n ≥ 2, we need to study the remaining negative eigenvalues β̃ i (n, p), i = 1, . . . , n − 1 and, since there are exactly n radial simple negative eigenvalues of L̃ np , we have to prove that they produce exactly N(n − 1) negative nonradial eigenvalues μ̃ j (n, p) by the formula (4.85) (counted with their multiplicity). Since by Proposition (4.13), we have lim inf β̃ i (n, p) ≥ −(N − 1), p→ps

for any i = 1, . . . , n − 1

(4.92)

and observing that λk ≥ 2N > N − 1 for all k ≥ 2, it follows that for p sufficiently close to ps β̃ i (n, p) + λk > 0,

for any i = 1, . . . , n − 1, for all k ≥ 2.

(4.93)

By (4.93) and the estimate (4.87), we immediately have that for p close to ps , β̃ i (n, p) + λ1 < 0,

for any i = 1, . . . , n − 1.

(4.94)

Indeed, since there are exactly n radial simple negative eigenvalues of L̃ np , by (4.87) there must be at least N(n − 1) negative nonradial eigenvalues of L̃ np (counted with their multiplicity). By (4.93), for p close to ps , these nonradial eigenvalues must be obtained by the formula (4.85) for i = 1, . . . , n − 1 and k = 1 (for k = 0 only radial eigenvalues may be constructed). Hence, observing that the multiplicity of λ1 is N (by (4.86)), we deduce that, if (4.94) does not hold, then (4.87) cannot be true. In conclusion, by (4.93) and (4.94) we get that, for p close to ps there are exactly N(n − 1) negative nonradial eigenvalues of L̃ np counted with their multiplicity, given by the formula (4.94). This proves the assertion (4.83). as

We end this section by a few comments about the formula (4.83). It can be written m(unp ) = n(N + 1) − N

which shows that the Morse index m(unp ) grows linearly with respect to the number n of nodal domains, which, in turns, corresponds to the number of negative radial eigenvalues of the linearized operator Lunp . This is somehow surprising since, in general,

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142 | 4 Morse index of radial solutions of Lane–Emden problems one would expect many more negative nonradial eigenvalues than the negative radial ones. Indeed, if we look at the distribution of the radial and nonradial eigenvalues of the Laplace operator (−Δ) in H01 (B), we observe that: (i) on one side by results of Brüning-Heintze and Donnelly [51, 52, 111] we get that λr,n ∼ Cn2

as n → +∞

where λr,n is the nth radial eigenvalues of (−Δ), which implies that the number kr (n2 ) of the radial eigenvalues of (−Δ) bounded by n2 is n, more precisely kr (n2 ) ∼ n

as n → +∞

(ii) on the other side by the classical Weil law (see, e. g., [210]): k(n2 ) ∼ CnN

as n → +∞

(N is the dimension)

where k(n2 ) is the number of all eigenvalues of (−Δ) in H01 (B) less than or equal to n2 . In an equivalent way, we can observe that if we consider a radial eigenfunction of (−Δ) in H01 (B) with n nodal regions, i. e., corresponding to the eigenvalue λr,n , then its Morse index is just the number of the eigenvalues less that λr,n which, by (i) and (ii), grows at a rate of order nN and so faster than n (if N ≥ 2) as n → +∞. So Lunp represents an example of a linear, Schrödinger-type, operator determined by the potential Vpn (x) = p|unp (x)|p−1 , for p approaching ps , for which (i) and (ii) do not hold, at least for negative eigenvalues. Another interesting consequence could be derived studying (4.1) as p → 1. In this case, it is reasonable to conjecture the convergence of the Morse index m(unp ) to the Morse index of the Dirichlet radial eigenfunction of (−Δ) with n nodal regions (i. e., the eigenfunction corresponding to the radial eigenvalue λr,n ) possibly augmented by the multiplicity of λr,n which is 1. Indeed, suitable normalizations of solutions of (4.1) converge to eigenfunctions of the Laplacian as p → 1 (see [37, 135]). Therefore, the previous considerations indicate that for large n the Morse index m(unp ) for p close to 1 is of order nN , hence it is much larger that n + N(n − 1), which is, by (4.83), the Morse index of unp for p close to ps . So bifurcations from unp should appear (as for the problems described in Chapter 5), as p ranges from 1 to ps , showing that the structure of the solution set of (4.1) is richer than one could imagine. Finally, we would like to point out another interesting fact: the formula (4.83) does not hold in dimension N = 2, as p → ps = +∞, as we will show in the next section.

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4.4 Computation of the Morse index in dimension N = 2

|

143

4.4 Computation of the Morse index in dimension N = 2 As in Section 4.2.2, we only consider the nodal radial solution of (4.1) with the least energy that we denote by up and assume up (0) > 0. By the results of Section 4.1, since up has only two nodal regions and its radial Morse index is two, we have to estimate the first two eigenvalues β̃ ih , i = 1, 2, of the linear weighted operator L̃ hup ,rad in the interval ( h1 , 1) (see (4.16)). Making explicit the dependence on p we denote by β̃ ih (p) the eigenvalues β̃ ih , i = 1, 2. The first estimate which holds for β̃ h (p) is the following. 2

+ Proposition 4.15. Let h ∈ ℕ , p ∈ (1, +∞). Then there exists h󸀠󸀠 p ∈ ℕ such that +

β̃ 2h (p) > −1,

for any h ≥ h󸀠󸀠 p.

The proof is the same as that of Proposition 4.12, a bit simpler because there is only one nodal radius. As for the case N ≥ 3, it is convenient to choose h depending on p as follows: + hp = max{h󸀠p , h󸀠󸀠 p , [(ϵp ) ] + 1} −2

(4.95)

+ −2 where h󸀠p is as in Proposition 4.6, h󸀠󸀠 p as in Proposition 4.15 and (ϵp ) as in (4.58). h Since Proposition 4.15 gives an estimate for β̃ (p), we only have to study the eigen2

h

values β̃ 1 p (p) that we denote by β̃ 1 (p), having fixed h = hp . The corresponding linear operator will be denoted by L̃ p,rad instead of L̃ hup ,rad . The crucial result to estimate these eigenvalues for large values of p is the following. Proposition 4.16. We have l2 + 2 ≈ −26.9 lim β̃ 1 (p) = − p→+∞ 2

(4.96)

where the number l is defined in (4.67). Let us immediately observe that (4.96) is different from the estimate (4.74) which holds in the higher dimensional case N ≥ 3. This is due to the different asymptotic behavior of the solutions up , as p → +∞, with respect to the case N ≥ 3 where p → N+2 ps = N−2 , as explained in Section 4.2.2. The proof of Proposition 4.16 is very long and difficult, therefore, we indicate its main steps, referring to [105] for the details. The strategy is the same as for the case N ≥ 3, i. e., we consider the normalized eigenfunction φp corresponding to β̃ 1 (p), p ∈ (1, +∞), which is radial and positive. The φ convenient normalization is ‖ |y|p ‖L2 (Ap ) = 1 where Ap is the annulus Ap = Ahp = { y ∈ ℝ2 :

1 < |y| < 1 } . hp

(4.97)

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144 | 4 Morse index of radial solutions of Lane–Emden problems It satisfies

φp N −1 󸀠 { −φ󸀠󸀠 φp − p|up |p−1 φp = β̃ 1 (p) 2 { p − { r r { { { { {φ ( 1 ) = φ (1) = 0 p p hp {

in (

1 , 1) hp

(4.98)

Again, we cannot pass to the limit, as p → +∞, in (4.98) since the term p|up |p−1 is not bounded. Hence it is convenient to rescale φp . The solutions up have two nodal domains and we have seen in Section 4.2, Proposition 4.11, that the asymptotic behavior of the rescaling of the positive part u+p and of the negative part u−p (i. e., zp+ and zp− defined in (4.59) and (4.60)) with the parameters ϵp± defined in (4.58) are different. In particular, their limits are different (see (4.62), (4.63)). Hence we have two possible scalings for the eigenfunction φp : ̂±p (x) = φp (ϵp± x) φ

(4.99)

which satisfy ̂p (x) φ { { ̂±p (x) − Vp± (x)φ ̂±p (x) = β̃ 1 (p) −Δφ { { { |x|2 { { { ± { {φ ̂p = 0 { ±

where

󵄨󵄨 up (ϵp+ x) 󵄨󵄨p−1 󵄨 󵄨󵄨 Vp+ (x) = 󵄨󵄨󵄨 󵄨 , 󵄨󵄨 up (0) 󵄨󵄨󵄨

in

Ap ϵp±

on 𝜕(

Ap ϵp±

󵄨󵄨 up (ϵp− x) 󵄨󵄨p−1 󵄨󵄨 󵄨 Vp− (x) = 󵄨󵄨󵄨 󵄨 󵄨󵄨 up (sp ) 󵄨󵄨󵄨

(4.100) )

(4.101)

where sp is defined in (4.57). For both rescalings, we have by (4.69) and (4.95) that the limit domain is ℝ2 \ {0}, as p → +∞. Moreover, by (4.62) and (4.63) we have 󵄨󵄨 zp+ 󵄨󵄨󵄨p−1 󵄨 0 (ℝ2 ), Vp+ = 󵄨󵄨󵄨1 + 󵄨󵄨󵄨 󳨀→ V + = eU in Cloc 󵄨󵄨 p 󵄨󵄨 󵄨󵄨 zp− 󵄨󵄨󵄨p−1 󵄨 0 Vp− = 󵄨󵄨󵄨1 + 󵄨󵄨󵄨 󳨀→ V − = eZl in Cloc (ℝ2 \ {0}). p 󵄨󵄨 󵄨󵄨

(4.102) (4.103)

̂±p (see [105]) allows to pass to the limit in (4.100). HowThis, together with bounds of φ ever, the difficulty underlined in the case N ≥ 3 in Section 4.3 could appear, i. e., both ̂±p could vanish in the limit and nothing can be deduced by the limit eigenfunctions φ value problem. Here is the main difference with respect to the higher dimensional ̂−p does not converge to zero. Indeed, it holds case, since in [105] it is proved that φ lim inf

̂−p (x)2 φ

∫

p→+∞

{

1 ≤|x|≤k k

}

|x|2

dx > 0

(4.104)

for some k > 1 (see [105], Proposition 6.6).

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4.4 Computation of the Morse index in dimension N = 2

| 145

The proof of (4.104) is nontrivial and requires several delicate estimates. Now we are ready to outline the proof of (4.96). Sketch of proof of Proposition 4.16. Let us define the Hilbert space L21 (ℝN ) = { v ∈ ℝN 󳨀→ ℝ : |x|

endowed with the scalar product (u, v) = ∫ℝN

v ∈ L2 (ℝN ) } , |x| u(x)v(x) |x|2

N≥2

(4.105)

dx. Then we define the space

N 2 N Drad (ℝN ) = D1,2 rad (ℝ ) ∩ L 1 (ℝ ) |x|

(4.106)

where D1,2 (ℝN ) is the subspace of the radial functions in D1,2 (ℝN ) which, in turn, is rad 1

the closure of Cc∞ (ℝN ) with respect to the Dirichlet norm |v|D1,2 (ℝN ) = (∫ℝN |∇v(x)|2 dx) 2 . ̂−p } is bounded in Drad (ℝ2 ), and hence, It is not difficult to see that the sequence {φ ̂ in Drad (ℝ2 ), as p → up to a subsequence, weakly converge to a nonnegative function φ +∞. ̂ ≢ 0. This allows to pass to the limit in By (4.104), it is possible to prove that φ (4.100), proving that there exists β̃ 1 < 0 such that ̂(x)∇ρ(x) dx − ∫ V − (x)φ ̂(x)ρ(x) dx ∫ ∇φ ℝ2 \{0}

= β̃ 1 ∫ ℝ2 \{0}

φ(x)ρ(x) |x|2

ℝ2

dx

(4.107)

where V − (x) = eZl , Zl as in (4.66), (4.67) and ρ is any test function in C0∞ (ℝ2 \ {0}). ̂ is a weak (and also classical) nontrivial nonnegative solution to the limit Namely, φ equation ̂󸀠󸀠 (s) − −φ

̂(s) ̂󸀠 (s) φ φ ̂(s) = β̃ 1 2 , − V − (s)φ s s

s ∈ (0, +∞).

(4.108)

2

̂(δ( √s ) 2+γ ), δ as in (4.67) so that by (4.108) η As in [128], it is convenient to set η(s) = φ 2 2 satisfies − η󸀠󸀠 − with

4β̃ 1 (γ+2)2

< 0 and V(s) = (

the problem

4β̃ 1 η η󸀠 − V(s)η = s (γ + 2)2 s2

1 )2 . 1+ 81 s2

This means that

− Δη − V(|x|)η = λ

η

|x|2

in (0, +∞) 4β̃ 1 (γ+2)2

(4.109)

is the first eigenvalue β̃ ∗ for

in ℝ2 \ {0}.

(4.110)

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146 | 4 Morse index of radial solutions of Lane–Emden problems To know the value of β̃ ∗ is then crucial to compute β̃ 1 , and hence the Morse index of up as shown in the next theorem. In Section 4.5, we will show that β̃ ∗ = −1.

(4.111)

Then using (4.111) we have, by (4.109), that 4β̃ 1 = −1 (γ + 2)2 which, by the definition of γ in (4.67), implies (4.96) since l ≈ 7.1979 (see (4.67)), We can now prove the result on the Morse index of the solution up . Theorem 4.17. Let up be the least energy nodal radial solution of (4.1) with up (0) > 0. Then m(up ) = 12

for p sufficiently large.

Proof. The proof follows the same procedure as in Theorem 4.14. Let hp be defined as in (4.95). By Proposition 4.6, we know that determining the h Morse index m(u ) is equivalent to counting the number k̃ = k̃ p of the negative eigenp h

h

p

p

values μ̃ i (p) = μ̃ i p , i ∈ ℕ+ , of the operator L̃ p = L̃ upp , defined in (4.15). Hence we have to show that k̃p = 12,

for p sufficiently large.

(4.112)

By (4.21), we have that μ̃ j (p) = β̃ i (p) + λk ,

for i, j ∈ ℕ+ and k ∈ ℕ

(4.113)

where λk = k 2 , by (4.20). Moreover, by (4.22), (4.23) the eigenspace associated to λ0 has dimension one, while the eigenspace associated to each λk has dimension two, for k ∈ ℕ+ . By (4.27), we already know that the only negative eigenvalues β̃ i (p) are β̃ 1 (p) ≤ ̃β (p) < 0, so that these are the only eigenvalues which can make μ̃ (p) < 0 in (4.113). 2 j By Proposition 4.15, we know that β̃ 2 (p) > −1 and this implies that β̃ 2 (p) + λk > 0,

for k > 0

(4.114)

while ̃ β̃ 2 (p) + λ0 = β(p) 0

for k ≥ 6

β̃ 1 (p) + λk < 0

for k ≤ 5.

while (4.116)

Remembering that λk has multiplicity two for k ≥ 1 and λ0 has multiplicity one for k = 0, from (4.115) and (4.116) we get exactly 12 negative eigenvalues of the linear operator L̃ p . Hence (4.112) is proved. It is interesting to observe that Theorem 4.17 shows that the formula (4.83) for the Morse index of radial solutions of (4.1) in dimension N ≥ 3, as p → ps , does not hold in dimension N = 2 as p → +∞. Indeed, since the least energy radial solution to (4.1) has two nodal regions inserting N = 2 and n = 2 in (4.83), one would get m(up ) = 4 while Theorem 4.17 gives m(up ) = 12. The reason for this difference relies on the behavior of the radial solutions of (4.1) in dimension N ≥ 3 which is different from that in dimension N = 2. Indeed, while in dimension N ≥ 3 the rescalings of the solutions in each nodal region converge to the same function U defined in (4.29), in dimension N = 2 it happens that the rescalings of the negative and positive parts of up converge to two different functions, namely the ones defined in (4.62) and (4.63). In particular, the rescaling zp− of u−p converge to a solution Zl of the singular Liouville problem in ℝ2 (see Proposition 4.11). The proof of Theorem 4.17, by means of Proposition 4.16, shows clearly that the main contribution to the Morse index of up comes from the nodal domain where up is negative.

4.5 A weighted eigenvalue problem in ℝN In this section, we compute the first eigenvalue of the weighted linear operator L̃ ∗ (v) = |x|2 (−Δv − V(x)v),

x ∈ ℝN , N ≥ 2

(4.117)

where the potential V is defined as 2

1 { { eU(x) = ( ) { { { 1 + 81 |x|2 { V(x) = { N−2 { { 2 { N +2 N(N − 2) p −1 { s {ps U (x) = ( ) 2 N − 2 N(N − 2) + |x| {

if N = 2 (4.118) if N ≥ 3

where U(x) is defined in (4.29) for N ≥ 3 and (4.64) for N = 2.

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148 | 4 Morse index of radial solutions of Lane–Emden problems As in Section 4.4, we consider the Hilbert space Drad (ℝN ) defined in (4.106) endowed with the scalar product (u, v) = ∫ ∇u(x)∇v(x) dx + ∫

u(x)v(x)

ℝN

ℝN

|x|2

dx.

(ℝN ) and in L21 (ℝN ) (see Obviously, Drad (ℝN ) is continuously embedded both in D1,2 rad |x|

(4.106)). Moreover, by the Hardy inequality [139, 140, 184] Drad (ℝN ) = D1,2 (ℝN ) when rad 1,2 2 2 N ≥ 3, while it is well known that Drad (ℝ ) ⊊ Drad (ℝ ). Let us set β̃ ∗ =

inf

v∈Drad (ℝN ) v=0 ̸

R̃ ∗ (v)

(4.119)

where R̃ ∗ (v) =

Q̃ ∗ (v) , v 2 ‖L2 (ℝN ) ‖ |x|

󵄨 󵄨2 Q̃ ∗ (v) = ∫ (󵄨󵄨󵄨∇v(x)󵄨󵄨󵄨 − V(x)v(x)2 ) dx. ℝN

Since the function V(x)|x|2 is bounded, Q̃ ∗ (v) and R̃ ∗ (v) are well-defined for v ∈ Drad (ℝN ). Indeed, ∫ V(x)v(x)2 dx ≤ sup(V(x)|x|2 ) ∫ ℝN

ℝN

ℝN

v(x) |x|2

dx < +∞.

The aim of this section is to compute exactly the value of the first eigenvalue β̃ ∗ of the operator L̃ ∗ defined in (4.117). To do this, an important step is the following result on the spectrum of L̃ ∗ . Proposition 4.18. Let λ ≤ 0 and η ∈ C 2 (ℝN \ {0}) ∩ Drad (ℝN ), η ≥ 0, η ≢ 0, be a radial solution to − Δη − V(x)η = λ

η

|x|2

,

in ℝN \ {0}

(4.120)

Then λ = −(N − 1). The proof of Proposition 4.18 relies on some properties of regular functions in the space Drad (ℝN ) which are stated in the following lemmas.

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4.5 A weighted eigenvalue problem in ℝN

|

149

Lemma 4.19. Let N ≥ 3 and η ∈ C 2 (ℝN \ {0}) ∩ Drad (ℝN ), then |x|N−1 η(x) → 0

as |x| → 0

η(x) →0 |x|

and

Lemma 4.20. Let f ∈ L∞ (ℝ2 ), f ≥ 0 be such that

as |x| → +∞.

1 f( x ) |x|4 |x|2

∈ L∞ (ℝ2 ) and let α ≥ 0. If

η ∈ C 2 (ℝ2 \ {0}) ∩ Drad (ℝ2 ), η ≥ 0 is a nontrivial radial solution of − Δη − f (x)η = −α2

η

|x|2

in ℝ2 \ {0}

(4.121)

then |x|η(x) → 0

as |x| → 0

and

η(x) →0 |x|

as |x| → +∞.

The proof of Lemma 4.19 and Lemma 4.20 are contained in [105] (see Lemma 8.1 and Lemma 8.2 therein). Proof of Proposition 4.18. It is easy to see that the function |x| { { 1 2 { { { 1 + 8 |x| η1 (x) = { |x| { { { N { |x|2 2 { (1 + N(N−2) )

if N = 2 (4.122)

if N ≥ 3

satisfies the equation − Δη1 − V(x)η1 = λ1

η1

|x|2

in ℝN \ {0}

(4.123)

with λ1 = −(N − 1) and V(x) as in (4.118). Let us assume that there exists a function η2 ∈ C 2 (ℝN \ {0}) ∩ Drad (ℝN ) radial and nonnegative for which − Δη2 − V(x)η2 = λ2

η2

|x|2

in ℝN \ {0}

(4.124)

for some λ2 ≤ 0. Since ∫ℝN |∇η2 |2 dx < +∞, then there exist two sequences of radii rn → 0 and Rn → +∞ such that 󵄨 󵄨2 rnN 󵄨󵄨󵄨∇η2 (rn )󵄨󵄨󵄨 󳨀→ 0

󵄨 󵄨2 and RNn 󵄨󵄨󵄨∇η2 (Rn )󵄨󵄨󵄨 󳨀→ 0

as n → +∞.

As a consequence, 󵄨 󵄨 rnN 󵄨󵄨󵄨∇η2 (rn )󵄨󵄨󵄨 󳨀→ 0

󵄨 󵄨 and RNn 󵄨󵄨󵄨∇η2 (Rn )󵄨󵄨󵄨 󳨀→ 0

as n → +∞.

(4.125)

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150 | 4 Morse index of radial solutions of Lane–Emden problems Applying Lemma 4.19 and Lemma 4.20, we get that rnN−1 η2 (rn ) → 0

η2 (Rn ) → 0, Rn

and

as n → +∞.

(4.126)

Next, multiplying (4.123) by η2 and (4.124) by η1 , adding them and integrating over BRn (0) \ Brn (0) we get (λ1 − λ2 )

η1 (x)η2 (x)

∫

|x|2

BRn (0)\Brn (0)

dx =

∫

η1 ∇η2 ⋅ 𝜐 ds

𝜕B Rn (0) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ =:An

+

−

∫

η2 ∇η1 ⋅ 𝜐 dS −

∫

η1 ∇η2 ⋅ 𝜐 ds

𝜕B Rn (0) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝜕B rn (0) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=:Bn

=:Cn

∫

η2 ∇η1 ⋅ 𝜐 dS

(4.127)

𝜕B rn (0) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ =:Dn

where 𝜐 is the outer normal to 𝜕BRn (0). Then using the explicit expression of η1 and (4.125), (4.126), we can estimate An , Bn , Cn and Dn as follows: 󵄨󵄨 󵄨󵄨 󵄨 N−1 −(N−1) 󵄨󵄨 |An | ≤ cn RN−1 󵄨󵄨∇η2 (Rn )󵄨󵄨󵄨 󳨀→ 0, n η1 (Rn )󵄨󵄨∇η2 (Rn )󵄨󵄨 ≤ cN Rn Rn

c η (R ) 󵄨󵄨 󵄨󵄨 N−1 N−1 |Bn | ≤ cn RN−1 = N 2 n 󳨀→ 0, n η2 (Rn )󵄨󵄨∇η1 (Rn )󵄨󵄨 ≤ cN Rn η2 (Rn )Rn Rn 󵄨󵄨 󵄨󵄨 󵄨󵄨 N−1 N−1 󵄨󵄨 |Cn | ≤ cN rn η1 (rn )󵄨󵄨∇η2 (rn )󵄨󵄨 ≤ cN rn rn 󵄨󵄨∇η2 (rn )󵄨󵄨 󳨀→ 0, 󵄨 󵄨 |Dn | ≤ cN rnN−1 η2 (rn )󵄨󵄨󵄨∇η1 (rn )󵄨󵄨󵄨 ≤ cN rnN−1 η2 (rn ) 󳨀→ 0, where in the above estimates the limits are for n → +∞ and we have denoted by cN a generic constant depending only on N. Thus, passing to the limit in (4.127), we get (λ1 − λ2 ) ∫ ℝN

η1 (x)η2 (x) |x|2

dx = 0,

which implies that λ2 = λ1 = −(N − 1) because η1 > 0 in ℝN \ {0} and, by assumption, η2 ≥ 0, η2 ≢ 0. A first estimate for the first eigenvalue β̃ ∗ is easily obtained by using the function η1 defined in (4.122) to evaluate the infimum in (4.119). Indeed, since η1 ∈ Drad (ℝN ) and satisfies (4.123), multiplying both sides of (4.123) by η1 and integrating we get R̃ ∗ (η1 ) = −(N − 1).

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4.5 A weighted eigenvalue problem in ℝN

| 151

Hence we have β̃ ∗ ≤ −(N − 1).

(4.128)

We will show that equality holds in (4.128). Indeed, we have the following. Theorem 4.21. For any N ≥ 2, β̃ ∗ = −(N − 1) and it is achieved by the function η1 defined in (4.122). Proof. We first show a coercivity property. For any v ∈ Drad (ℝN ) which satisfies 2 ∫ℝN v(x)2 dx = 1, one has |x|

2

v(x) 󵄨 󵄨2 Q̃ ∗ (v) = ∫ 󵄨󵄨󵄨∇v(x)󵄨󵄨󵄨 dx − ∫ V(x)|x|2 dx |x|2 N N ℝ

ℝ

v(x)2 󵄨 󵄨2 ≥ ∫ 󵄨󵄨󵄨∇v(x)󵄨󵄨󵄨 dx − sup(V(x)|x|2 ) ∫ dx |x|2 ℝN N N ℝ

(4.129)

ℝ

󵄨 󵄨2 = ∫ 󵄨󵄨󵄨∇v(x)󵄨󵄨󵄨 dx − C, ℝN

where (0 −∞. v Let {vn }n ⊂ Drad (ℝN ) be a minimizing sequence for (4.119) with ‖ |x|n ‖L2 (ℝN ) = 1. We can assume without loss of generality that vn ≥ 0 (because otherwise we could consider |vn |). By the coercivity property (4.129), it follows that vn is bounded in v D1,2 (ℝN ), and hence in Drad (ℝN ), because ‖ |x|n ‖L2 (ℝN ) = 1. Therefore, by the reflexivity rad of Drad (ℝN ), there exists v ∈ Drad (ℝN ) such that, up to a subsequence: vn ⇀ v

in Drad (ℝN ),

vn → v

in Lq (BR ), 1 < q < +∞ if N = 2; 1 < q
0 then 󵄨󵄨 󵄨󵄨 󵄨󵄨 ∫ 󵄨󵄨

{ |x|>R }

󵄨󵄨 󵄨 V(x)(vn (x)2 − v(x)2 ) dx󵄨󵄨󵄨 󵄨󵄨 vn (x)2

≤ sup (V(x)|x|2 )[ ∫ |x|>R

≤

as n → +∞.

{ |x|>R }

C ϵ < 2 2 R

|x|2

dx +

∫

v(x)2

{ |x|>R }

|x|2

dx]

choosing R sufficiently large. On the other hand, fixing the same R, since vn → v in L2 (BR ), also 1

1

in L2 (BR ),

V 2 vn → V 2 v and hence

∫ V(x)vn (x)2 dx 󳨀→ ∫ V(x)v(x)2 dx. BR

BR

Therefore, for n large 󵄨󵄨 󵄨 ϵ 󵄨󵄨 2 2 󵄨󵄨 󵄨󵄨 ∫ V(x)vn (x) dx − ∫ V(x)v(x) 󵄨󵄨󵄨 < , 󵄨󵄨 󵄨󵄨 2 BR

BR

thus proving (4.132). By (4.130), (4.132) and (4.127), it follows that 󵄨 󵄨2 Q̃ ∗ (v) = ∫ (󵄨󵄨󵄨∇v(x)󵄨󵄨󵄨 − V(x)v(x)2 ) dx ℝN

󵄨 󵄨2 ≤ lim inf ∫ (󵄨󵄨󵄨∇vn (x)󵄨󵄨󵄨 − V(x)vn (x)2 ) dx n

ℝN

= β̃ ∗ ≤ −(N − 1) < 0,

(4.133)

in particular, Q̃ ∗ (v) < 0 and so v ≠ 0.

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4.5 A weighted eigenvalue problem in ℝN

| 153

Next, we show that 󵄩󵄩 v 󵄩󵄩 󵄩󵄩 󵄩󵄩 = 1. 󵄩󵄩 󵄩󵄩 󵄩󵄩 |x| 󵄩󵄩L2 (ℝN )

(4.134)

By the definition of β̃ ∗ and (4.133), we have β̃ ∗ ≤ R̃ ∗ (v) =

β̃ ∗ (v) Q̃ ∗ (v) ≤ . v 2 v 2 ‖ |x| ‖L2 (ℝN ) ‖ |x| ‖L2 (ℝN )

(4.135)

Since β̃ ∗ < 0, then necessarily 󵄩󵄩 v 󵄩󵄩 󵄩󵄩 󵄩󵄩 ≥ 1, 󵄩󵄩 󵄩󵄩 󵄩󵄩 |x| 󵄩󵄩L2 (ℝN ) which together with (4.131) gives (4.134). As a consequence from (4.135), we get R̃ ∗ (v) = β̃ ∗ , namely the infimum in (4.119) is attained at v. Finally, since v ≥ 0, v ≠ 0, is a radial solution to v(x) −Δv(x) − V(x)v(x) = β̃ ∗ 2 , |x|

x ∈ ℝN ,

with β̃ ∗ < 0, we can apply Proposition 4.18 obtaining that β̃ ∗ = −(N − 1).

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5 Bifurcation from radial solutions In this chapter, we show how Morse theory can be used to get some bifurcation results. After recalling preliminary theorems in Section 5.1, we will focus on bifurcation from radial positive solutions of the Lane–Emden Dirichlet problem (3.17) in an annulus. We will consider two cases. In the first one, we will take the exponent of the nonlinearity as a bifurcation parameter and will show the existence of many “branches” of nonradial solutions. In the second case, we will study the bifurcation from radial solutions, as the radii of the annulus vary. Most of the results we describe are taken from [129] where the more general case of a linear perturbation added to the nonlinear term is considered. As compared with [129], we will present simplified or more detailed proofs. Finally, let us mention that bifurcation results can also be obtained when the domain is a ball or for sign changing solutions, again using Morse index estimates (see [10, 11, 130]).

5.1 Preliminaries We consider the Lane–Emden Dirichlet problem that we rewrite in the case when the domain is an annulus and for positive solutions: p

−Δu = u { { { u>0 { { { {u = 0

in A (5.1)

in A on 𝜕A

where p > 1 and A = { x ∈ ℝn , a < |x| < b }, b > a > 0, N ≥ 2. We recall that for any p > 1, problem (5.1) has only one radial solution u (see [148, 182]) which is also nondegenerate in the space of radial functions. This means that if we denote, as in (3.21), by Lu the linearized operator at u: Lu = −Δ − pup−1

(5.2)

then zero is not an eigenvalue of Lu to which a radial eigenfunction corresponds. A direct easy proof of this property of u is given in the next proposition. Proposition 5.1. Let u be the radial solution of (5.1). Then the linearized problem at u: −Δv − pup−1 v = 0

in A

v=0

in A

{

(5.3)

does not admit any nontrivial radial solution. https://doi.org/10.1515/9783110538243-005

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156 | 5 Bifurcation from radial solutions Proof. Arguing by contradiction, let us assume that v̄ ≠ 0 is a radial solution of (5.3). Since the positive radial solution of (5.1) is unique, it is obvious that u is the least energy radial solution, and hence it has radial Morse index one, as shown in Sec1 tion 3.2.2 (arguing in the space H0,rad (A)). Then, denoting by βi , i ∈ ℕ+ , the radial eigenvalues of Lu , as in Section 4.1, we have that the contradiction assumption implies that β2 = 0. Thus the corresponding eigenfunction φ2 has only two nodal regions, by Courant’s nodal domain theorem. We denote them by A1 = { x ∈ ℝN : a < |x| < d } and A2 = { x ∈ ℝN : d < |x| < b }, d ∈ (a, b), and in each of them the first eigenvalue of the linearized operator Lu is zero. It is easy to see that the radial function z = x ⋅ ∇u +

2 u p−1

satisfies −Δz − pu { { { z>0 { { { {z < 0

p−1

z=0

in A if |x| = a if |x| = b

Hence z changes sign in A. We claim that z has only two nodal domains which are annuli. Indeed if it had more than two nodal regions then, by the boundary behavior it should have at least four nodal regions Bj , j = 1, . . . , k, k ≥ 4. Thus, for at least one region Bj it should happen that Bj ⊂ Ai , for some i ∈ {1, 2}, and obviously z = 0 on 𝜕Bj . This implies that 0 = λ1 (Lu , Bj ) > λ1 (Lu , Ai ) = 0 which is a contradiction (here λ1 (Lu , X) denotes the first eigenvalue of the operator Lu in the domain X). Hence the function z has only two nodal regions B1 and B2 and, necessarily, Ai ⊆ Bj , for some i, j = 1 or 2. Using z as a test function, it is easy to see that λ1 (Lu , Bj ) > 0 against the fact that λ1 (Lu , Bj ) ≤ λ1 (Lu , Ai ) = 0. In order to study possible bifurcations from the radial solution u, as described in Section 5.2 and Section 5.3, we need to analyze the spectrum of the linearized operator Lu to detect the values of the exponent p or of the radii of A at which Lu admits zero as an eigenvalue, in the whole space H01 (A). To do this, we use the same spectral decomposition performed in Section 4.1, for the case of radial nodal solutions in a ball. However, the procedure is a bit easier when the domain is an annulus since we do not have to worry about the singularity at the origin of the auxiliary weighted eigenvalue problem. Let us summarize what is needed in the next sections.

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5.1 Preliminaries | 157

We consider the weighted linear operator L̃ u : H 2 (A) ∩ H01 (A) 󳨀→ L2 (A) L̃ u (v) := |x|2 (−Δv − pu(x)p−1 v),

x∈A

(5.4)

and the associated radial operator L̃ u,rad : H 2 ((a, b)) ∩ H01 ((a, b)) 󳨀→ L2 ((a, b)) N −1 󸀠 L̃ u,rad (v) := r 2 (−v󸀠󸀠 − v − pup−1 (r)v), r

r ∈ (a, b).

(5.5)

By Lemma 4.4, we have the spectral decomposition σ(L̃ u ) = σ(L̃ u,rad ) + σ(−ΔSN−1 )

(5.6)

and, by Lemma 4.3, we have that the Morse index m(u) of the solution u is equal to the number of the negative eigenvalues of L̃ u , while the radial Morse index mrad (u) is equal to the number of negative eigenvalues of the operator L̃ u,rad . To get bifurcation results, we need to understand whether the linearized operator Lu can have zero as an eigenvalue. Denoting by μ̃ i the eigenvalues of L̃ u , by β̃ i those of L̃ u,rad , i ∈ ℕ+ , and by λk the eigenvalues of ΔSN−1 , k ∈ ℕ, we get the following. Lemma 5.2. Let u be the radial solution of (5.1). The linearized operator Lu admits zero as an eigenvalue if and only if there exists k ≥ 1 such that β̃ 1 + λk = 0.

(5.7)

Proof. We start observing that a function v is a nontrivial solution of (5.3) if and only if it satisfies L̃ u (v) = 0 and v = 0 on 𝜕A. Hence zero is an eigenvalue of Lu if and only if zero is an eigenvalue of L̃ u . By (5.6), all eigenvalues μ̃ j of L̃ u , j ∈ ℕ+ are given by μ̃ j = β̃ i + λk ,

i, j ∈ ℕ+ , k ∈ ℕ.

(5.8)

Thus we have to check for what i and k the eigenvalue μ̃ j is zero in (5.8). Since λk ≥ 0 (by (4.20)), we have to analyze when β̃ i ≤ 0, for i ∈ ℕ+ . By Lemma 4.3, the number of the negative eigenvalues of L̃ u,rad is the same as the number of the negative radial eigenvalues of Lu . We have already observed in the proof of Proposition 5.1 that only the first radial eigenvalue β1 of Lu is negative. Hence β̃ i < 0 only for i = 1. On the other side, some β̃ i can be zero if and only if (5.3) has a nontrivial radial solution, which is excluded by Proposition 5.1. Thus the assertion is proved because λ0 = 0 (by (4.20)). Note that in (5.7) λk depends only on the dimension N, while β̃ 1 = β̃ 1 (u) depends on the solution u which, in turn, depends on the data of the Dirichlet problem (5.1) which are the exponent p > 1 and the annulus A. In the next sections, we analyze the behavior of β̃ 1 with respect to p and to A in order to deduce the bifurcation results.

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158 | 5 Bifurcation from radial solutions

5.2 Bifurcation with respect to the exponent p In this section, we consider the exponent p in (5.1) as a bifurcation parameter, while the annulus A is fixed. To emphasize the dependence on p of the unique radial solution of (5.1), we will denote it by up . For the same reason, the first eigenvalue β̃ 1 of the operator L̃ up ,rad which appears in (5.7) will be denoted by β̃ 1 (p). To solve the equation (5.7), we need to analyze the behavior of β̃ 1 (p), as p varies, which, by the definition of the operator L̃ u ,rad , depends on the behavior of the radial p

solution up , with respect to p.

5.2.1 Asymptotic analysis of the radial solution The exponent p of the nonlinear term in (5.1) belongs to the interval (1, +∞) and we wish to analyze the behavior of the corresponding radial solution up , as p → 1 or p → +∞. The first case is easier. Proposition 5.3. Let up be the radial solution of (5.1). Then lim‖up ‖p−1 ∞ = λ1

p→1

where λ1 is the first Dirichlet eigenvalue of −Δ in A. Proof. Let us start by showing that ‖up ‖p−1 ∞ is bounded, as p → 1. To do this, we can follow the proof of Lemma 3.9, for λ = 0, arguing by contradiction and so assuming that for a sequence pn ↘ 1 it holds Mnpn −1 = ‖un ‖p∞n −1 = [upn (xpn )]

pn −1

󳨀→ +∞.

Then, if xpn stay away from 𝜕A as in the case of Lemma 3.9, where the domain is supposed to be convex, then we reach a contradiction exactly as in the proof of Lemma 3.9. If instead the distance dn of xpn to the boundary 𝜕A tends to zero, as n → +∞, then we distinguish two cases: either

dn 󳨀→ +∞, (Mn )pn −1

or

dn 󳨀→ δ ≥ 0. (Mn )pn −1

In the first case, considering the rescaled function ũ n =

1 u ( Mn n

x

pn −1 2

Mn

+ xp ) it is easy to

see that it converges locally uniformly to a function ũ which is a nontrivial solution of the same linear problem in ℝN as in Lemma 3.9 and then a contradiction is reached in the same way.

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5.2 Bifurcation with respect to the exponent p

| 159

In the second case, exactly as in the proof of Case 2 of Theorem 1.1 of [125] we can prove that δ > 0, so that the rescaled functions ũ n converge locally uniformly to a function ũ which solves −Δũ = ũ in H = { x = (x1 , . . . , xN ), xN > −δ } { { { { { in H {ũ > 0 { { ̃ u(0) =1 { { { { on 𝜕H {ũ = 0

(5.9)

This is not possible by Theorem 2 in [87] which shows that a solution of (5.9) must be strictly monotone increasing in the xN direction. Hence ‖up ‖p−1 ∞ is bounded, as p → 1 and we denote by μ its limit. Then, exactly as in the proof of Lemma 3.9 (with λ = 0), we obtain that μ = λ1 . Now we pass to the study of up , as p → +∞. This is much more difficult, and we state below the results obtained in [134]. Proposition 5.4. Let up be the radial solution of (5.1). Then, as p → ∞, in C 0 (A),

up (|x|) 󳨀→ ω(|x|)

(5.10)

where, if N ≥ 3, ω(|x|) =

a2−N − |x|2−N 2 { − b2−N |x|2−N − b2−N

for a ≤ |x| ≤ r0

log|x| − log a 2 { log b − log a log b − log|x|

for a ≤ |x| ≤ r0

a2−N

for r0 ≤ |x| ≤ b

while, if N = 2 ω(|x|) =

for r0 ≤ |x| ≤ b

with r0 given by 1

{ a2−N + b2−N 2−N { {( ) 2 { { { √ { ab

if N ≥ 3 if N = 2

and we recall that a, b are the radii of the annulus A. Moreover, there exists C > 0 such that |∇up | ≤ C

∀p > 1.

(5.11)

Proof. For (5.10), we refer to Theorem 1.1 of [134] and for (5.11) to Proposition 2.4 of [134].

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160 | 5 Bifurcation from radial solutions To study the local behavior of up at the radius of A where it achieves its maximum we consider the rescalings, in polar coordinates: ũ p (r) =

p (u (ϵ r + rp ) − ‖up ‖∞ ) ‖up ‖∞ p p

(5.12)

where ϵp−2 = p‖up ‖p−1 ∞ ,

r = |x|

and

up (rp ) = ‖up ‖∞ .

Note that ϵp is defined analogously as in (4.58). Proposition 5.5. We have 1 in Cloc (ℝ)

ũ p 󳨀→ U where U(r) = log

4e

√2r

(1 + e√2r )2

(5.13)

is the only solution of the ODE problem −z 󸀠󸀠 = ez { z(0) = z 󸀠 (0) = 0

in ℝ

(5.14)

Proof. See Proposition 3.2 in [134]. The last asymptotic estimate we need is the following one. Proposition 5.6. Let up be the radial solution of (5.1). Then, for p sufficiently large and some positive constants c1 , c2 , ũ p ≤ c1 U + c2 in the interval [ap , bp ] with ap =

a−rp , ϵp

bp =

b−rp ϵp

and the function U is as in (5.12).

Proof. See Lemma 4.2 of [129].

5.2.2 Asymptotic behavior of the radial eigenvalue β̃ 1 (p) We start by showing the following result. Proposition 5.7. The map p 󳨃󳨀→ β̃ 1 (p) is real analytic.

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(5.15)

5.2 Bifurcation with respect to the exponent p

|

161

Proof. The real analyticity of β̃ 1 (p) follows by a result of Kato [147], once we prove that up is real analytic in p. Let φ1 be the first positive Dirichlet normalized eigenfunction of −Δ in A. We define 󵄩󵄩 u 󵄩󵄩 󵄩 󵄩 Cφ1 = { u ∈ C0 (A) : u is radial and 󵄩󵄩󵄩 󵄩󵄩󵄩 ≤ k, for some k > 0 } . 󵄩󵄩 φ1 󵄩󵄩∞ It was proved by Dancer in [88] that if p0 and u0 are such that there exists μ > 0 for which u ≥ μφ1 , then the map (p, u) ∈ (1, +∞) × Cφ1 󳨀→ (−Δ)−1 (up ) ∈ Cφ1 is analytic near (p0 , u0 ). We claim that ∀p > 1, the function up belongs to Cφ1 and

∃μ > 0 such that up ≥ μφ1 in A.

(5.16)

Let us assume that (5.16) holds. Then the function up is the only solution of the equation F(p, u) = 0 where F: (1, +∞) × Cφ1 󳨀→ Cφ1 is defined as F(p, u) = u − (−Δ)−1 (up ). We observe that, by the result of [88] quoted before, the map F is analytic in a neighborhood of (p0 , up0 ) for any p0 > 1. Moreover, the map Fu (p0 , up0 ) is an isomorphism since up0 is nondegenerate in the space of radial functions, for any p0 > 1. Thus, from the analytic version of the implicit function theorem, it follows that the map p 󳨃󳨀→ up is real analytic. It remains to prove the claim (5.16). By the symmetry of the domain, we know that the first eigenfunction φ1 is radial. u We set ψ = φp > 0, in A. By the Hopf boundary lemma, we have that limr→a ψ(r) = u󸀠p (a) φ󸀠1 (a)

1

> 0 and the same holds at b. This implies that there exist constants c, C > 0 such

that C ≥ ψ(r) ≥ c > 0 in a neighborhood of 𝜕A and the same holds (possibly changing the constants) in the interior of A. This completes the proof. In the next proposition, we study the behavior of β̃ 1 (p), as p → 1. Proposition 5.8. We have β̃ 1 (p) 󳨀→ 0,

as p → 1.

(5.17)

Proof. Let us consider the positive eigenfunction w1,p corresponding to the eigenvalue β̃ 1 (p), with ‖w1,p ‖∞ = 1. It satisfies β̃ 1,p N −1 󸀠 󸀠󸀠 { {−w1,p − w1,p − pup−1 w p w1,p = r r 2 1,p { { {w1,p (a) = w1,p (b) = 0

in (a, b)

(5.18)

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162 | 5 Bifurcation from radial solutions Moreover, by the variational characterization of β̃ 1 (p) we have b

b

2 N−1 󸀠 2 N−1 dr )r dr − p ∫a up−1 ∫a (w1,p p w1,p r

β̃ 1 (p) =

b

2 r N−3 dr ∫a w1,p

≥

b

2 N−1 dr −p‖up ‖p−1 ∞ ∫a w1,p r

≥

b

2 r N−3 dr ∫a w1,p

≥ −Cp

(5.19)

so that β̃ 1 (p) is bounded, as p → 1, and hence, up to a sequence, β̃ 1 (p) 󳨀→ β̃ 1 ≤ 0, as p → 1. By standard regularity theorem we have that w1,p 󳨀→ w1 , as p → 1 in C 2 (A). Therefore, passing to the limit in (5.18) and using Proposition 5.3 we get that w1 satisfies β̃ N −1 󸀠 { { −w1󸀠󸀠 − w1 = λ1 w1 + 21 w1 { { r { r { w > 0 { 1 { { { {w1 (a) = w1 (b) = 0

in (a, b) in (a, b)

i. e., β̃ { {−Δw1 − λ1 w1 = 12 |x| { { w = 0 { 1

in A on 𝜕A

(5.20)

Then, multiplying by φ1 , which is the first positive normalized eigenfunction of −Δ in A, and integrating we get wφ ∫ ∇w1 ∇φ1 − λ1 ∫ w1 φ1 = β̃ 1 ∫ 1 21 . |x|

(5.21)

A

A

A

But, by the definition of φ1 , using w1 as a test function, we have that ∫ ∇w1 ∇φ1 − λ1 ∫ w1 φ1 = 0. A

A

Hence, from (5.21) we deduce that β̃ 1 = 0 and then w1 = φ1 , by (5.20). Now we prove an asymptotic expansion of β̃ 1 (p), as p → +∞. Proposition 5.9. We have lim

p→+∞

and

‖up ‖p∞ p

= lim− r→r0

ω󸀠 (r)2 =M>0 2

1 β̃ 1 (p) = − Mr02 p2 + o(p2 ), 2

as p → +∞

where ω and r0 are defined as in Proposition 5.4.

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(5.22)

(5.23)

5.2 Bifurcation with respect to the exponent p

|

163

Proof. Let us start by proving (5.22). It is convenient to write the equation for up in radial coordinates, i. e., − u󸀠󸀠 p −

N −1 󸀠 up = upp r

in (a, b).

(5.24)

Denoting again by rp the radius where up achieves its maximum, multiplying (5.24) by u󸀠p and integrating on the interval (a, rp ) we get rp

rp

󸀠 − ∫ u󸀠󸀠 p up dr a

− (N − 1) ∫ a

(u󸀠p )2 r

rp

dr = ∫ up u󸀠 dr. a

This implies u󸀠p (a)2 2

rp

− (N − 1) ∫

(u󸀠p )2 r

a

dr =

up+1 (rp ) p+1

=

‖up ‖p+1 ∞ p+1

.

By Proposition 5.4, we have that ‖up ‖∞ 󳨀→ 1, as p → +∞ and (5.11) holds. Hence we can pass to the limit obtaining lim

p→+∞

‖up ‖p∞ p

= lim

p→+∞

r0

‖up ‖p+1 ∞

1 (ω󸀠 )2 = ω󸀠 (a)2 − (N − 1) ∫ dr. p+1 2 r

(5.25)

a

On the other side, the function ω satisfies N −1 󸀠 ω = 0 in (a, r0 ) {−ω󸀠󸀠 − r { {ω(a) = 0, ω(r0 ) = 1

(5.26)

as well as an analogous problem in (r0 , b). Thus, multiplying the equation (5.26) by ω󸀠 and integrating on the interval (a, r0 ) we get r0

ω󸀠 (r )2 ω󸀠 (a)2 (ω󸀠 )2 − − 0 + − (N − 1) ∫ dr = 0 2 2 r

(5.27)

a

where ω󸀠− (r0 ) = limr→r0− ω󸀠 (r). Combining (5.27) and (5.25), we obtain (5.22). To prove (5.23), we consider, as in Proposition 5.8, the positive, L∞ -normalized eigenfunction w1,p , corresponding to the eigenvalue β̃ 1 (p). We recall that it satisfies (5.18) and the variational characterization (5.19) holds. Then, by (5.22) we get β̃ 1 (p) ≥

b

2 N−1 −p‖up ‖p−1 dr ∞ ∫a w1,p r b

2 r N−3 dr ∫a w1,p

≥ −Cp2

(5.28)

for a constant C ≥ 0.

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164 | 5 Bifurcation from radial solutions Now we define w̃ 1,p (t) = w1,p (ϵp t + rp ), with ϵp as in (5.12), which solves 󸀠󸀠 − w̃ 1,p − ϵp

ϵp2 β̃ 1 (p) ũ p p N −1 󸀠 w̃ 1,p = (1 + ) w̃ 1,p + w̃ . ϵp s + rp p (ep s + rp )2 1,p

(5.29)

Since ‖w̃ 1,p ‖∞ = 1, we have that w̃ 1,p 󳨀→ ψ̃ 1 ≥ 0, as p → +∞ uniformly on compact sets of ℝ. Passing to the limit in (5.29), by Proposition 5.5, we get −ψ̃ 1 − eU ψ̃ 1 = β̃ 1 ψ̃ 1 { ψ̃ 1 ≥ 0

in ℝ

(5.30)

where U is defined in (5.13) and β̃ 1 = lim

p→+∞

ϵp2 β̃ 1 (p) rp2

(5.31)

.

Note that the limit in (5.31) is finite by (5.22), (5.28) and the definition of ϵp . We would like to prove that ψ̃ 1 ≠ 0 so that it is a first eigenfunction for (5.30). To do this, we proceed as follows. With the change of variable r = √2 log s and Z(s) = φ1 (r), we get that the function Z satisfies 2β̃ 1 Z󸀠 8 { {−Z 󸀠󸀠 − Z + Z = s (1 + s2 )2 s2 { { {‖Z‖∞ ≤ 1 and Z ≥ 0

in (0, +∞)

(5.32)

To show that Z ≠ 0, we denote by ηp the point where w1,p achieves its maximum, i. e., w1,p (ηp ) = 1, and set η̃ p =

ηp −rp . ϵp

󸀠 w1,p (ηp ) = 0,

󸀠󸀠 w1,p (ηp ) ≤ 0

󸀠 w̃ 1,p (η̃ p ) = 0,

󸀠󸀠 w̃ 1,p (η̃ p ) ≤ 0.

Then w̃ 1,p (η̃ p ) = 1,

From (5.29), we deduce 󸀠󸀠 − w̃ 1,p (η̃ p ) = (1 +

ũ p (η̃ p ) p

p

) + ϵp2 β̃ 1 (p)

1 . η2p

We claim that η̃ p 󳨀→ η0 ∈ ℝ.

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(5.33)

5.2 Bifurcation with respect to the exponent p

|

165

By contradiction, let us assume that η̃ p 󳨀→ +∞ (the case when η̃ p 󳨀→ −∞ is similar). By (5.15), we get (1 +

ũ p (η̃ p ) p

p

) ≤ Ceup (ηp ) ≤ Ce−Cηp 󳨀→ 0. ̃

̃

̃

Observing that ϵp2 β̃ 1 (p) 󳨀→ β̃ 1 r02 < 0, as p → +∞ and ηp ∈ (a, b), from (5.33) it follows 󸀠󸀠 −w̃ 1,p (η̃ p ) < 0

which is not possible. Hence η̃ p 󳨀→ η0 ∈ ℝ and Z(η0 ) = 1, so that Z ≠ 0. Therefore, Z is a first eigenfunction for (5.32), and hence, by Theorem 4.21 of Section 4.5 (the case N = 2) we have that 2β̃ 1 = β̃ ∗ = −1. Since β̃ 1 = − 21 , by the limit (5.31) and (5.22) we deduce ϵp2 β̃ 1 (p) β̃ 1 (p) β̃ 1 (p) 1 = − + o(1) = = 2 p−1 2 2 rp p‖up ‖∞ rp2 p (M + o(1))(r02 + o(1)) which gives (5.23).

5.2.3 Bifurcation results We start with a definition (see [66, 150, 195] and the references therein for classical results in bifurcation theory). Definition 5.10. Let p̄ > 1 and let up̄ be the radial solution of (5.1) in the annulus A, for p = p.̄ We say that a nonradial bifurcation occurs at (up̄ , p)̄ if in every neighborhood of (up̄ , p)̄ in C 1,α (A) × (1, +∞) there exists a point (vp , p) where vp is a nonradial solution of (5.1). Definition 5.11. If, in the previous definition, it happens that for any neighborhood of (up̄ , p)̄ in C 1,α (A) × (1, +∞) there exist at least m distinct points (upi , pi ), i = 1, . . . , m with upi nonradial solution of (5.1), then we say that a m-nonradial bifurcation occurs at ̄ (up̄ , p). The bifurcation results we present are the following. Theorem 5.12. There exists a sequence of exponents pk , k ∈ ℕ+ , such that pk 󳨀→ +∞, as k → +∞, and a nonradial bifurcation occurs at (upk , pk ). Moreover, for k even, a [ N2 ]-nonradial bifurcation occurs at (upk , pk ) ([ N2 ] is the integer part of N2 ). To prove this theorem, we go back to equation (5.7) and show the following result.

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166 | 5 Bifurcation from radial solutions Proposition 5.13. For any k ≥ 1, there exists an exponent pk such that β̃ 1 (pk ) + λk = 0.

(5.34)

Moreover, pk 󳨀→ +∞ as k → +∞. Proof. It follows by Proposition 5.7, Proposition 5.8 and Proposition 5.9. As a consequence of (5.23) and using Lemma 4.3 and Lemma 4.5 (where Ah is just A), we easily get the following. Proposition 5.14. Let up be the radial solution of (5.1). The Morse index m(up ) goes to +∞, as p → +∞. Now we observe that the equation in (5.1) can be written as u = Tp (u), where Tp (w) = (−Δ)−1 (|u|p−1 u) and Tp is a compact operator from C01,α (A) into C01,α (A). If D is an open bounded set in C01,α (A) such that I − Tp ≠ 0 on 𝜕D, then the Leray– Schauder degree for the map I − Tp , i. e., deg(I − Tp , D, 0) is well-defined. Now let us observe that, by Proposition 5.7, the set of numbers p which satisfy (5.7), for a given k ≥ 1, is at most finite. Hence the exponents pk , defined by (5.34) are isolated and so, for p in a suitable neighborhood of pk the operator I − Tp󸀠 (up ) is invertible. Hence for p ≠ pk , we have deg(I − Tp , B, 0) = deg(I − Tp󸀠 (up ), D, 0) = (−1)mp

(5.35)

if B is a neighborhood of up in C01,α (A) such that up is the only solution of (I − Tp )(v) = 0 in B and mp = m(up ) is the Morse index of up . Proof of Theorem 5.12. We start by pointing out that the equation (5.1) is invariant with respect to the action of the orthogonal group O(N) and its subgroups. Then we consider the subspace ℱ of C 1,α (A) given by ℱ := { v ∈ C

1,α

(A) : v(x󸀠 , xN ) = v(g(x 󸀠 ), xN ), ∀g ∈ O(N − 1) }

where x󸀠 = (x1 , . . . , xN−1 ) and O(N − 1) is the orthogonal group in ℝN−1 . By a result of [204] (Proposition 5.2 therein) we have that, for any k ≥ 1, the eigenspace Vk spanned by the eigenfunctions corresponding to the eigenvalue λk of the Laplace–Beltrami operator on SN−1 , which are O(N − 1) invariant, is one-dimensional. Then let {pk } be the sequence of exponents defined in Proposition 5.13. Since the Morse index of upk tends to +∞, by Proposition 5.14, we have that, up to a subsequence, that we still denote by {pk }, at each pk the number of the negative eigenvalues of the linearized operator Lup increases, and in the space ℱ can only increase by one because of (5.6) and the fact that the eigenfunctions of L̃ u are a product of the eigenfunction of L̃ u,rad by an p

eigenfunction of −ΔSN−1 . Hence the Morse index of up , in the space ℱ , increases by one, crossing pk , i. e., m(upk +ϵ ) = m(upk −ϵ ) + 1, if ϵ is small enough.

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5.3 Bifurcation with respect to the radius of the annulus | 167

Then deg(I − Tpk −ϵ , Bpk −ϵ , 0) = (−1)deg(I − Tpk +ϵ , Bpk +ϵ , 0) where Bp is a neighborhood of up in the space ℱ . This implies that there is a change in the degree at the point (upk , pk ) and then a bifurcation must occur, by the classical results in bifurcation theory (see, e. g., [150]). Moreover, the bifurcating solutions are nonradial since the radial solution up is nondegenerate in the space of radial functions. Finally, by the maximum principle, the bifurcating solutions are positive. To get a [ N2 ]-nonradial bifurcation result when k is even, we consider the subgroup od O(N) defined by 𝒢h = O(h) × O(N − h),

for 1 ≤ h ≤ [

N ]. 2

In [205], it is shown that if k is even then the eigenspace relative to the eigenvalue λk of −ΔSN−1 , restricted to the functions invariant by the action of 𝒢h , has dimension one. Moreover, any solution u of (5.1) that is invariant with respect to the action of both 𝒢h1 and 𝒢h2 with h1 ≠ h2 , must be radial. Hence nonradial solutions which are invariant for the action of different groups 𝒢h are actually distinct. So if we repeat the previous proof considering, instead of ℱ the subspaces of functions which are invariant with respect to the action of G̃ h , for 1 ≤ h ≤ [ N2 ], we get the existence of [ N2 ] distinct nonradial positive solutions of (5.1) bifurcating from upk , for k even.

5.3 Bifurcation with respect to the radius of the annulus In this section, we consider the case of expanding annuli AR := {x ∈ ℝN , R < |x| < R + 1}, R > 0 and we will consider the radius R as the bifurcation parameter. To stress the dependence on R, we will denote by uR the unique radial solution (5.1) in AR and by β̃ 1 (R) the first eigenvalue of the operator L̃ uR ,rad which appears in (5.7). As in Section 5.2, we have to analyze the behavior of uR and of β̃ 1 (R), as R → +∞. 5.3.1 Asymptotic estimates for the radial solution The radial positive solution uR satisfies the following ODE problem: N −1 󸀠 { −u󸀠󸀠 uR = upR in (R, R + 1) { R − { r { { uR > 0 in (R, R + 1) { { { {uR (R) = uR (R + 1) = 0

(5.36)

for any fixed p > 1.

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168 | 5 Bifurcation from radial solutions As recalled in Section 5.1, there exists only one solution of (5.36) and it is nondegenerate in the space of radial functions (Proposition 5.1). To study the behavior of uR , as R → +∞, we consider the function ũ R (t) = uR (t + R),

t ∈ [0, 1]

(5.37)

for which we prove the following results. Proposition 5.15. We have: (i) ũ R 󳨀→ ũ 0 uniformly in (0, 1) where ũ 0 is the unique positive solution of −u󸀠󸀠 = up in (0, 1) { { { u>0 in (0, 1) { { { {u(0) = u(1) = 0

(5.38)

(ii) the function R 󳨃󳨀→ ũ R (t) is continuously differentiable with respect to R, for any t ∈ (0, 1) and it holds 1 󵄨󵄨 𝜕ũ 󵄨󵄨q 󵄨 󵄨 lim Rq ∫󵄨󵄨󵄨 R 󵄨󵄨󵄨 dt = 0, 󵄨󵄨 𝜕R 󵄨󵄨 R→∞

∀q > 1.

(5.39)

0

Proof. Since uR is the unique positive solution of (5.1) or (5.36) it is easy to see, by constrained minimization, that it achieves the following infimum: R+1

βR =

inf

u∈H01 (R,R+1) u=0 ̸

∫R (u󸀠 )2 r N−1 dr

R+1

2

(∫R |u|p+1 r N−1 dr) p+1

.

Taking a function φ ∈ C0∞ ((R, R + 1)) and making a change of variable, we get 1

R+1

βR ≤

∫R (φ󸀠 )2 r N−1 dr R+1

(∫R

≤ CR

2

φp+1 r N−1 dr) p+1

2 (N−1)(1− p+1 )

≤

∫0 (φ󸀠 )2 (R + t)N−1 dr 1

2

(∫0 φp+1 (R + t)N−1 dr) p+1

.

Thus, from the equation (5.36) we get R+1

p+1

N−1 dr = βRp−1 ≤ CRN−1 . ∫ up+1 R r R

Then for the function ũ R (t) defined in (5.37) we obtain 1

∫ ũ p+1 R dt 0

R+1

= ∫ R

up+1 R dr

≤

1

RN−1

R+1

N−1 dr ≤ C. ∫ up+1 R r R

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(5.40)

5.3 Bifurcation with respect to the radius of the annulus | 169

On the other side, from (5.36), multiplying by uR and integrating we get R+1

R+1

2

N−1 dr. ∫ (u󸀠R ) r N−1 dr = ∫ up+1 R r R

(5.41)

R

By (5.40) and (5.41), we derive 1

2 ∫(ũ 󸀠R ) dt

≤

0

=

1

1

RN−1 1

2

∫(ũ 󸀠R ) (t + R)N−1 dt = 0

(5.42)

R+1

RN−1

∫ R

2 (u󸀠R ) r N−1 dt

≤ C.

Now we observe that ũ R satisfies N −1 󸀠 { −ũ 󸀠󸀠 ũ = ũ pR { R − { t+R R { { {ũ R > 0 { { {ũ R (0) = ũ R (1) = 0

in (0, 1) in (0, 1)

(5.43)

Thus, since, by (5.42), ũ R is bounded in H01 (0, 1), it is also bounded in C 2 (0, 1)∩L∞ (0, 1), and hence ũ R 󳨀→ ũ 0 uniformly, as R → +∞, where ũ 0 satisfies (5.38). Note that u0 ≠ 0 because ‖ũ R ‖∞ = ‖uR ‖∞ ≥ α > 0 where α is a constant independent of R. Indeed multiplying the equation in (5.36) by the first eigenfunction ψ1,R of −Δ in AR , with corresponding eigenvalue μ1 (R), and integrating we get ∫ ∇uR ∇ψ1,R dx = ∫ upR ψ1,R dx = μ1 (R) ∫ uR ψ1,R dx.

AR

AR

AR

From this, we derive ‖uR ‖p−1 ∞ ∫ uR ψ1,R dx ≥ μ1 (R) ∫ uR ψ1,R 1, dx AR

AR

so that 1

‖uR ‖∞ ≥ [μ1 (R)] p−1 . Since it is easy to see that μ1 (R) 󳨀→ μ1 > 0, where μ1 is the first eigenvalue of the problem −u󸀠󸀠 = μu

in (0, 1),

u(0) = u(1) = 0

we get the bound from below for ‖uR ‖∞ . This proves assertion (i).

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170 | 5 Bifurcation from radial solutions Since the solution uR is nondegenerate in the space of radial functions, applying the implicit function theorem to the function N −1 󸀠 ψ + ψp t+R

F(ψ, R) = ψ󸀠󸀠 +

it is easy to see that ũ R is continuously differentiable with respect to R. The function ũ R satisfies V(⋅, R) = 𝜕𝜕R N −1 󸀠 N −1 󸀠 󸀠󸀠 { V + ũ = pũ p−1 {−V − R V t+R (t + R)2 R { { {V(0) = V(1) = 0

in (0, 1)

We claim that 󵄩 󵄩 R󵄩󵄩󵄩V(⋅, R)󵄩󵄩󵄩H 1 ((0,1)) ≤ C 0

(5.44)

for some constant C > 0. Indeed, if (5.44) does not hold, then, for a sequence Rn → +∞ we have 󵄩 󵄩 Rn 󵄩󵄩󵄩V(⋅, Rn )󵄩󵄩󵄩H 1 ((0,1)) 󳨀→ +∞,

as n → +∞.

0

The function zn =

V(⋅,Rn ) ‖V(⋅,Rn )‖H 1 ((0,1)) 0

satisfies

(N − 1)Rn ũ 󸀠Rn N −1 󸀠 { 󸀠󸀠 { z + = pũ p−1 {−zn − Rn zn t + Rn n (t + Rn )2 Rn ‖V(⋅, Rn )‖H 1 ((0,1)) { 0 { { {zn (0) = zn (1) = 0

(5.45)

and it converges weakly in H01 ((0, 1)) and strongly in Lq ((0, 1)), for any q > 1 to a function z0 . Moreover, ũ 󸀠Rn = u󸀠Rn is bounded in L∞ ((0, 1)). Indeed by the equation in (5.36), integrating between the point aR ∈ (R, R + 1) where uR achieves its maximum and r, we get r

r

−u󸀠R (r)r N−1 = − ∫(u󸀠R r N−1 ) dr = ∫ tRp r N−1 dr ≤ CRN−1 󸀠

ar

ar

by (5.42). Hence 󵄨󵄨 󸀠 󵄨󵄨 N−1 ≤ CRN−1 . 󵄨󵄨uR (r)󵄨󵄨R So, passing to the limit in (5.45) we get −z0󸀠󸀠 = pũ p−1 0 z0

{

in (0, 1)

z(0) = z(1) = 0

where ũ 0 is the limit function defined in (i).

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5.3 Bifurcation with respect to the radius of the annulus | 171

Then z0 ≡ 0, because ũ 0 is a nondegenerate solution of (5.38) as it is easy to see, repeating, for example, the proof of Proposition 5.1. Indeed the unique solution ũ 0 of (5.38) has Morse index one and so the proof of Proposition 5.1 shows that the second eigenvalue of the linearized operator cannot be zero. On the other side, z0 cannot vanish. Indeed, multiplying the equation for zn in (5.45) by zn and integrating we have 1

1=

2 lim ∫(zn󸀠 ) dr n→+∞ 0

1

1

2 ̃ p−1 2 = lim p ∫ ũ p−1 R ⋅ zn dr = p ∫ u0 z0 dr. n→+∞

n

0

0

So we have reached a contradiction which shows that (5.44) holds. This implies that the functions RV(⋅, R) converge weakly in H01 ((0, 1)) and strongly in Lq ((0, 1)), for any q > 1 as R → +∞, to a function Ṽ which solves as before the linearized problem ̃ −Ṽ 󸀠󸀠 = pũ p−1 in (0, 1) 0 V { ̃ ̃ V(0) = V(1) = 0 We have just observed that this problem has only the solution Ṽ ≡ 0, so that (5.39) holds. 5.3.2 Asymptotic analysis of the eigenvalue β̃ 1 (R) We start with an expansion of β̃ 1 (R). Proposition 5.16. We have β̃ 1 (R) = β̃ 1 R2 + o(R2 )

as R → +∞

(5.46)

where β̃ 1 is the first eigenvalue for the linear problem −v󸀠󸀠 − pũ p−1 0 b = βv

{

in (0, 1)

v(0) = v(1) = 0

and ũ 0 is the limit of ũ R , as before. Proof. To estimate β̃ 1 (R), we use, as in Section 5.2.2, its variational characterization, i. e., β̃ 1 (R) =

R+1

inf

R+1

∫R (v󸀠 )2 r N−1 dr − p ∫R

2 N−1 up−1 dr R v r

R+1 2 N−3 v r dr

v∈H01 ((0,1)) v=0 ̸

∫R

.

(5.47)

Inserting a test function φ ∈ C0∞ ((R, R + 1)), φ ≥ 0 and making a change of variable we have β̃ 1 (R) ≤

1

1

2 N−1 dt ∫0 (φ󸀠 )2 (R + t)N−1 dt − p ∫0 ũ p−1 R φ (R + t) 1

∫0 φ̃ 2 (R + t)N−3 dt

≤ CR2 .

(5.48)

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172 | 5 Bifurcation from radial solutions In order to get the reverse inequality, let us denote by w1,R the eigenfunction associated to β̃ 1 (R) with ‖w1,R ‖∞ = 1. Plugging w1,R into (5.47) we have, for a positive constant C = C(N, p), β̃ 1 (R) =

R+1

R+1

∫R (w1,R )2 r N−1 dr − p ∫R R+1

∫R

R+1

≥

−p‖uR ‖p−1 ∞ ∫R R+1

∫R

2 N−1 up−1 dr R w1,R r

2 RN−3 dr w1,R

2 N−1 w1,R r dr

2 r N−3 dr w1,R

≥ −CR

≥ (5.49)

2

because we showed that ‖uR ‖∞ ≤ K. Now we define w̃ 1,R (t) = w1,R (t + R) in (0, 1) which solves w̃ 1,R N −1 󸀠 󸀠󸀠 { w̃ 1,R − pũ p−1 w̃ 1,R = β̃ 1 (R) {−w̃ 1,R − R t+R (t + R)2 { { {w̃ 1,R (0) = w̃ 1,R (1) = 0

in (0, 1)

(5.50)

Since ‖w̃ 1,R ‖∞ = 1, we have that w̃ 1,R 󳨀→ φ1 ≠ 0, as R → ∞, uniformly in (0, 1), and φ1 ≥ 0 solves p−1 ̃ −φ󸀠󸀠 1 − pũ 0 φ1 = β1 φ1 { φ1 (0) = φ1 (1) = 0

in (0, 1)

(5.51)

β̃ (R) where β̃ 1 = lim 1R2 , by (5.48) and (5.49). Hence β̃ 1 is the first eigenvalue for the problem (5.51) and obviously β̃ 1 < 0. Then we have, with a change of variable, from (5.49),

β̃ 1 (R) =

1

1

󸀠 2 N−1 ̃2 ) (t + R)N−1 dt − p ∫0 ũ p−1 dt ∫0 (w̃ 1,R R w1,R (t + R) R+1

∫R

2 (t + R)N−3 dt w̃ 1,R

.

Thus passing to the limit, as R → +∞, we get 1

1

p−1 2 󸀠 2 β̃ 1 (R) ∫0 (φ ) dt − p ∫0 ũ 0 φ1 dt = + o(1) 1 R2 ∫ φ2 dt 0

1

which, together with (5.51) gives the assertion. Now we make a deeper analysis on the behavior of β̃ 1 (R) showing that, for large R, it is a strictly decreasing function of the radius R. More precisely, we show the following. Proposition 5.17. The eigenvalue β̃ 1 (R) is differentiable with respect to R and 𝜕β̃ 1 (R) = 2β̃ 1 (R) + o(R) as R → ∞ 𝜕R where β̃ 1 is the first eigenvalue of the linear problem (5.51).

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(5.52)

5.3 Bifurcation with respect to the radius of the annulus | 173

Proof. As in Proposition 5.16 we consider the eigenfunction w̃ 1,R corresponding to the eigenvalue β̃ 1 (R) and recall that the functions w̃ 1,R converge, as R → +∞, to the function φ1 > 0 which is a solution of (5.51). By results of Kato (see [147, p. 380]), we have 𝜕w̃ that both w̃ 1,R and β̃ 1 (R) depend analytically on R. Thus the function Φ = Φ(t, R) = 𝜕R1,R satisfies 𝜕ũ R N −1 󸀠 N −1 󸀠 Φ + w̃ 1,R − p(p − 1)ũ p−2 w̃ − pũ p−1 R Φ R 2 t+R 𝜕R 1,R (t + R) β̃ (R) β̃ (R) 𝜕β̃ (R) 1 w̃ + 1 Φ − 2 1 3 w̃ 1,R . = 1 𝜕R (t + R)2 1,R (t + R)2 (t + R)

− Φ󸀠󸀠 −

Multiplying this equation by w̃ 1,R and integrating, we get 1

∫Φ

󸀠

0

󸀠 w̃ 1,R (t

N−1

+ R)

1

󸀠 󸀠 dt + (N − 1) ∫ w̃ 1,R w̃ 1,R (t + R)N−3 dt

− p(p − 1) ∫ ũ p−2 R 0

1

1

0

𝜕ũ R 2 w̃ (t + R)N−1 dt 𝜕R 1,R

N−1 ̃ − p ∫ ũ p−1 dt R w1,R Φ(t + R) 0

1

1

𝜕β̃ (R) 2 (t + R)N−3 dt + β̃ 1 (R) ∫ Φw̃ 1,R (t + R)N−3 dt = 1 ∫ w̃ 1,R 𝜕R 0

0

1

2 − 2β̃ 1 (R) ∫ w̃ 1,R (t + R)N−4 dt. 0

Multiplying (5.50) by Φ and integrating, we get 1

1

󸀠 N−1 ̃ (t + R)N−1 dt − p ∫ ũ p−1 dt ∫ Φ󸀠 w̃ 1,R R w1,R Φ(t + R) 0

1

0

= β̃ 1 (R) ∫ Φw̃ 1,R (t + R)N−3 dt. 0

Subtracting the last two equations, we deduce 1

1

󸀠 (N − 1) ∫ w̃ 1,R w̃ 1,R (t + R)N−3 dt − p(p − 1) ∫ ũ p−2 R 0

1

0

𝜕ũ R 2 w̃ (t + R)N−1 dt 𝜕R 1,R

1

𝜕β̃ (R) 2 2 = 1 (t + R)N−3 dt − 2β̃ 1 (R) ∫ w̃ 1,R (t + R)N−4 dt. ∫ w̃ 1,R 𝜕R 0

0

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174 | 5 Bifurcation from radial solutions Since 1

󸀠 (N − 1) ∫ w̃ 1,R w̃ 1,R (t + R)N−3 dt = − 0

1

(N − 1)(N − 3) 2 (t + R)N−4 dt = O(RN−4 ) ∫ w̃ 1,R 2 0

and by (5.39) 1

p(p − 1) ∫ ũ p−2 R (R 0

N−1 𝜕ũ R 2 (t + R) )w̃ 1,R dt = o(RN−2 ) 𝜕R R

we have 1

1

0

0

N−4 N−3 𝜕β̃ 1 (R) 2 (t + R) ̃ (R) ∫ w̃ 2 (t + R) dt = 2 β dt + o(R). ∫ w̃ 1,R 1 1,R 𝜕R RN−3 RN−3

Finally, using the convergence of w̃ 1,R to φ1 as in (5.51) we get 1

1

0

0

𝜕β̃ 1 (R) (∫ φ21 dt + o(1)) = 2β̃ 1 R(∫ φ21 dt + o(1)) + o(R) 𝜕R so that (5.52) holds. Hence, by Proposition 5.17 we have that β̃ 1 (R) is strictly decreasing, as a function of the radius R, for R large. Thus we have the following. Corollary 5.18. There exists k̄ ≥ 1 such that for any k ≥ k̄ there exists only one radius Rk such that β̃ 1 (Rk ) solves (5.7). Moreover, the sequence Rk behaves asymptotically as Rk = √

−k(k + N − 2) + o(1) as k → ∞. β̃ 1

(5.53)

As a consequence, the Morse index m(uR ) tends to +∞, as R → +∞. Proof. As observed, by Proposition 5.17 we have the existence of R̄ > 0 such that β̃ 1 (R) is strictly decreasing for R > R.̄ Then, by (5.46), there exists k̄ such that, for any k ≥ k,̄ the equation (5.7) has a solution β = β̃ 1 (Rk ) for only one radius Rk . Moreover, by (5.46) we get (β̃ 1 + o(1))R2k = −k(k + N − 2) which gives (5.53). Finally (5.52), together with (5.7) and Lemma 4.3 implies that, as R crosses Rk , the Morse index of the radial solution uR increases and diverges to +∞ as k → +∞. Remark 5.19. Let us point out that the radii Rk , which are the only ones for which the linearized operator LuR can be degenerate, are well distant from each other. Indeed, from (5.53) we derive τ := lim (Rk+1 − Rk ) = k→+∞

1 . √|β1 |

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5.3 Bifurcation with respect to the radius of the annulus | 175

5.3.3 Bifurcation results In this section, we prove the existence of nonradial solutions, bifurcating from the radial solution uR . Since the radial solutions, and hence the linearized operators LuR are defined in variable domains we first reduce the problems in a fixed annulus. This will allow to build a functional setting independent of R as in [159]. Let A be a fixed annulus with radii 0 < a < b, as in (5.1). For any R > 0 we define the diffeomorphism hR : AR 󳨀→ A defined by hR (r, θ1 , . . . , θN−1 ) = (a + (b − a)(r − R), θ1 , . . . , θN−1 ) (having used the polar coordinates in ℝN , N ≥ 2). The function hR induces the map h∗R : C01,α (AR ) 󳨀→ C01,α (A),

h∗R (u)(x) = u(h−1 R (x))

(5.54)

for any x ∈ A. Then the equation (5.1), written in AR , becomes p

DR (u) = u { { { u>0 { { { {u = 0

in A in A

(5.55)

on 𝜕A

where DR (u) = h∗R (−Δ((h∗R )−1 (u))). Now we can give definitions analogous to those of Section 5.2.3. Definition 5.20. Let R̄ > 0 and uR̄ be the radial solution of (5.1) in AR̄ . We say that a nonradial bifurcation occurs at (uR̄ , R)̄ if in every neighborhood of (h∗R̄ (uR̄ ), R)̄ in C 1,α (A) × (0, +∞) a nonradial solution (wR , R) of (5.55) exists. Definition 5.21. If in the previous definitions it happens that for any neighborhood of (h∗R̄ (uR̄ ), R)̄ in C 1,α (A) × (0, +∞) at least m distinct nonradial solutions (wRi , Ri ), i = ̄ 1, . . . , m of (5.55) exist, we say that an m-nonradial bifurcation occurs at (uR̄ , R). The bifurcation results we obtain are as follows. Theorem 5.22. There exists a strictly increasing sequence of radii Rk , k ∈ ℕ+ , and Rk → +∞ such that a nonradial bifurcation occurs at (uRk , Rk ). Moreover, if k is even and sufficiently large a [ N2 ]-nonradial bifurcation occurs at (uRk , Rk ). To prove this theorem, we proceed as in Section 5.2.3. We start observing that the equation in (5.55) can be written as w = TR (w), with TR (w) = (DR )−1 (|w|p−1 w). The operator TR is well-defined from C01,α (A) into C01,α (A) and is a compact operator. If D is an open bounded set in C01,α (A) such that I − TR ≠ 0 on 𝜕D, then the Leray– Schauder degree for the map I − TR , i. e., deg(I − TR , D, 0) is well-defined. Applying a

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176 | 5 Bifurcation from radial solutions result of [183, Proposition 2, p. 243], we have that deg(I − TR , D, 0) = deg(I − PR , (h∗R ) (D), 0) −1

(5.56)

where PR (v) = (−Δ)−1 (|v|p−1 v) and it is defined in C01,α (AR ). The operator PR is differentiable at uR and, by Corollary 5.18 and Remark 5.19 we have that the linearized operator I − PR󸀠 (uR ) is invertible, for any R ≠ Rk , where {Rk } is the sequence defined in Corollary 5.18. Hence, for any R ≠ Rk , deg(I − PR , B, 0) = deg(I − PR󸀠 (uR ), B, 0) = (−1)mR

(5.57)

where B is a neighborhood of uR in C01,α (A) such that uR is the only solution of (I − PR )(v) = 0 in B and mR = m(uR ) is the Morse index of uR . Proof of Theorem 5.22. It is similar to that of Theorem 5.12. Since we need to use the map h∗R , we detail the main steps for the reader’s convenience. We consider the subspace ℱR of C 1,α (AR ) of the functions which are O(N − 1)-invariant with respect to the first (N − 1) variables. As in the proof of Theorem 5.12, we can invoke Proposition 5.2 of [204] to claim that the eigenspace Vk corresponding to the eigenvalue λk of −ΔSN−1 and spanned by the O(N − 1)-invariant eigenfunctions, is one-dimensional for any k ≥ 1. By Corollary 5.18, arguing as in the proof of Theorem 5.12 we then have that the Morse index m(uR ), in the space ℱR increases by one, as R crosses the values Rk , i. e., m(uRk +ϵ ) = m(uRk −ϵ ) + 1, if ϵ is small enough. Then, by (5.56) and (5.57) we get deg(I − TRk −ϵ , BRk −ϵ , 0) = (−1)deg(I − TRk +ϵ , BRk +ϵ , 0) if BR is a neighborhood of wR = h∗R (uR ) in the space of functions which are O(N − 1)-invariant in A. The change of degree at the point (wRk , Rk ) implies that a bifurcation occurs and the bifurcating solutions are nonradial since the solutions uR are nondegenerate in the space of radial function. By maximum principle, the bifurcating solutions are positive. The [ N2 ]-bifurcation, when k is even, is proved exactly as in Theorem 5.12. We end mentioning that in [159] a nonradial bifurcation result is obtained for problem (5.1) in annuli with inner radius a < 1 and fixed outer radius b = 1. More precisely, the author proves the existence of k̃ such that, for any k ≥ k̃ there exists a radius ak for which nonradial bifurcation occurs. The result of Theorem 5.22, which is taken from [129], is an improvement of that of [159] since it shows that for k large there is only one radius Rk at which a nonradial bifurcation occurs, so these radii Rk are the only ones for which bifurcation can take place.

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6 Morse index and symmetry for semilinear elliptic equations in bounded domains In this chapter, we show connections between the Morse index of solutions of semilinear elliptic equations and their symmetry properties. More precisely, we will prove that, under some convexity hypotheses on the nonlinearity, the solutions which have Morse index not greater than the dimension of the space have an axial symmetry property. This kind of results were obtained in [186, 190] and extended in [35] to a class of fully nonlinear equations. We describe them in an unified context simplifying some proofs and adding some recent developments obtained in [78] for nonlinear mixed boundary value problems. Finally, we mention that similar symmetry results can be obtained for variational problems by using some symmetrization and reflection methods (see [24, 48–50, 152, 161, 165–167, 172, 203, 220–223] and the survey paper [224]). To point out the range of applications, we start by presenting in Section 6.1 the famous symmetry theorems of Gidas, Ni and Nirenberg [122] obtained by the method of moving planes, together with some counterexamples.

6.1 Symmetry and monotonicity of positive solutions 6.1.1 Moving planes and symmetry In this section, we recall the famous result by Gidas, Ni and Nirenberg [122] about the radial symmetry of positive solutions of semilinear elliptic equations in balls. It is based on the Alexandrov–Serrin moving planes method that goes back to Alexandrov (see [8]) and has been introduced in the context of partial differential equations by Serrin in [199]. In the proof of next theorem and in the sequel, we will use mostly the Berestycki– Nirenberg version of the moving planes method (see [31]), based on the strong maximum principle together with the weak maximum principle in small domains proved in Chapter 1. This technique allows us to prove a variant of the method, known as rotating planes method (see Theorem 6.3) as well as analogous symmetry results for solutions of semilinear elliptic equations in cylindrically symmetric domains and for solutions of semilinear elliptic systems. Nevertheless, we will also sketch the original proof in [122] (see Remark 6.2). Theorem 6.1. Let B = BR = {x ∈ ℝN : |x| < R}, R > 0, be a ball centered at the origin and assume that u ∈ H01 (B) ∩ C 0 (B) is a weak solution of the problem −Δu = f (|x|, u) in B { { { u>0 in B { { { on 𝜕B {u = 0

(6.1)

https://doi.org/10.1515/9783110538243-006

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178 | 6 Morse index and symmetry for semilinear elliptic equations in bounded domains where f = f (r, s) : [0, +∞) × [0, +∞) → ℝ is a continuous function which is locally Lipschitz continuous in the s-variable uniformly with respect to r, and nonincreasing in r for any fixed s ∈ [0, +∞). Then u is radial and radially decreasing, i. e., u(x) = v(|x|) for some function v = v(r) : [0, R] → ℝ and v󸀠 (r) < 0. Proof. Since f is continuous by standard regularity results, the solution belongs at least to C 1 (B) ∩ C 0 (B). We will prove symmetry with respect to every hyperplane orthogonal to any direction e ∈ SN−1 . For simplicity of notation, we fix the hyperplane orthogonal to the e1 = (1, 0, . . . , 0) direction. The proof is the same for any other direction. We set, for −R < λ ≤ 0, Hλ = {x ∈ ℝN : x1 = λ}

Bλ = {x = (x1 , . . . , xN ) ∈ B : x1 < λ}

xλ = Rλ (x) = Rλ (x1 , . . . xN ) = (2λ − x1 , . . . xN ) i. e., Rλ is the reflection through the hyperplane Hλ . Then we define uλ (x) = u(xλ ) for x ∈ Bλ Note that uλ satisfies the equation −Δuλ = f (|xλ |, uλ ) in Bλ and, since 0 = u ≤ uλ on 𝜕Bλ \ Hλ and u = uλ on Hλ ∩ B, we have that u ≤ uλ on 𝜕Bλ . Moreover, since f (|x|, u) is nonincreasing in |x| and |x| ≥ |xλ | when λ ≤ 0, we get that f (|xλ |, uλ ) ≥ f (|x|, uλ ). Summing up, we have, for −R < λ < 0: −Δu = f (|x|, u); −Δuλ ≥ f (|x|, uλ ) { u ≤ uλ

in Bλ

on 𝜕Bλ

(6.2)

We will prove the following inequality: u ≤ uλ

in Bλ for any λ ∈ (−R, 0)

(6.3)

Let us first show that if (6.3) holds then we also have u < uλ

in Bλ ,

∀λ ∈ (−R, 0)

(6.4)

and ux1 (x) > 0, ∀x ∈ Hλ ∩ B

and

∀λ ∈ (−R, 0)

(6.5)

To prove (6.4) and (6.5), let us observe that if λ < 0 then by (6.2) and (6.3) and by the strong comparison principle (Theorem 1.31) we get that u < uλ in Bλ . Indeed, when λ < 0, there are points on the boundary of the ball where the strict inequality 0 = u < uλ holds (because these points are reflected inside the ball, where u is positive). This proves (6.4) and, again by Theorem 1.31, we get that ux1 > (uλ )x1 = −ux1 on Hλ ∩ B, since

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6.1 Symmetry and monotonicity of positive solutions | 179

u = uλ on Hλ ∩ B ⊂ 𝜕Bλ and e1 is an outer direction with respect to the cap Bλ . Hence (6.5) follows. Let us show that once (6.3) is proved we can conclude the proof of the theorem. Indeed, if (6.3) holds, by continuity we obtain that u ≤ u0 in B0 = [x1 < 0]. By considering the opposite direction −e1 , we deduce u ≡ u0 in B0

(6.6)

Moreover, since (6.5) holds, it follows that ux1 > 0

in B0

(6.7)

To get the radial symmetry, it is enough to repeat the argument leading to (6.6) and (6.7) for every direction e ∈ SN−1 . Let us now prove (6.3) in two step. Step 1, starting the moving planes: Let us set A = ‖u‖L∞ (B) . Then for any λ ∈ (−R, 0) we have that ‖u‖L∞ (Bλ ) ≤ A,

‖uλ ‖L∞ (Bλ ) ≤ A

Let δ = δ(A) as in Theorem 1.21 and observe that δ is independent of λ and u, uλ satisfy (6.2). If λ > −R is close to −R, then the measure of Bλ can be made arbitrarily small, in particular meas (Bλ ) < δ, so that we get from (6.2) and Theorem 1.21 that u ≤ uλ in Bλ . Hence we get that ∃α > 0 : u < uλ in Bλ and ux1 > 0 in Bλ , ∀λ ∈ (−R, −R + α) Step 2, continuing the moving planes: Let us set λ0 = sup{λ ∈ (−R, 0) : u ≤ uμ in Bμ , ∀μ ∈ (−R, λ)}. To prove (6.3), we have to prove that λ0 = 0. Suppose by contradiction that λ0 < 0. Then by continuity u ≤ uλ0 in Bλ0 , and by the strong comparison principle (Theorem 1.31) we get, as before, that u < uλ0 in Bλ0 . Take

now a compact set K ⊂ Bλ0 such that |Bλ0 \ K| < δ2 , where δ is again as in Theorem 1.21, and observe that minK (uλ0 − u) = m > 0 since (uλ0 − u) is continuous and positive in K. By continuity for λ > λ0 and close to λ0 , we still have that λ < 0 and |Bλ \ K| < δ,

min(uλ − u) ≥ K

m >0 2

Moreover, setting Bλ 󸀠 = (Bλ \ K) we have that u ≤ uλ on 𝜕Bλ 󸀠 . Applying again Theorem 1.21, we get that u ≤ uλ in B󸀠λ , and since u ≤ uλ in K, we deduce that u ≤ uλ in Bλ for any λ greater than λ0 and sufficiently close to λ0 . This contradicts the definition of λ0 , and hence necessarily λ0 = 0.

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180 | 6 Morse index and symmetry for semilinear elliptic equations in bounded domains Remark 6.2. Let us sketch here the original proof in [122], which is entirely based on the strong maximum principle and Hopf’s lemma. This proof works for a C 2 (B) solution and relies deeply on the smoothness of the domain B. We also assume that f (R, 0) ≥ 0 (for the case when f (R, 0) < 0 a variant of Hopf’s lemma can be used, and we refer to [122], or to [115] for a simple proof). Since |x| ≤ R and f (r, s) is nonincreasing in r, we get that u satisfies −Δu = f (|x|, u) − f (|x|, 0) + f (|x|, 0) ≥ f (|x|, u) − f (|x|, 0) + f (R, 0) ≥ f (|x|, u) − f (|x|, 0)

Therefore, u ≥ 0 satisfies the inequality −Δu + c(x)u ≥ 0, with f (|x|,u(x))−f (|x|,0) u(x)

−c(x) = { 0

if u(x) ≠ 0

if u(x) = 0

,

which is bounded since f (r, s) is Lipschitz continuous in s. Since u = 0 on 𝜕B and e1 = (1, . . . , 0) is an inner normal in every point x of 𝜕B where x1 < 0, by Hopf’ s lemma (Theorem 1.28) we get 𝜕u (x) > 0 𝜕x1

on 𝜕B ∩ [x1 ≤ λ] for any λ < 0

(6.8)

By (6.8), we can start the moving planes procedure and get that ∃α > 0 : u < uλ in Bλ and ux1 > 0 in Bλ ,

∀λ ∈ (−R, −R + α)

Indeed, if this is not the case, there exists a sequence λn → −R and a sequence of points yn = ((yn )1 , yn󸀠 ) ∈ Bλn , such that, up to a subsequence, yn → (−R, 0, . . . , 0), and u(yn ) ≥ uλn (yn ) = u((2λn − (yn )1 , yn󸀠 )). By the mean value theorem, there exists a sequence zn = ((zn )1 , zn󸀠 ) with (yn )1 < (zn )1 < 2λn − (yn )1 , zn󸀠 = yn󸀠 (so that again up to a subsequence zn → (−R, 0, . . . , 0)) and ux1 (zn ) ≤ 0. As n → ∞ we get ux1 ((−R, 0, . . . , 0)) ≤ 0, which contradicts (6.8). So the set {λ ∈ (−R, 0) : u ≤ uμ in Bμ , ∀μ ∈ (−R, λ)} is not empty and contains an interval (−R, −R + α), α > 0. Setting λ0 = sup{λ ∈ (−R, 0) : u ≤ uμ in Bμ , ∀μ ∈ (−R, λ)} we have to prove that λ0 = 0. Let us first observe that, by continuity, u ≤ uλ0 in Bλ0 . Exactly as in the previous proof, using Theorem 1.31, we get that if we suppose, by contradiction, that λ0 < 0, then u < uλ0 in Bλ0

and

ux1 > 0 on Hλ0 ∩ B

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6.1 Symmetry and monotonicity of positive solutions | 181

This, together with (6.8), gives ux1 > 0

on 𝜕Bλ0

(6.9)

The inequality (6.9) in turn implies that there exists ε > 0 such that λ0 + ε < 0 and the inequality (6.3) still holds for λ0 < λ < λ0 + ε, contradicting the definition of λ0 . Indeed, if this is not the case, there exists a sequence λn = λ0 + εn → λ0 , a sequence of points yn = ((yn )1 , yn󸀠 ) ∈ Bλn with u(yn ) ≥ uλn (yn ) = u((2λn − (yn )1 , yn󸀠 )) and a sequence zn of points in the segment joining yn and (yn )λn with ux1 (zn ) ≤ 0. For a subsequence,

we then have that yn → y0 ∈ B0 and, by continuity, we get u(y0 ) ≥ uλ0 (y0 ), so that y0 ∈ Hλ0 ∩ B, because u < uλ0 in Bλ0 ∪ 𝜕B \ Hλ0 . Hence also zn → y0 and by continuity ux1 (y0 ) ≤ 0, contradicting (6.9).

A variant of the moving planes method is the so called rotating planes method (exploited, e. g., in [190],) which is similar to the one used in the proof of Theorem 6.1, but rotating the planes instead of moving them in a fixed direction. Let Ω be a bounded rotationally symmetric domain in ℝN , i. e., a ball or an annulus. For a unit vector e ∈ SN−1 , we consider the hyperplane H(e) = {x ∈ ℝN : x ⋅ e = 0}

(6.10)

orthogonal to the direction e and the open half-domain Ω(e) = {x ∈ Ω : x ⋅ e > 0}

(6.11)

We then set σe (x) = x − 2(x ⋅ e)e

for every x ∈ Ω

(6.12)

i. e., σe : Ω → Ω is the reflection with respect to the hyperplane H(e). Note that H(−e) = H(e)

and

Ω(−e) = σe (Ω(e)) = −Ω(e)

for every e ∈ SN−1 .

(6.13)

Finally, if u : Ω → ℝ is a continuous function we define the reflected function uσ(e) : Ω → ℝ by uσ(e) (x) = u(σe (x))

(6.14)

Now let us consider solutions, possibly sign-changing, of the following problem: −Δu = f (|x|, u) in Ω

{

u=0

(6.15)

on 𝜕Ω

where Ω is a bounded rotationally symmetric domain and f = f (r, s) : [0, ∞) × ℝ → ℝ is a continuous function which is locally Lipschitz continuous in the variable s ∈ ℝ. In the next theorem, we denote by eϑ the directions defined by eϑ = (cos(ϑ), sin(ϑ), 0, . . . , 0),

ϑ∈ℝ

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182 | 6 Morse index and symmetry for semilinear elliptic equations in bounded domains Theorem 6.3 (Rotating planes method ). Let Ω and f be as in (6.15) and let u ∈ H01 (Ω) ∩ C 0 (Ω) be a weak solution of (6.15). If there exists ϑ0 ∈ ℝ such that u < uσ(eϑ0 ) in Ω(eϑ0 ), then there exists ϑ1 > ϑ0 such that u ≡ uσ(eϑ1 ) in Ω(eϑ1 ),

u < uσ(eϑ ) in Ω(eϑ ), ∀ϑ ∈ (ϑ0 , ϑ1 )

Proof. Let us observe that the functions uσ(eϑ ) satisfy the same equation as u, namely −Δuσ(eϑ ) = f (|x|, uσ(eϑ ) ) in Ω, and both ‖u‖L∞ (Ω(eϑ )) and ‖uσ(eϑ ) ‖L∞ (Ω(eϑ )) are bounded by ‖u‖L∞ (Ω) =: A. Let us fix δ = δ(A) as in Theorem 1.21 and observe that δ is independent of ϑ, and the functions u, uσ(eϑ ) satisfy −Δu = f (|x|, u); −Δuσ(eϑ ) = f (|x|, uσ(eϑ ) ) { u = uσ(eϑ )

in Ω(eϑ )

on 𝜕Ω(eϑ )

(6.16)

Let us set Θ = {ϑ ≥ ϑ0 : u < uσ(eϑ󸀠 ) in Ω(eϑ󸀠 ) ∀ϑ󸀠 ∈ (ϑ0 , ϑ)} and let us show that the set Θ is nonempty and contains an interval [ϑ0 , ϑ0 +ε) for ε > 0 sufficiently small. Indeed we can take a compact set K ⊂ Ω(eϑ0 ) such that |Ω(eϑ0 )\K| ≤ δ2 and m = minK (uσ(eϑ0 ) −u) > 0. By continuity if ϑ is close to ϑ0 , we have that K ⊂ Ω(eϑ ), (uσ(eϑ ) − u) ≥ m2 > 0 in K, |Ω(eϑ ) \ K| ≤ δ and (uσ(eϑ ) − u) ≥ 0 on 𝜕(Ω(eϑ ) \ K). Then by the weak comparison principle in small domains (Theorem 1.21) we get that u ≤ uσ(eϑ ) in Ω󸀠 = Ω(eϑ ) \ K, and hence in Ω(eϑ ). Moreover, u < uσ(eϑ ) in Ω(eϑ ) by the strong comparison principle (Theorem 1.31). So the set Θ is nonempty, and is bounded from above by ϑ0 + π, since, considering the opposite direction, the inequality between u and the reflected function gets reversed. Let us set ϑ1 = sup Θ. We claim that u ≡ uσ(eϑ1 ) in Ω(eϑ1 ). Indeed, if this is not the case, we get u < uσ(eϑ1 ) in Ω(eϑ1 ) by the strong comparison principle (Theorem 1.31), since by continuity u ≤ uσ(eϑ1 ) in Ω(eϑ1 ). Then, using again the weak comparison principle in small domains and the previous technique we get u < uσ(eϑ ) in Ω(eϑ ) for ϑ > ϑ1 and close to ϑ1 , contradicting the definition of ϑ1 . 6.1.2 Monotonicity by the method of moving planes The bibliography on the moving planes method is huge; we refer, e. g., to [28–31, 33, 47, 49, 53, 56, 68, 69, 75, 76, 80–85, 87, 94, 95, 118, 119, 122, 123, 155, 156, 199, 216, 217, 219] and the references therein for many applications to different elliptic problems. Let us observe that the moving planes method yields not only symmetry results but also monotonicity results in general domains as we recall below. Let us start with some notation. Let Ω be a bounded domain and e a direction in N ℝ . For a real number λ, we define Hλe = {x ∈ ℝN : x ⋅ e = λ}

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(6.17)

6.1 Symmetry and monotonicity of positive solutions | 183

xλe

Ωeλ = {x ∈ Ω : x ⋅ e > λ}

=

Reλ (x)

= x + 2(λ − x ⋅ e)e,

(6.18) N

x∈ℝ

(6.19)

(i. e., Reλ is the reflection through the hyperplane Hλe ) a(e) = sup x ⋅ e x∈Ω

(6.20)

If λ < a(e), then Ωeλ is nonempty, thus we set (Ωeλ ) = Reλ (Ωeλ ) 󸀠

(6.21)

If Ω is smooth and λ < a(e), λ close to a(e), then the reflected cap (Ωeλ )󸀠 is contained in Ω and will remain in it, at least until one of the following situations occurs: (i) (Ωeλ )󸀠 becomes internally tangent to 𝜕Ω at some point not on Hλ𝜐 (ii) Hλe is orthogonal to 𝜕Ω at some point Let Λ1 (e) be the set of those λ󸀠 < a(e) such that for each λ ∈ (λ󸀠 , a(e)] none of the conditions (i) and (ii) holds and define λ1 (e) = inf Λ1 (e)

(6.22)

In the terminology of [122], Ωeλ1 (e) is the maximal cap. In some situations, e. g. in a rectangle, it may happen that for λ < λ1 (e) the reflected cap is still contained in Ω, i. e., (Ωeλ )󸀠 ⊂ Ω. We consider the set Λ2 (e) of those λ < a(e) such that (Ωeμ )󸀠 ⊂ Ω for each μ ∈ (λ, a(e)] and define λ2 (e) = inf Λ2 (e)

(6.23)

In the terminology of [122], Ωeλ2 (e) is the optimal cap. Let us consider the problem −Δu = f (x, u) { { { u>0 { { { {u = 0

in Ω in Ω

(6.24)

on 𝜕Ω

One of the main results by Gidas, Ni and Nirenberg [122] is the following. Theorem 6.4. Let Ω be a bounded smooth domain, e a direction and u ∈ C 2 (Ω) be a solution of (6.24), where f = f (x, s) is a locally Lipschitz continuous function in Ω × [0, +∞) and it is monotone in the direction e w. r. t. the first variable, in the sense that for any λ in the interval (λ1 (e), a(e)) it holds f (x, s) ≤ f (xλe , s) if x ∈ Ωeλ , s ∈ [0, +∞)

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184 | 6 Morse index and symmetry for semilinear elliptic equations in bounded domains Then for λ in the interval (λ1 (e), a(e)) we have u(x) ≤ u(xλe ) ∀x ∈ Ωeλ

(6.25)

𝜕u (x) < 0 ∀x ∈ Ωeλ1 (e) 𝜕e

(6.26)

Moreover,

The proof is essentially the same sketched in Remark 6.2 where we have fixed the direction e = −e1 . The radiality of the solutions in balls is just a corollary of Theorem 6.4, which is however a monotonicity result, in particular in a neighborhood of the boundary of Ω. It has been exploited in many situations, e. g., in a priori estimates using the blowup technique (see [62, 71, 96, 124, 125] and the references therein). The proof of Theorem 6.4 given in [122] fails for simple domains with corners as a rectangle, while the proof that we have presented in the case of the ball (Theorem 6.1), due to Berestycki and Nirenberg [31], also works in these cases (and also assuming that u is a C 1 (Ω) weak solution). Moreover, (6.25) and (6.26) hold in the optimal cap defined by λ2 (e). The proof is essentially the same as the one of Theorem 6.1 given above, where we have fixed the direction e = −e1 . Note however that in [31] much more general equations are considered, in particular, the paper focuses on fully nonlinear elliptic equations. 6.1.3 Counterexamples to radial symmetry If one of the hypotheses of Theorem 6.1 fails, or the ball is replaced by an annulus, the solutions need not to be radial anymore. In particular, this can happen when Ω is a rotationally symmetric bounded domain, e. g., a ball or an annulus, but either one of the following situations occurs: a) the nonlinearity f (r, s) is not locally Lipschitz continuous in the variable s b) the nonlinearity f (r, s) is not monotone decreasing in the variable r c) the domain is an annulus d) the solution changes sign in Ω Let us discuss briefly, following the survey paper [187], some counterexamples to the radial symmetry in each of these cases. a) In the case when the nonlinearity is not Lipschitz continuous, there can be some compactly supported positive solutions which exhibit a combination of “bumps.” As an example, if p > 2 the function (1 − |x|2 )p

w(x) = {

0

if |x| ≤ 1

if |x| > 1

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(6.27)

6.1 Symmetry and monotonicity of positive solutions | 185

satisfies in ℝN the equation − Δw = f (w)

(6.28)

with the Hölder continuous function f with exponent (1 − p2 ) defined by 1− p2

f (s) = −2p(p − 2)s

1− p1

+ 2p(N + 2p − 2)s

If x0 is any point with |x0 | = 3, the function u(x) = w(x) + w(x − x0 ) satisfies the Dirichlet problem associated to (6.28) in Ω = B5 but it is not radial. b) The problem (6.1) when f (|x|, u) = |x|α up is the Hénon problem (see [144]) already considered in Chapter 3. Note that if α > 0 then the monotonicity hypothesis on the function r 󳨃→ f (r, s) in Theorem 6.1 is not satisfied. if N ≥ 3. Let us assume that α > 0 and 1 < p < +∞ if N = 2, 1 < p < N+2 N−2 A solution of (6.1) can be found using a constrained minimization method. Indeed, if we define Sα =

inf

∫B |∇v|2 dx

v∈H01 (B), v≡0 ̸

2

(∫B |x|α |v|p+1 dx) p+1

(6.29)

since the embedding of H01 (B) in Lp+1 (B) is compact, we have that Sα is achieved by a function v that can be assumed to be nonnegative (taking its modulus) and by the strong maximum principle it follows that v > 0 in B. 1 In the same way, we can minimize among the radial function, i. e., if H0,rad (B) is the subspace of the radial functions, we can consider the infimum Zα =

inf

∫B |∇v|2 dx

1 v∈H0,rad (B), v≡0 ̸

2

(∫B |x|α |v|p+1 dx) p+1

(6.30)

As before, the infimum is achieved by a radial function that solves (6.1). It is clear that Sα ≤ Zα and it is proved in [202] that there exists α∗ > 0 such that Sα < Zα for any α > α∗ . Obviously, for any such α there exists a nonradial solution of (6.1) in the ball. c) An interesting counterexample was given in the classical paper [45] in the presence of a critical nonlinearity. Let us consider the problem N+2

−Δu = u N−2 + λu in Ω { { { u>0 in Ω { { { on 𝜕Ω {u = 0

(6.31)

where Ω is a bounded smooth domain in ℝN , N ≥ 4 and λ > 0.

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186 | 6 Morse index and symmetry for semilinear elliptic equations in bounded domains In [45], it is proved that for λ ∈ (0, λ1 ), where λ1 is the first eigenvalue of the operator −Δ in H01 (Ω), there is a solution to problem (6.31). This is obtained by proving that the infimum Sλ =

inf

∫ |∇v|2 dx − λ ∫ |v|2 dx,

v∈H01 (Ω) ‖v‖ 2∗ =1 Ω L

(Ω)

2∗ =

Ω

2N N −2

is achieved. If Ω is an annulus A = [r < |x| < R], we can consider also the infimum Zλ =

inf

∫ |∇v|2 dx − λ ∫ |v|2 dx,

1 v∈H0,rad (Ω) ‖v‖ 2∗ =1 Ω L

(Ω)

Ω

2∗ =

2N N −2

1 among the radial functions. By the compactness of the embedding of H0,rad (A)

in L2 (A), the infimum Zλ is also achieved when λ = 0, while the infimum S0 is never achieved in any bounded domain. This remark is crucial in [45] to prove that Sλ < Zλ for λ close to zero, so that the least energy solution in an annulus is not radial. d) It is very easy to construct a sign changing solution in a ball which is not radial. For example, the second eigenfunction of the Laplacian in a ball is an antisymmetric function with respect to a hyperplane passing through the origin. More generally, if f (u) is an odd function and u is a positive solution of the problem ∗

−Δu = f (u) in B− { u=0 on 𝜕B−

(6.32)

in the half-ball B− = {x = (x1 , . . . , xN ) ∈ BR : x1 < 0}, then by reflecting u by oddness in B+ = {x = (x1 , . . . , xN ) ∈ BR : x1 > 0} we get a sign-changing solution in BR which is not radial.

6.2 Foliated Schwarz symmetry and related properties Let us now define a particular kind of axial symmetry, known as foliated Schwarz symmetry. This name was introduced in [203] and refers to foliations by spheres with the same center. We will focus mainly on rotationally symmetric bounded domains Ω, i. e., either a ball or an annulus in ℝN and that we always assume with the center at the origin. Nevertheless, the definition that we give and some of its properties that we discuss here also work for unbounded rotationally symmetric domains, such as the whole space ℝN or the complement of a ball, as it will be sketched in Chapter 8. Definition 6.5. Let Ω be a rotationally symmetric domain in ℝN , N ≥ 2. We say that a function v ∈ C 0 (Ω) is foliated Schwarz symmetric (briefly FSS) if there is a unit vector

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6.2 Foliated Schwarz symmetry and related properties | 187

x ⋅ p), and p ∈ ℝN such that v(x) only depends on r = |x| and (if x ≠ 0) on θ := arccos( |x| v is nonincreasing in θ.

Remark 6.6. A foliated Schwarz symmetric function v is axially symmetric with respect to the axis containing the vector p. This means that v is symmetric with respect to every hyperplane orthogonal to any direction e which is orthogonal to p. Indeed, in this case, the points x and σe (x) (σe defined in (6.12)) have the same modulus and the angle between x and p or σe (x) and p is the same. Note however that besides the axial symmetry, Definition 6.5 requires the monotonicity with respect to the angular variable ϑ. A radial function is a particular case of a FSS function. We will see that in general, e. g., for solutions of semilinear elliptic equations, a nonradial FSS function is actually strictly decreasing in the angle variable θ (see Remark 6.13, point 2). Let us recall here some definitions from the previous section. For a unit vector e ∈ S = SN−1 , we consider the hyperplane H(e) = {x ∈ ℝN : x ⋅ e = 0} orthogonal to the direction e and the open half-domain Ω(e) = {x ∈ Ω : x ⋅ e > 0}. We then set σe (x) = x − 2(x ⋅ e)e for every x ∈ Ω i. e. σe : Ω → Ω is the reflection with respect to the hyperplane H(e). Finally, if u : Ω → ℝ is a continuous function we define the reflected function σ(e) u : Ω → ℝ as uσ(e) (x) = u(σe (x)) Let us now consider solutions of the following semilinear problem: −Δu = f (|x|, u) in Ω

{

u=0

on 𝜕Ω

(6.33)

A sufficient condition for the foliated Schwarz symmetry of a solution of (6.33) is given in the following proposition, proved in [48] for the case of an annulus (but the proof carries over to the case of a ball) and also implicitly contained in [24]. Proposition 6.7. Let Ω be a (bounded or unbounded) rotationally symmetric domain and u ∈ H01 (Ω) ∩ C 0 (Ω) a weak solution of (6.33) where f = f (r, s) : [0, ∞) × ℝ → ℝ is a continuous function which is locally Lipschitz continuous in the s-variable. Assume that for every unit vector e ∈ SN−1 it holds: either u(x) ≥ u(σe (x)) ∀x ∈ Ω(e) or

u(x) ≤ u(σe (x)) ∀x ∈ Ω(e)

(6.34)

Then u is foliated Schwarz symmetric. Proof. Let r > 0 and Sr = {x ∈ ℝN : |x| = r} ⊂ Ω, and let p ∈ SN−1 be such that u(rp) = maxSr u. We define Tp+ = {e ∈ SN−1 : e ⋅ p > 0} and Tp− = {e ∈ SN−1 : e ⋅ p < 0}. To prove that u is foliated Schwarz symmetric with respect to p, it is enough to show that u(x) ≥ u(σe (x)) for all x ∈ Ω(e)

(6.35)

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188 | 6 Morse index and symmetry for semilinear elliptic equations in bounded domains whenever e ∈ Tp+ . Indeed this would imply the reverse inequality for e ∈ Tp− , so that if e is any direction orthogonal to p then u(x) = u(σe (x)), for all x ∈ Ω(e). In other words, u would be axially symmetric about the axis with direction p, and the angular monotonicity would follow also by the same inequalities. So for fixed e ∈ Tp+ we consider the difference w = u − uσ(e) , and observe that w satisfies the linear equation − Δw + c(x)w = 0

in Ω

(6.36)

with σ(e)

(|x|,u { f (|x|,u(x))−f u(x)−uσ(e) (x) −c(x) = { 0 {

(x))

if u(x) ≠ uσ(e) (x)

if u(x) = uσ(e) (x)

and obviously c ∈ L∞ (Ω). By assumption, either w ≥ 0, or w ≤ 0. In the first case, (6.35), is satisfied. In the second case, since w ≤ 0 satisfies a linear equation, by the strong maximum principle (Theorem 1.28), we get that either w < 0 or w ≡ 0 in Ω(e). The first case is not possible, since rp ∈ Ω(e) and w(rp) ≥ 0 because u(rp) is the maximum of u on Sr . Hence (if w ≤ 0 then) w ≡ 0 in Ω(e) and (6.35) is satisfied. – –

In the rest of this section: Ω is either a ball or an annulus centered at the origin in ℝN , N ≥ 2 u ∈ C 2 (Ω) is a classical solution of the semilinear problem (6.33), with f ∈ C 1 ((0, +∞) × ℝ).

Let f 󸀠 (|x|, s) =

𝜕f (|x|, s) 𝜕s

and let us set Vu (x) = f 󸀠 (|x|, u(x)),

x∈Ω

(6.37)

We consider the quadratic form Qu (v; Ω) = ∫(|∇v|2 − Vu |v|2 ) dx, Ω

v ∈ H01 (Ω)

(6.38)

corresponding to the linear operator Lu = −Δ − Vu and denote its eigenvalues and eigenfunctions by λk = λk (−Δ − Vu ; Ω);

ϕk = ϕk (−Δ − Vu ; Ω)

(6.39)

If e is a direction, we also consider the corresponding quadratic form Qu (v; Ω(e)) = ∫ (|∇v|2 − Vu |v|2 ) dx,

v ∈ H01 (Ω(e))

Ω(e)

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(6.40)

6.2 Foliated Schwarz symmetry and related properties | 189

while we denote the eigenvalues and the eigenfunctions of the operator −Δ − Vu in the cap Ω(e) by λke = λk (−Δ − Vu ; Ω(e));

φek = φk (−Δ − Vu ; Ω(e))

(6.41)

Observing that we can write σ(e)

f (u(x)) − f (u

1

(x)) = ∫[f 󸀠 (|x|, tu(x) + (1 − t)uσ(e) (x))](u(x) − uσ(e) (x)) dt 0

we define, for any direction e ∈ SN−1 , the potential 1

Ve (x) = ∫ f 󸀠 (|x|, tu(x) + (1 − t)uσ(e) (x)) dt 0 σ(e)

(|x|,u { f (|x|,u(x))−f u(x)−uσ(e) (x) ={ f 󸀠 (|x|, u(x)) {

(x))

if u(x) ≠ uσ(e) (x) if u(x) = uσ(e) (x)

,

x ∈ Ω(e)

(6.42)

Note that the difference we (x) = u(x) − uσ(e) (x) satisfies in Ω(e) the Dirichlet problem −Δwe − Ve we = 0 in Ω(e) { e w =0 on 𝜕Ω(e)

(6.43)

Now we describe other sufficient conditions for the foliated Schwarz symmetry of a solution u of (6.33). To this end, we begin with some geometric considerations about cylindrical coordinates with respect to a plane spanned by two orthogonal directions η1 , η2 . Using cylindrical coordinates with respect to this plane, we will write x = (x1 , . . . , xN ) as x = r cos(ϑ)η1 + r sin(ϑ)η2 + x̃

(6.44)

where r = r(η1 , η2 ) = √(x ⋅ η1 )2 + (x ⋅ η2 )2 ,

̃ 1 , η2 ) = x − (x ⋅ η1 )η1 − (x ⋅ η2 )η2 x̃ = x(η

ϑ = ϑ(η1 , η2 ) ∈ [0, 2π) : x ⋅ η1 = r cos(ϑ),

x ⋅ η2 = r sin(ϑ)

(6.45) (6.46)

Let us denote by uϑ = uϑ(η1 ,η2 ) the derivative of a function u ∈ C 1 (Ω) whit respect to the angular coordinate ϑ, trivially extended where it is not well-defined, i. e., the function 𝜕u

uϑ (r, ϑ, x)̃ = { 𝜕ϑ 0

(r, ϑ, x)̃

if r ≠ 0

if r = 0

(6.47)

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190 | 6 Morse index and symmetry for semilinear elliptic equations in bounded domains For simplicity, we omit the dependence on η1 , η2 if the directions are fixed. Moreover, to simplify the notation, in the calculation that follows we will assume that η1 = e1 = (1, 0, . . . , 0), η2 = e2 = (0, 1, 0, . . . , 0). Let eβ = (cos(β), sin(β), 0, . . . , 0) be a unit vector in the x1 -x2 plane for β ∈ ℝ. Then we consider as before the hyperplane H(eβ ) and the half-domains Ω(eβ ) and, for simplicity, we will use the notation Ωβ = Ω(eβ ), Hβ = H(eβ ), σβ = σeβ and uσβ for the reflection of a function u with respect to the hyperplane Hβ . It is easy to see that the reflection σβ with respect to Hβ can be written as σβ (r cos ϑ, r sin ϑ, x)̃ = (r cos(2β − ϑ + π), r sin(2β − ϑ + π), x)̃ = (r cos(2β − ϑ − π), r sin(2β − ϑ − π), x)̃

(6.48)

since 2β − ϑ + π = ϑ + 2(β + π2 − ϑ) and the angular variable is defined up to a multiple of 2π. This can also be proved analytically writing the usual reflection in Cartesian coordinates and using simple trigonometric formulas. Let us set h±β (ϑ) = 2β − ϑ ± π

(6.49)

Note that if we choose an interval [ϑ0 , ϑ0 + 2π) to which the angular coordinate ϑ belongs, the images 2β − ϑ ± π could not belong to the same interval. Nevertheless, we observe that for a fixed β̃ ∈ ℝ, if we take ϑ ∈ [β̃ − π2 , β̃ + 32 π] then h+β̃ (ϑ) belongs to the same interval, whereas if we take ϑ ∈ [β̃ − 3 π, β̃ + π ] then h− (ϑ) belongs to the same 2

interval. More precisely, we have

2

β̃

π π π 3 ϑ ∈ [β̃ − , β̃ + ] ⇒ 2β̃ − ϑ + π = h+β̃ (ϑ) ∈ [β̃ + , β̃ + π], 2 2 2 2 π π π 3 ϑ ∈ [β̃ − π, β̃ − ] ⇒ 2β̃ − ϑ − π = h−β̃ (ϑ) ∈ [β̃ − , β̃ + ]. 2 2 2 2

(6.50) (6.51)

This can be easily verified evaluating h±β̃ on the boundary of the intervals of definition, since the mappings h±β̃ are decreasing.

Proposition 6.8. Let Ω be a rotationally symmetric domain in ℝN centered at the origin, β̃ ∈ ℝ, and assume that u ∈ C 1 (Ω) satisfies: a) u is symmetric with respect to the hyperplane Hβ̃ . b) uϑ ≥ 0 in Ωβ̃ = [x ⋅ eβ̃ > 0] where uϑ is as in (6.47) Then for any β ∈ [β̃ − π, β]̃ and for any x ∈ Ωβ = [x ⋅ eβ > 0] we have u(x) ≤ u(σβ (x)), while for every β ∈ [β,̃ β̃ + π] we have u(x) ≥ u(σβ (x)) in Ωβ . In other words, if a) and b) holds, then (6.34) holds for any direction e in the (x1 , x2 ) plane that we consider. To prove Proposition 6.8, we need the following simple lemma.

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6.2 Foliated Schwarz symmetry and related properties | 191

Lemma 6.9. Suppose that the assumptions of Proposition 6.8 hold and let ϑ0 ∈ (β̃ − π ̃ , β + π2 ]. Then 2 u(r, ϑ󸀠 , x)̃ ≥ u(r, ϑ0 , x)̃ ∀ϑ󸀠 ∈ [ϑ0 , 2β̃ − ϑ0 + π].

(6.52)

Proof. Since u is symmetric, uϑ is antisymmetric with respect to Hβ̃ . By hypothesis uϑ ≥ 0 in Ωβ̃ = [x ⋅ eβ̃ > 0] = {(r, ϑ, x)̃ : β̃ − π2 ≤ ϑ󸀠 ≤ β̃ + π2 } so that uϑ (r, ϑ󸀠 , x)̃ ≤ 0 if β̃ + π ≤ ϑ󸀠 ≤ β̃ + 3 π. Moreover, by (6.50), if β̃ − π < ϑ < β̃ + π then h+ (ϑ ) = 2β̃ − ϑ + π ∈ 2

2

2

0

2

β̃

0

0

[β̃ + π2 , β̃ + 32 π]. This means that u(r, ⋅, x)̃ increases in [ϑ0 , β̃ + π2 ], then decreases in ̃ [β̃ + π , β̃ + 3 π], in particular in [β̃ + π , 2β̃ − ϑ + π]. Since u(r, ϑ , x)̃ = u(r, 2β̃ − ϑ + π, x), 2

2

2

(6.52) follows.

0

0

0

Proof of Proposition 6.8. Let β ∈ [β̃ − π, β]̃ and let x ∈ Ωβ = [x ⋅ eβ > 0], equivalently x = (r cos ϑ, r sin ϑ, x)̃ with β − π2 < ϑ < β + π2 . We have to show that u(x) ≤ uσβ (x) if β − π2 < ϑ < β + π2 , β̃ − π ≤ β ≤ β.̃ Let us observe that, since u is symmetric with respect to Hβ̃ , uσβ (x) = u(σβ̃ (σβ (x))) = u(r cos(ϑ + 2(β̃ − β)), r sin(ϑ + 2(β̃ − β)), x)̃ because 2β̃ − (2β − ϑ + π) + π = ϑ + 2(β̃ − β). Let us first assume that x ∈ Ωβ ∩ Ωβ̃ , i. e. x = (r cos ϑ, r sin ϑ, x)̃ with (β − that

π π π π ≤)β̃ − < ϑ < β + (≤ β̃ + ). 2 2 2 2

Then we can apply Lemma 6.9 taking ϑ0 = ϑ, ϑ󸀠 = ϑ + 2(β̃ − β), because we have π β̃ − < ϑ ≤ ϑ + 2(β̃ − β) ≤ 2β̃ − ϑ + π 2

Indeed β̃ − β ≥ 0, and the last equality is equivalent to ϑ < β + π2 , which is true, since x ∈ Ωβ . So from Lemma 6.9 it follows that uσβ (x) = u(r, ϑ + 2(β̃ − β), x)̃ ≥ u(r, ϑ, x)̃ = u(x). Assume instead that x = (r, ϑ, x)̃ ∈ Ωβ \ Ωβ̃ , i. e., x = (r cos ϑ, r sin ϑ, x)̃ with 3 π π π (β̃ − π ≤)β − < ϑ ≤ β̃ − (≤ β + ). 2 2 2 2 Since u ≡ uσβ̃ , we have that u(x) = u(r, ϑ, x)̃ = u(r, 2β̃ − ϑ − π, x)̃ while as before we get that uσβ (x) = u(r, ϑ + 2(β̃ − β), x)̃

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192 | 6 Morse index and symmetry for semilinear elliptic equations in bounded domains Setting ϑ0 = 2β̃ − ϑ − π, ϑ󸀠 = ϑ + 2(β̃ − β), we have to prove the inequality u(r, ϑ󸀠 , x)̃ ≥ u(r, ϑ0 , x)̃ = u(x) This follows again by Lemma 6.9 provided we show that π π ϑ0 := (2β̃ − ϑ − π) ∈ [β̃ − , β̃ + ] 2 2 which follows from (6.51) (because β̃ − 32 π ≤ ϑ ≤ β̃ − π2 ), and ϑ0 ≤ ϑ󸀠 ≤ 2β̃ − ϑ0 + π i. e. 2β̃ − ϑ − π ≤ ϑ + 2(β̃ − β) ≤ 2β̃ − ϑ0 + π = ϑ + 2π The inequality ϑ + 2(β̃ − β) ≤ ϑ + 2π follows from the relation β̃ − π ≤ β ≤ β,̃ while 2β̃ − ϑ − π ≤ ϑ + 2(β̃ − β) is equivalent to ϑ > β − π2 , which is true since x ∈ Ωβ . A sufficient condition for the nonnegativity of the derivative uϑ in a symmetry position (which was one of the hypotheses of Proposition 6.8) is given in the following. Proposition 6.10. Let β̃ ∈ ℝ and assume that u is symmetric with respect to Hβ̃ . Assume further that there exists β < β̃ such that for any β ∈ (β , β)̃ we have u ≤ uσβ in Ω . Then uϑ =

𝜕u 𝜕ϑ

≥ 0 in Ωβ̃ .

1

1

β

Proof. We can write the angular derivative as uϑ (r, ϑ, x)̃ = lim+ α→0

u(r, ϑ + α, x)̃ − u(r, ϑ, x)̃ . α

With the change of variable α = 2(β̃ − β), β = β̃ − α2 , we have that β → β̃ − . If α is small, ̃ and, if x ∈ Ω ̃ , then x ∈ Ω definitively for β → β̃ − . Since, as observed, then β ∈ (β1 , β), β β σβ ̃ u (r, ϑ, x)̃ = u(r, ϑ + 2(β − β), x)̃ we obtain uϑ (r, ϑ, x)̃ = lim

β→β̃ −

= lim

β→β̃ −

u(r, ϑ + 2(β̃ − β), x)̃ − u(r, ϑ, x)̃ 2(β̃ − β)

uσβ (r, ϑ, x)̃ − u(r, ϑ, x)̃ ≥ 0. 2(β̃ − β)

Other properties of the angular derivative are given in the following lemma. Lemma 6.11. Let Ω be a rotationally symmetric domain in ℝN , N ≥ 2, u ∈ C 2 (Ω) a solution of (6.33) and η1 , η2 orthogonal directions. Let uϑ = uϑ(η1 ,η2 ) be as in (6.47) the angular derivative with respect to cylindrical coordinates relative to this pair of directions. Then:

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6.2 Foliated Schwarz symmetry and related properties | 193

(i) uϑ weakly satisfies the Dirichlet problem −Δuϑ − Vu uϑ = 0 { uϑ = 0

in Ω

(6.53)

on 𝜕Ω

where Vu is defined in (6.37). (ii) If e ∈ span (η1 , η2 ) is a direction of symmetry for the solution, i. e., u ≡ uσ(e) in Ω, then uϑ weakly satisfies in the half domain Ω(e) the Dirichlet problem −Δuϑ − Vu uϑ = 0 in Ω(e)

{

uϑ = 0

(6.54)

on 𝜕Ω(e)

Proof. (i) Since the proof involves only two directions we assume for simplicity that N = 2. Writing the Laplacian in polar coordinate, we have that Δu = uρρ + ρ−1 uρ + ρ−2 uϑϑ and if the function u is sufficiently regular, say of class C 3 (Ω), it is easy to check that the angular derivative satisfies the linearized equation −Δuϑ − Vu uϑ = 0 in Ω pointwisely. To see that if u ∈ C 2 (Ω) then u weakly satisfies the equation let us consider the diffeomorphism T = T(ρ, ϑ) = (x(ρ, ϑ), y(ρ, ϑ)) from U = (0, +∞) × (0, 2π) and V = ℝ2 \{(x, y) ∈ ℝ2 : y = 0, x ≥ 0} which defines the polar coordinates: x = ρ cos(ϑ), y = ρ sin(ϑ). Then the Jacobian matrix of T is given by xρ yρ

xϑ cos(ϑ) )=( yϑ sin(ϑ)

JT (ρ, ϑ) = (

−ρ sin(ϑ) ) ρ cos(ϑ)

so that its inverse, as a function of (ρ, ϑ), is given by ρx ϑx

JT −1 (x(ρ, ϑ), y(ρ, ϑ)) = (

ρy cos(ϑ) ) = ( −1 ϑy −ρ sin(ϑ)

Let φ ∈ Cc∞ (Ω \ {0}) be a test function, φϑ = solution of (6.33). Then we have

𝜕φ 𝜕ϑ

sin(ϑ) ) ρ cos(ϑ) −1

and assume that u ∈ C 2 (Ω) is a

sin ϑ cos ϑ , uy = uρ sin ϑ + uϑ , ρ ρ sin ϑ cos ϑ (φϑ )x = φρϑ cos ϑ − φϑϑ , (φϑ )y = φρϑ sin ϑ + φϑϑ ρ ρ ux = uρ cos ϑ − uϑ

(6.55)

so that ∇u ⋅ ∇φϑ = (uρ cos ϑ − uϑ

sin ϑ sin ϑ )(φρϑ cos ϑ − φϑϑ ) ρ ρ

+ (uρ sin ϑ + uϑ

cos ϑ cos ϑ 1 )(φρϑ sin ϑ + φϑϑ ) = uρ φρϑ + 2 uϑ φϑϑ ρ ρ ρ (6.56)

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194 | 6 Morse index and symmetry for semilinear elliptic equations in bounded domains Exchanging the role of u and φ, we get 1 u φ ρ2 ϑ ϑϑ 1 ∇uϑ ⋅ ∇φ = uρϑ φρ + 2 uϑϑ φϑ ρ ∇u ⋅ ∇φϑ = uρ φρϑ +

(6.57)

Taking ψ = φϑ as a test function in (6.33) we have that ∫ ∇u ⋅ ∇φϑ dx dy = ∫ f (u)φϑ dx dy Ω

Ω

Integrating in polar coordinates and using (6.57), we get ∞

2π

∫ ∇u ⋅ ∇φϑ dx dy = ∫ ρdρ ∫ dϑ(uρ φρϑ + 0

Ω

0 ∞

1 u φ ) ρ2 ϑ ϑϑ

2π

= − ∫ ρdρ ∫ dϑ(uρϑ φρ + 0

0

1 u φ ) = − ∫ ∇uϑ ⋅ ∇φ dx dy ρ2 ϑϑ ϑ Ω

and analogously ∫Ω f (u)φϑ dx dy = − ∫Ω f 󸀠 (|x|, u)uϑ φ dx dy so that ∫ ∇uϑ ⋅ ∇φ dx dy = ∫ f 󸀠 (|x|, u)uϑ φ dx dy Ω

(6.58)

Ω

for any test function φ ∈ Cc∞ (Ω \ {0}). By Proposition 1.46, we can extend (6.58) to all test functions φ ∈ Cc∞ (Ω),1 i. e., uϑ weakly solves −Δuϑ − Vu uϑ = 0 in Ω. Moreover, uϑ = 0 on 𝜕Ω. ii) By the symmetry of u with respect to the hyperplane H(e), we have that uϑ is antisymmetric with respect to H(e) and, therefore, vanishes on H(e). Since it vanishes on 𝜕Ω as well, it vanishes on 𝜕Ω(e), i. e., it satisfies the boundary condition in (6.54). Let us now prove some useful sufficient condition for the foliated Schwarz symmetry of a solution to the Dirichlet problem (6.33). Theorem 6.12 (Sufficient conditions for FSS). Let Ω be a bounded rotationally symmetric domain and u ∈ C 2 (Ω) a solution of (6.33), where f ∈ C 1 ([0, ∞)×ℝ). Then u is foliated Schwarz symmetric provided one of the following conditions holds: (i) there exists a direction e ∈ SN−1 such that u ≡ uσ(e) in Ω and λ1e ≥ 0, with λ1e as in (6.41). (ii) there exists a direction e ∈ SN−1 such that u < uσ(e) or u > uσ(e) in Ω(e). 1 If N ≥ 3, we start with test functions φ ∈ Cc∞ (Ω \ F), with F = {x = (x1 , . . . , xn ) ∈ ℝN : x1 = x2 = 0}, and then exploit Remark 1.49.

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6.2 Foliated Schwarz symmetry and related properties | 195

Proof. (i) For any direction η orthogonal to e the corresponding angular derivative uϑ = uϑ(e,η) satisfies (6.54) in Ω(e). Since λ1e ≥ 0 either λ1e > 0 and then (by the maximum principle, which holds by Theorem 1.50) uϑ = 0, or λ1e = 0 and then, if it does not vanish, uϑ is a first eigenfunction of the operator −Δ − Vu in Ω(e) with Dirichlet boundary condition and, therefore, it does not change sign. In any case by Proposition 6.8, we have that (6.34) holds for any direction in the plane spanned by e and η and since η is an arbitrary direction orthogonal to e we get that (6.34) holds for any direction and, by Proposition 6.7, u is foliated Schwarz symmetric. (ii) If η is any direction orthogonal to e, by Theorem 6.3 we get that rotating the hy󸀠 perplanes there exists a direction e󸀠 in the plane spanned by e and η such that u ≡ uσ(e ) 󸀠󸀠 in Ω(e󸀠 ) and u ≤ uσ(e ) for any direction e󸀠󸀠 obtained while rotating e to reach e󸀠 . Therefore, the corresponding angular derivative (with respect to e and η) is nonnegative in Ω(e󸀠 ) by Proposition 6.10. Hence the assertion follows as in (i). Remark 6.13. 1. In the proof of (ii) of Theorem 6.12, we can avoid using Proposition 6.10 by show󸀠 ing that λ1e = 0 in the final direction of symmetry obtained by rotating the planes (see the proof of Theorem 6.12), and observing that the corresponding first eigenfunction uϑ does not change sign in Ω(e󸀠 ), so that the hypotheses of case (i) are satisfied. Indeed, using the notation of Theorem 6.3, we have that for every ϑ ∈ (ϑ0 , ϑ1 ) the function wϑ = u − uσ(eϑ ) is strictly negative in Ω(eϑ ) and, therefore, it is the first eigenfunction, with eigenvalue zero, of the operator −Δ − Veϑ in Ω(eϑ ). When σe

(u ) tends to the potential Vu = f 󸀠 (|x|, u), and, ϑ → ϑ1 , the potential Veϑ = f (u)−f σ u−u eϑ by continuity (see Theorem 1.44 v)), the first eigenvalue of −Δ − Vu in Ω(eϑ1 ) is zero (with uϑ as a first eigenfunction). Let u be a foliated Schwarz symmetric solution of (6.33) and e a direction such that u ≡ uσ(e) in Ω and λ1e ≥ 0. As we have seen in the proof of Theorem 6.12 for any direction η orthogonal to e the corresponding angular derivative uϑ = uϑ(e,η) (with respect to the plane spanned by e, η) either vanishes or is strictly positive (or negative) in Ω(e). In particular, if u is not radial and η is the direction of the vector p which appears in Definition 6.5 (i. e., the direction of the symmetry axis of u), then the derivative e,η uϑ(e,η) = uϑ is not zero, and it is strictly positive in Ω(e) since u(rp) = maxSr u for r > 0 (see the proof of Proposition 6.7). Equivalently, interchanging the directions, we get that uϑ(η,e) < 0 in Ω(e). Observe that if Π is the plane spanned by η and e then, for any x ∈ Π ∩ Ω(e), the x angular variable θ = arccos( |x| ⋅ p) which appears in Definition 6.5, coincides with the angular variable ϑ(η, e) in the plane Π. By the axial symmetry, the direction e can be substituted by any direction orthogonal to η, so that u is strictly decreasing x with respect to the angular variable θ = arccos( |x| ⋅ p) if it is not radial. ϑ

2.

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196 | 6 Morse index and symmetry for semilinear elliptic equations in bounded domains

6.3 Foliated Schwarz symmetry of low Morse index solutions of elliptic Dirichlet problems In this section, we begin to study the relation between the Morse index of a solution to a semilinear elliptic problem and its symmetry.

6.3.1 Convex nonlinearities We suppose that Ω is either a ball or an annulus centered at the origin in ℝN , N ≥ 2, and u ∈ C 2 (Ω) is a solution of the semilinear problem −Δu = f (|x|, u) in Ω { u=0 on 𝜕Ω

(6.59)

f ∈ C 1 ((0, +∞) × ℝ)

(6.60)

where

We will use the definitions and notation (6.37)–(6.43). The first result relating the Morse index of a solution to its FSS symmetry was proved in [186] and we discuss it below. Theorem 6.14. Suppose that the nonlinearity f = f (|x|, s) in (6.59) is convex in the second variable. Then any solution u ∈ C 2 (Ω) of (6.59) with Morse index m(u) ≤ 1 is foliated Schwarz symmetric. Moreover, if u has Morse index zero, then it is radial. The proof is based on the following proposition, which relates the convexity of the function f to the Morse index of the solution in the caps Ω(e). It will be exploited again in the sequel. Proposition 6.15. Let the function f = f (|x|, s) in (6.59) be convex in the second variable and let u ∈ C 2 (Ω) be a solution of (6.59). If e ∈ SN−1 is a direction such that λ1e ≥ 0, then one of the following alternatives holds: 1. u ≡ uσ(e) in Ω(e) 2. u < uσ(e) in Ω(e) 3. u > uσ(e) in Ω(e) If either f is strictly convex or if λ1e > 0 then (3) cannot hold. Proof. Since f is convex, defining v(x) = uσ(e) (x) we have that if u(x) > v(x) then

f (|x|, u(x)) − f (|x|, v(x)) ≤ f 󸀠 (|x|, u(x)) u(x) − v(x)

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(6.61)

6.3 FSS of solutions of Dirichlet problems | 197

with strict sign if f is strictly convex. It follows that for any u, v 2

2

Ve ((u − v)+ ) ≤ Vu ((u − v)+ )

(6.62)

with Vu and Ve as defined in (6.37) and (6.42) and the strict sign holds if f is strictly convex and (u − v)+ does not vanish. + Testing the equation (6.43) with the function w+ = (we ) = (u − uσ(e) )+ , we obtain 󵄨 󵄨2 󵄨 󵄨2 0 = ∫ (󵄨󵄨󵄨∇w+ 󵄨󵄨󵄨 − Ve |w|2 ) dx ≥ ∫ (󵄨󵄨󵄨∇w+ 󵄨󵄨󵄨 − Vu |w|2 ) dx Ω(e)

(6.63)

Ω(e)

Since λ1e ≥ 0, we deduce that either w+ ≡ 0 or w+ is the first eigenfunction corresponding to the eigenvalue 0 = λ1 (−Δ − Vu ; Ω(e)). Note that the second alternative cannot happen if f is strictly convex in the second variable or if λ1e > 0, in which case we deduce that w+ ≡ 0. In any case, if w+ is positive in Ω(e), then u > uσ(e) in Ω(e), otherwise w+ ≡ 0 so that u ≤ uσ(e) in Ω(e). In the latter case, since w satisfies (6.43), by the strong maximum principle either u < uσ(e) or u ≡ uσ(e) in Ω(e). Corollary 6.16. Under the hypotheses of Proposition 6.15 if, in addition, also λ1−e ≥ 0, then u ≡ uσ(e) in Ω(e) provided one of the following holds: 1. λ1e > 0, λ1−e > 0 2. f is strictly convex in the second variable Remark 6.17. Proposition 6.15 and Corollary 6.16 hold more generally if Ω is a domain symmetric with respect to the direction e (see [186]). Proof of Theorem 6.14. Let us assume that u is either positive or sign changing and let x0 ∈ Ω be such that u(x0 ) = maxΩ u (if u is negative we take the minimum point and the same proof applies). Then we consider any direction e orthogonal to the direction p of the axis passing through the origin and the point x0 . Since m(u) ≤ 1, by the variational characterization of the eigenvalues (Theorem 1.42), either λ1e ≥ 0 or λ1−e ≥ 0. Indeed, if this would not be the case, then the quadratic form Qu (v) = ∫Ω (|∇v|2 − Vu |v2 |) dx would be negative definite on the 2-dimensional space spanned by the trivial extensions of the eigenfunctions φe1 and e −e φ−e 1 , and hence m(u) ≥ 2. We can assume that λ1 ≥ 0, otherwise we consider λ1 . σ(e) σ(e) σ(e) Since f is convex, by Proposition 6.15 either u ≡ u or u < u or u > u in e σ(e) Ω(e). However, only the first possibility can hold, because if w (x) = u(x) − u (x) had a strict sign in Ω(e), since it satisfies (6.43), by Hopf’s lemma it would have a positive or negative normal derivative on the boundary. This is impossible, since ∇u(x0 ) = 0 because x0 is the maximum of the function u. So (u − uσ(e) ) ≡ 0 in Ω(e) and λ1e = λ1−e ≥ 0.

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198 | 6 Morse index and symmetry for semilinear elliptic equations in bounded domains Since e is an arbitrary direction e orthogonal to p, we get that u is axially symmetric with respect to the axis passing through x0 and by the sufficient condition given by Theorem 6.12 u is foliated Schwarz symmetric. Finally, if the Morse index is zero then λ1 (−Δ − Vu ; Ω) ≥ 0, so that for any direction e we have that λ1e = λ1 (−Δ − Vu ; Ω(e)) > 0. By Corollary 6.16, this implies that we ≡ 0 ∀ e ∈ SN−1 , i. e., u is radial. Remark 6.18. If m(u) = 0, i.e. if u is a stable solution of (6.59), the radial symmetry actually follows as in Theorem 1.5. of [131], without requiring any convexity on f . To generalize the result of Theorem 6.14 to higher Morse index solutions, we will use another, more indirect, method where the symmetry axis will be determined using also the rotating planes procedure. Moreover, we will exploit the well-known Borsuk– Ulam theorem. We recall here a version of it referring, e. g., to Corollary 4.2. in [107] for the proof. Theorem 6.19 (Borsuk–Ulam theorem). Let D be a bounded domain in ℝN containing the origin and symmetric with respect to it, and let g : 𝜕D → ℝM , with M < N, be a continuous function. Then g(x) = g(−x) for some x ∈ 𝜕D. In particular, if g : SN−1 → ℝM is continuous and odd (i. e., g(−x) = −g(x) for any x ∈ SN−1 ) then it has a zero, i. e., there exists e ∈ SN−1 such that g(e) = 0 ∈ ℝM . Let us now prove the following generalization of Theorem 6.14 to the case of solutions with Morse index not exceeding the dimension N. Theorem 6.20. Let the function f = f (|x|, s) in (6.59) be convex in the second variable. Then any solution u ∈ C 2 (Ω) of (6.59) with Morse index m(u) ≤ N is foliated Schwarz symmetric. The proof will be based on the following. Proposition 6.21. Let u be a solution of problem (6.59) with Morse index m(u) ≤ N. Then there exists a direction e ∈ SN−1 such that λ1e ≥ 0. Proof. The assertion is immediate if the Morse index of the solution satisfies m(u) ≤ 1. Indeed, in this case for any direction e, at least one among λ1e and λ1−e must be nonnegative, as observed at the beginning of the proof of Theorem 6.14. So let us assume that 2 ≤ j = m(u) ≤ N. Denote by Φk the L2 (Ω) normalized eigenfunctions of the operator Lu = −Δ − Vu in Ω, with Φ1 positive in Ω, and for any direction e ∈ SN−1 let us consider the function (φ−e ,Φ1 )

1

1 e L (Ω) 2 { {( (φe1 ,Φ1 )L2 (Ω) ) φ1 (x) e ψ (x) = { { (φe1 ,Φ1 )L2 (Ω) 21 −e −( ) φ1 (x) { (φ−e 1 ,Φ1 )L2 (Ω) 2

if x ∈ Ω(e) if x ∈ Ω(−e)

where φe1 is as in (6.41).

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6.3 FSS of solutions of Dirichlet problems | 199

The mapping e 󳨃→ ψe is odd and continuous from SN−1 to H01 (Ω) and, by construction, (ψe , Φ1 )L2 (Ω) = 0

(6.64)

The function h : SN−1 → ℝj−1 defined by h(e) = ((ψe , Φ2 )L2 (Ω) , . . . , (ψe , Φj )L2 (Ω) )

(6.65)

is also odd and continuous. Since (j−1) < N, by the Borsuk–Ulam theorem it must have a zero. This means that there exists a direction e ∈ SN−1 such that ψe is orthogonal to all the eigenfunctions Φ1 , . . . , Φj . Since m(u) = j, by Theorem 1.42 (ii) we deduce that Qu (ψe ; Ω) ≥ 0, which in turn implies that either Qu (φe1 ; Ω(e)) ≥ 0 or Qu (φ−e 1 ; Ω(−e)) ≥ 0, i. e. either λ1e or λ1−e is nonnegative, so the assertion is proved. Proof of Theorem 6.20. Once we have a direction e such that λ1e ≥ 0 we get from Proposition 6.15 that either we > 0, or we < 0 in Ω(e), or we ≡ 0 in Ω(e) (with λ1e ≥ 0). Thus by the sufficient conditions given by Theorem 6.12 u is foliated Schwarz symmetric. 6.3.2 Nonlinearities with a convex derivative The results in the previous section apply to several problems, in particular they apply to the model superlinear nonlinearity, f (|x|, u) = g(|x|)up , with g(r) > 0, p > 1, when the solutions are positive. A corresponding model nonlinearity for sign changing solutions is the function f (|x|, u) = g(|x|)|u|p−1 u, g > 0, p > 1, which is not convex with respect to the second variable. In this section, we study, following [190], another case, namely we suppose that the derivative f 󸀠 (|x|, s) = 𝜕f (|x|, s) is convex. 𝜕s In particular, if f (|x|, s) = g(|x|)|s|p−1 s then the derivative is f 󸀠 (|x|, s) = g(|x|)p|s|p−1 , which is convex when p ≥ 2. We assume, as in the previous section, that Ω is either a ball or an annulus centered at the origin in ℝN , N ≥ 2, and u ∈ C 2 (Ω) is a solution of the semilinear problem (6.59) where f ∈ C 1 ((0, +∞) × ℝ). We will use the definitions and notation of the previous section, in particular (6.37)–(6.43). It is important now to introduce another potential for any direction e ∈ SN−1 , which is the e-symmetric part of the potential Vu : Ves (x) = V(e,s)(u) (x) =

f 󸀠 (|x|, u(x)) + f 󸀠 (|x|, u(σe (x))) 2

(6.66)

and the corresponding quadratic forms Qes (v; Ω) = ∫(|∇v|2 − Ves |v|2 ) dx, Ω

v ∈ H01 (Ω)

(6.67)

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200 | 6 Morse index and symmetry for semilinear elliptic equations in bounded domains Qes (v; Ω(e)) = ∫ (|∇v|2 − Ves |v|2 ) dx,

v ∈ H01 (Ω(e))

(6.68)

Ω(e)

Let us denote by λk (e, Ves )

(6.69)

the eigenvalues of the operator −Δ−Ves (x) in the half domain B(e), with homogeneous Dirichlet boundary conditions. Note that Qes and Qu coincide on functions which are either even or odd with respect to the reflection σe as it is easy to check with a change of variable in the integrals: Qes (v; Ω) = Qu (v; Ω) if v = ±vσ(e) in Ω

(6.70)

We will prove in this section the following result. Theorem 6.22. Assume that the nonlinearity f = f (|x|, s) has a derivative f 󸀠 (|x|, s) = 𝜕f (|x|, s) convex in the s-variable, for every x ∈ Ω. Then every solution of (6.59) with 𝜕s Morse index j ≤ N is foliated Schwarz symmetric. Let us remark that in fact there are only two possibilities under the assumptions of Theorem 6.22: either the solution is radially symmetric or it is strictly monotone in the polar angle by Remark 6.13. We also show an interesting consequence of Theorem 6.22 concerning the geometrical properties of the nodal set of a sign changing solution of (6.59) which relies on some results of [7]. Theorem 6.23. Suppose that the nonlinearity f in (6.59) does not depend on x, i. e., f = f (s) and its derivative f 󸀠 is convex. Then the nodal set of any sign changing solutions of (6.59) with Morse index less than or equal to N intersects the boundary of Ω. Note that the assumptions of Theorem 6.23 are satisfied for the power type nonlinearity f (s) = |s|p−1 s, p ≥ 2. In particular, under these assumptions, from Theorem 6.23 it follows that every sign changing solution u of (6.59) with Morse index less than or equal to N is nonradial. This was proved in [7] and we already discussed it in Chapter 3. The proofs of Theorem 6.22 and Theorem 6.23 are postponed after the following result which gives a sufficient condition for the existence of a symmetry hyperplane for a solution of (6.59). Proposition 6.24. Assume that f 󸀠 (|x|, s) is convex in the s-variable. Let u be a solution of (6.59), such that there exists a direction e ∈ SN−1 with λ1 (e, Ves ) ≥ 0. Then the following statements hold: (i) if either λ1 (e, Ves ) > 0 or f 󸀠 (|x|, s) is strictly convex in s, then u is symmetric with respect to the hyperplane H(e) and hence λ1 (e, Vu ) = λ1 (e, Ves ) ≥ 0. (ii) if λ1 (e, Ves ) = 0, f 󸀠 is only convex and u is not symmetric with respect to the hyperplane H(e) then either u < uσ(e) or u > uσ(e) .

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6.3 FSS of solutions of Dirichlet problems | 201

ing.

Combining this proposition with Theorem 6.12, we immediately obtain the follow-

Corollary 6.25. Under the assumptions of Proposition 6.24, the solution u is foliated Schwarz symmetric. Proof of Proposition 6.24. Let us denote by we the difference between u and its reflection with respect to the hyperplane H(e), i. e., we (x) = u(x) − u(σe (x)). It is easy to see that we solves the linear problem −Δwe − Ve (x)we = 0

{

e

w =0

in Ω(e), on 𝜕Ω(e),

(6.71)

where 1

Ve (x) = ∫ f 󸀠 (|x|, tu(x) + (1 − t)u(σe (x))) dt,

x ∈ Ω(e).

(6.72)

0

Since f 󸀠 is convex in the second variable, we have, for x ∈ Ω, 1

Ve (x) ≤ ∫[tf 󸀠 (|x|, u(x)) + (1 − t)f 󸀠 (|x|, u(σe (x)))] dt 0

1 = [f 󸀠 (|x|, u(x)) + f 󸀠 (|x|, u(σe (x)))] = Ves (x) 2

(6.73)

Here, the strict inequality holds if f 󸀠 is strictly convex and u(x) ≠ u(σe (x)). Hence, denoting by λk (e, Ve ) the eigenvalues of the linear operator −Δ − Ve (x) in Ω(e) with homogeneous Dirichlet boundary conditions, we have, by (6.73) and Theorem 1.44(iv), that λk (e, Ve ) ≥ λk (e, Ves ). In particular, λ1 (e, Ve ) ≥ λ1 (e, Ves ) ≥ 0 by hypothesis. If λ1 (e, Ve ) > 0, then we ≡ 0 because it satisfies (6.71), and hence we get the symmetry of u with respect to the hyperplane H(e). If λ1 (e, Ve ) = 0, then Ve ≡ Ves in Ω by the strict monotonicity of eigenvalues with respect to the potential, (Theorem 1.44(iv)). In the case when f 󸀠 is strictly convex, this implies that u(x) ≡ u(σe (x)) for every x ∈ Ω, so that again we get the symmetry of u with respect to the hyperplane H(e). It remains to consider the case when f 󸀠 is only convex and λ1 (e, Ve ) = 0. In this case, either we ≡ 0 or we does not change sign in Ω(e), and, by the strong maximum principle, we > 0 or we < 0 in Ω(e). Before passing to the proof of Theorem 6.22, we recall that the quadratic forms Qu and Qe,s are defined in (6.38) and (6.67), respectively. Proof of Theorem 6.22. Let u be a solution of (6.59) with Morse index m(u) = j ≤ N. Thus, for the Dirichlet eigenvalues λk of the linearized operator L = −Δ − Vu (x) in Ω we have λ1 < 0, . . . , λj < 0

and λj+1 ≥ 0.

(6.74)

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202 | 6 Morse index and symmetry for semilinear elliptic equations in bounded domains We distinguish two cases. Case 1: j ≤ N − 1 In this case, we obtain the foliated Schwarz symmetry of u, applying Corollary 6.25 in a simple way. Indeed, for any direction e ∈ S, let us denote by ge ∈ H01 (Ω) the odd extension in Ω of the positive L2 -normalized eigenfunction of the operator −Δ − Ves (x) in the half domain Ω(e) corresponding to λ1 (e, Ves ). It is easy to see that ge depends continuously on e in the L2 -norm. Moreover, g−e = −ge for every e ∈ S. Now we let ϕ1 , ϕ2 , . . . , ϕj ∈ H01 (Ω) denote L2 -orthonormal eigenfunctions of Lu corresponding to the eigenvalues λ1 , . . . , λj . By Theorem 1.42, we have inf

v∈H 1 (Ω)\{0} 0

v⊥ϕ1 ,...v⊥ϕj

Qu (v) = λj+1 ≥ 0. (v, v)L2 (Ω)

(6.75)

We consider the map h : S = SN−1 → ℝj ,

h(e) = [(ge , ϕ1 )L2 (Ω) , . . . , (ge , ϕj )L2 (Ω) ].

(6.76)

Since h is an odd and continuous map on the unit sphere S ⊂ ℝN and j ≤ N − 1, h must have a zero by the Borsuk–Ulam theorem. This means that there is a direction e ∈ S such that ge is L2 -orthogonal to all eigenfunctions ϕ1 , . . . , ϕj . Thus Qu (ge ) ≥ 0 by (6.75). But, since ge is an odd function, by (6.70) Qu (ge ) = Qes (ge ) = 2λ1 (e, Ves ), which yields that λ1 (e, Ves ) ≥ 0. Having obtained a direction e for which λ1 (e, Ves ) is nonnegative, Corollary 6.25 applies and we get the foliated Schwarz symmetry of u. Case 2: j = N The main difficulty in this case is that the map h, considered in (6.76), now goes from the (N − 1)-dimensional sphere S into ℝN , so that the Borsuk–Ulam theorem does not apply. We therefore use a different and less direct argument to find a symmetry hyperplane H(e󸀠 ) for u with λ1 (e󸀠 , Vu ) ≥ 0. Let S∗ ⊂ S be defined by S∗ = {e ∈ S : we ≡ 0 in Ω and λ1 (e, Vu ) < 0}, where, as before, we is given by we (x) = u(x) − u(σe (x)). Then S∗ is a symmetric set. If S∗ ≠ ⌀, we let k be the largest integer such that there exist k orthogonal directions e1 , . . . , ek ∈ S∗ . We then observe that k ∈ {1, . . . , N − 1}. Indeed, if we denote by 𝒱0 the closed subspace of L2 (Ω) consisting of the functions which are symmetric with respect to the hyperplanes H(e1 ), . . . , H(ek ), we have that u ∈ 𝒱0 (by the very definition of S∗ ) and also ϕ1 ∈ 𝒱0 , where ϕ1 denotes the unique positive L2 -normalized Dirichlet eigenfunction of the linearized operator Lu = −Δ − Vu (x) on Ω relative to the first eigenvalue

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6.3 FSS of solutions of Dirichlet problems | 203

λ1 . Now we consider, for l = 1, . . . , k, the functions gl ∈ H01 (Ω) defined as the odd extensions in Ω of the unique positive L2 -normalized eigenfunction of the operator Lu in the half-domain Ω(el ) corresponding to λ1 (el , Vu ) < 0. Then, since u is symmetric with respect to H(el ), gl is an eigenfunction of the operator −Δ − Vu (x) in the whole Ω corresponding to a certain eigenvalue λm (= λ1 (el , Vu ) < 0), m ≥ 2. Since each gl is odd with respect to the reflection at H(el ) and even with respect to the reflection at H(em ) for m ≠ l, we have, letting l vary from 1 to k a set {g1 , . . . , gk } of k orthogonal eigenfunctions of the operator −Δ − Vu (x), all corresponding to negative eigenvalues. Since the Morse index of u is equal to N and the first eigenfunction ϕ1 is also orthogonal to each gl we have that k ≤ N −1 as we claimed. The same argument shows that if we denote by L0 the self-adjoint operator which is the restriction of the operator L to the symmetric space 𝒱0 , i. e., L0 : H2 (Ω) ∩ H01 (Ω) ∩ 𝒱0 → 𝒱0 ,

L0 v = −Δv − Vu (x)v

and we denote by m0 the number of the negative eigenvalues of L0 (counted with multiplicity), we have 1 ≤ m0 ≤ N − k.

(6.77)

Note that the inequality on the left-hand side of (6.77) just follows from the fact that ϕ1 ∈ 𝒱0 is an eigenfunction of L relative to a negative eigenvalue. Now we first assume that S∗ ≠ ⌀ and consider the (N − k − 1)-dimensional sphere S∗ defined by S∗ = S ∩ H(e1 ) ∩ ⋅ ⋅ ⋅ ∩ H(ek ). By the maximality of k, we have S∗ ∩ S∗ = ⌀. We would like to prove that there exists a direction e ∈ S∗ such that we (x) ≥ 0

for every x ∈ Ω(e).

(6.78)

We argue by contradiction and suppose that we changes sign in Ω(e) for every e ∈ S∗ . We consider the functions w1e = (we )+ χΩ(e) + (we )− χΩ(−e) and w2e = (we )− χΩ(e) + (we )+ χΩ(−e) , where χA denotes the characteristic function of a set A. Since u ∈ 𝒱0 , we find that w1e , w2e ∈ 𝒱0 ∩ H01 (Ω) \ {0}, and both functions are nonnegative and symmetric with respect to H(e). Moreover, w1−e = w2e

and w2−e = w1e

for every e ∈ S∗ .

(6.79)

Recalling the definition of Ve in (6.72), since we satisfies the linear equation −Δwe − Ve (x)we = 0 in the whole domain Ω with we = 0 on 𝜕Ω, multiplying by (we )+ χB(e) − (we )− χΩ(−e) and integrating over Ω we obtain 0 = ∫ ∇we ⋅ ∇((we ) χΩ(e) − (we ) χΩ(−e) ) dx +

−

Ω

− ∫ Ve (x)we ((we ) χΩ(e) − (we ) χΩ(−e) ) dx +

−

Ω

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204 | 6 Morse index and symmetry for semilinear elliptic equations in bounded domains − + 󵄨2 󵄨2 󵄨 󵄨 = ∫(󵄨󵄨󵄨∇((we ) χΩ(e) )󵄨󵄨󵄨 + 󵄨󵄨󵄨∇((we ) χΩ(−e) )󵄨󵄨󵄨 ) dx Ω

2

2

− ∫ Ve (x)[((we ) χΩ(e) ) + ((we ) χΩ(e) ) ] dx +

−

Ω 2 󵄨 󵄨2 = ∫󵄨󵄨󵄨∇w1e 󵄨󵄨󵄨 dx − ∫ Ve (x)(w1e ) dx.

(6.80)

Ω

Ω

Now we can use the comparison between the potential Ve (x) and Ves (x), given by (6.73), and obtain from (6.80) 2 󵄨 󵄨2 0 ≥ ∫󵄨󵄨󵄨∇w1e 󵄨󵄨󵄨 dx − ∫ Ves (x)(w1e ) dx = Qes (w1e ) = Qu (w1e ), Ω

(6.81)

Ω

because w1e is a symmetric function with respect to σe . Similarly, we can show Qu (w2e ) ≤ 0.

(6.82)

Now, for every e ∈ S∗ , we let ψe ∈ 𝒱0 ∩ H01 (Ω) be defined by ψe (x) = (

(w2e , ϕ1 )L2 (Ω)

(w1e , ϕ1 )L2 (Ω)

1/2

) w1e (x) − (

(w1e , ϕ1 )L2 (Ω)

(w2e , ϕ1 )L2 (Ω)

1/2

) w2e (x).

Using (6.79), it is easy to see that e 󳨃→ ψe is an odd and continuous map from S∗ to 𝒱0 . By construction, ⟨ψe , ϕ1 ⟩ = 0 for all e ∈ S∗ . Moreover, since w1e and w2e have disjoint supports, (6.81) and (6.82) imply Qu (ψe ) ≤ 0

for all e ∈ S∗ .

(6.83)

Recalling (6.77), we first assume that m0 ≥ 2. Let λ1 , λ̃2 , . . . , λ̃m0 be the negative eigenvalues of the operator L0 in increasing order, and let ϕ1 , ϕ̃ 2 , . . . , ϕ̃ m0 ∈ 𝒱0 be corresponding L2 -orthonormal eigenfunctions. As in (6.75) we have inf

v∈H 1 (Ω)∩𝒱0 , v=0 ̸ 0

v⊥ϕ1 ,...v⊥ϕ̃ m0

Qu (v) ≥ 0. (v, v)L2 (Ω)

(6.84)

We now consider the map h : S∗ → ℝm0 −1 defined by h(e) = [(ψe , ϕ̃ 2 )L2 (Ω) , . . . , (ψe , ϕ̃ m0 )L2 (Ω) ] Since h is an odd and continuous map defined on a (N − k − 1)-dimensional sphere and m0 ≤ N − k, the map h must have a zero by the Borsuk–Ulam theorem. Hence there is e ∈ S such that (ψe , ϕ̃ k )L2 (Ω) = 0 for k = 2, . . . , m0 and, by construction, (ψe , ϕ1 )L2 (Ω) = 0.

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6.4 Symmetry of solutions of mixed boundary value problems | 205

Since ψe ∈ 𝒱0 and Qu (ψe ) ≤ 0, the function ψe is a mimimizer for the quotient in (6.84). Consequently, it must be an eigenfunction of the operator L0 corresponding to the eigenvalue zero. Hence ψe ∈ C 2 (Ω), and ψe solves −Δψe − Vu (x)ψe = 0 in Ω. Moreover, ψe = 0 on H(e) by the definition of ψe , and 𝜕e ψe = 0 on H(e), since ψe is symmetric with respect to the reflection at H(e). From this, it is easy to deduce that the function ψ̂ e defined by ψe (x),

ψ̂ e (x) = {

0,

x ∈ Ω(e),

x ∈ Ω(−e),

is also a (weak) solution of −Δψ̂ e − Vu (x)ψ̂ e = 0. This however contradicts the unique continuation theorem for this equation (see, e. g., [200, p. 519]). Now we assume that m0 = 1. Then, for any e ∈ S∗ , since ψe ∈ 𝒱0 , (ψe , ϕ1 )L2 (Ω) = 0 and Qu (ψe , ψe ) ≤ 0, the function ψe must be an eigenfunction of the operator L0 corresponding to the eigenvalue zero. This leads to a contradiction as in the previous case. Since in both cases we reached a contradiction, there must be a direction e ∈ S∗ such that we does not change sign on Ω(e). Hence (6.78) holds either for e or for −e. Now, to end the proof, we again distinguish two cases. Suppose first that we ≡ 0 on Ω. Since S∗ ∩ S∗ = ⌀, we then have λ1 (e, Vu ) ≥ 0. If instead we ≢ 0, we ≢ 0, then we > 0 in Ω(e) by (6.78) and the strong maximum principle, since we solves (6.71). Thus, by Theorem 6.12, u is foliated Schwarz symmetric. Finally if S∗ = ⌀ we repeat all the steps starting from (6.78), taking S∗ = S and replacing H01 (Ω) ∩ 𝒱0 by H01 (Ω). Again we reach a direction e󸀠 (possibly equal to e) such that we󸀠 ≡ 0, and thus λ1 (e󸀠 , Vu ) ≥ 0 because S∗ = ⌀. Hence u must be foliated Schwarz symmetric. Proof of Theorem 6.23. Let u be a solution with Morse index m(u) ≤ N. In the proof of Theorem 6.22, we have obtained a symmetry hyperplane H(e) for u such that λ1 (e, Vu ) ≥ 0. On the other hand, if the nodal set does not intersect the boundary, the proof of Theorem 3.19 shows that λ1 (e, Vu ) < 0. Hence the assertion holds.

6.4 Symmetry of solutions of mixed boundary value problems We study in this section elliptic problems with nonlinear mixed boundary conditions of the type −Δu = f (|x󸀠 |, xN , u) in Ω { { { { u=0 on Γ0 { { { 𝜕u { = g(|x 󸀠 |, u) on Γ { 𝜕𝜐

(6.85)

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206 | 6 Morse index and symmetry for semilinear elliptic equations in bounded domains where Ω is a cylindrically symmetric domain as in Definition 1.15 and its flat boundary Γ is defined in 1.33 (we refer to Chapter 1 for the corresponding notation, definitions and examples). We suppose that f : [0, +∞)×(0, +∞)×ℝ → ℝ and g : [0, +∞)×ℝ → ℝ are C 1 functions and a solution of (6.85) will be understood in a weak sense. We identify the (N − 2)-dimensional sphere in ℝN−1 in the following way: SN−2 = {v = (v1 , . . . , vN ) ∈ SN−1 : v ⋅ eN = 0}

(6.86)

where eN = (0, . . . , 1). We recall that in Chapter 1 we discussed the maximum principles and the construction of the eigenvalues for mixed boundary value problems in analogy with the case of Dirichlet problems. This allows to follow quite closely what done in the previous sections. The main differences are that the set of directions that we consider are only those in SN−2 , and the linearized problem is a mixed boundary value problem. In particular, the proof of Theorem 6.1 can be easily repeated to prove the cylindrical symmetry of positive solutions to (6.85) in cylindrical domains that are convex, namely half-balls or cylinders. Theorem 6.26. Let Ω be a half-ball or a cylinder and let u ∈ H01 (Ω ∪ Γ) ∩ C 0 (Ω) be a positive weak solution of (6.85), where f = f (|x 󸀠 |, xN , s) : [0, +∞) × ℝ × ℝ → ℝ, g = g(|x󸀠 |, s) : [0, +∞) × ℝ → ℝ are continuous functions that are C 1 with respect to s and nonincreasing w. r. t. the first variable. Then u has cylindrical symmetry, i. e., if x = (x 󸀠 , xN ), x 󸀠 ∈ ℝN−1 , then u(x 󸀠 , xN ) = 󸀠 v(|x |, xN ) for some function v = v(r, xN ). Moreover, vr (r, xN ) < 0. Proof. We have to prove the symmetry with respect to every hyperplane orthogonal to any direction in SN−2 (see (6.86)). The proof is exactly the same as the proof of Theorem 6.1 using Theorem 1.26 instead of Theorem 1.21, together with the strong comparison principle (Theorem 1.31), as in Theorem 6.1. If e ∈ SN−2 , we define as before H(e) = {x ∈ ℝN : x ⋅ e = 0} and

Ω(e) = {x ∈ Ω : x ⋅ e > 0}

with the corresponding boundaries Γ0 (e) = (Γ0 ∩ Ω(e)) ∪ (H(e) ∩ Ω(e)) Γ(e) = Γ ∩ (Ω(e) \ H(e))

(6.87)

The rotating planes technique (see Theorem 6.3) works exactly as in the Dirichlet case, and we get the following result. Theorem 6.27 (Rotating planes for cylindrically symmetric domains). Let Ω be a bounded cylindrically symmetric domain in ℝN , N ≥ 3 and u ∈ H01 (Ω ∪ Γ) ∩ C 0 (Ω) a weak solution of (6.85). Suppose that e = eϑ0 = (cos(ϑ0 ), sin(ϑ0 ), 0, . . . , 0) ∈ SN−2 such

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6.4 Symmetry of solutions of mixed boundary value problems | 207

that u < uσ(eϑ0 ) in Ω(eϑ0 ). Then there exists a direction eϑ1 = (cos(ϑ1 ), sin(ϑ1 ), 0, . . . , 0) with ϑ1 > ϑ0 such that u ≡ uσ(eϑ1 ) in Ω(eϑ1 )

and

u < uσ(eϑ ) in Ω(eϑ ) for any ϑ ∈ (ϑ0 , ϑ1 )

We now discuss a variant of the foliated Schwarz symmetry for solutions of (6.85). We will call it sectional foliated Schwarz symmetry. Since it is meaningful for N ≥ 3, we will not consider the case N = 2. Definition 6.28. Let Ω be a bounded domain with cylindrical symmetry in ℝN , N ≥ 3, and let u : Ω → ℝ be a continuous function. We say that u is sectionally foliated Schwarz symmetric if there exists a vector p󸀠 = (p1 , . . . , pN−1 , 0) ∈ ℝN , |p󸀠 | = 1, such 󸀠 that u(x) = u(x 󸀠 , xN ) depends only on xN , r = |x 󸀠 | and (if x󸀠 ≠ 0) on ϑ = arccos( |xx󸀠 | ⋅ p󸀠 ), and u is nonincreasing in ϑ. Referring to Definition 1.15, we have that the sectional foliated Schwarz symmetry of a function u just means that the h-sections of u, i. e., the functions x 󸀠 󳨃→ u(x 󸀠 , h) are either – radial for any h ∈ (0, b) or – nonradial but foliated Schwarz symmetric for any h ∈ (0, b) in the corresponding domain Ωh = Ω ∩ {xN = h}, with the same axis of symmetry. Let us consider the potentials Vu (x) = f 󸀠 (|x|, xN , u(x)) ∈ C 0 (Ω),

󵄨 󵄨 Wu (x󸀠 ) = g 󸀠 (󵄨󵄨󵄨x 󸀠 󵄨󵄨󵄨, u(x)) ∈ C 0 (Γ)

(6.88)

(where, as before, f 󸀠 (|x|, xN , s) is the derivative of the nonlinearity f = f (|x󸀠 |, xN , s) with respect to the third variable, and analogously g 󸀠 (|x 󸀠 |, s), is the derivative of the function g(|x󸀠 |, s) appearing in the nonlinear boundary conditions, with respect to the second variable). Then we consider the corresponding quadratic form Qu (v; Ω) = ∫(|∇v|2 − Vu |v|2 ) dx − ∫ Wu |v|2 dx󸀠 , Γ

Ω

v ∈ H01 (Ω ∪ Γ)

(6.89)

Let us denote by λk , Φk the eigenvalues and eigenfunctions of the problem −ΔΦ − Vu Φ = λΦ { { { { Φ=0 { { { 𝜕Φ { − Wu Φ = λΦ { 𝜕𝜐

in Ω on Γ0

(6.90)

on Γ

If e ∈ SN−2 is a direction orthogonal to eN , we consider also the corresponding quadratic form Qu (v; Ω(e)) = ∫ (|∇v|2 − Vu |v|2 ) dx − ∫ Wu |v|2 dx 󸀠 , Ω(e)

Γ

v ∈ H01 (Ω(e) ∪ Γ(e))

(6.91)

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208 | 6 Morse index and symmetry for semilinear elliptic equations in bounded domains while we denote by λke = λk (e; Vu ; Wu ) and φek = φk (e; Vu ; Wu ) the eigenvalues and eigenfunctions of the problem −Δφ − Vu φ = λφ { { { { φ=0 { { { { 𝜕φ − W φ = λφ u { 𝜕𝜐

in Ω(e) on (Γ0 (e))

(6.92)

on Γ(e)

where Γ0 (e) and Γ(e) are defined in (6.87). We also consider, for any direction e ∈ SN−2 , the potentials 1

Ve (x) = ∫ f 󸀠 (|x󸀠 |, xN , tu(x) + (1 − t)uσ(e) (x)) dt 0 󸀠

1

σ(e)

|,xN ,u { f (|x |,xN ,u(x))−f (|x u(x)−uσ(e) (x) ={ 󸀠 󸀠 {f (|x |, xN , u(x)) 󸀠

(x))

if u(x) ≠ uσ(e) (x)

if u(x) = uσ(e) (x)

,

x ∈ Ω(e)

(6.93)

,

x 󸀠 ∈ Γ(e)

(6.94)

We (x) = ∫ g 󸀠 (|x󸀠 |, tu(x) + (1 − t)uσ(e) (x)) dt 0 σ(e)

))−g(|x |,u { g(|x |,u(x u(x)−uσ(e) (x󸀠 ) ={ g 󸀠 (|x|, u(x)) { 󸀠

󸀠

󸀠

(x󸀠 ))

if u(x󸀠 ) ≠ uσ(e) (x 󸀠 ) σ(e)

if u(x ) = u 󸀠

󸀠

(x )

Note that the difference we (x) = u(x) − uσ(e) (x) satisfies in Ω(e) the problem −Δw − Ve w = 0 in Ω(e) { { { { w =0 on Γ0 (e) { { { 𝜕w { − We w = 0 on Γ(e) { 𝜕𝜐

(6.95)

−Δuϑ − Vu uϑ = 0 in Ω { { { { uϑ = 0 on 𝜕Γ0 { { { 𝜕u { ϑ − Wu uϑ = 0 on Γ { 𝜕𝜐

(6.96)

−Δuϑ − Vu uϑ = 0 in Ω(e) { { { { uϑ = 0 on Γ0 (e) { { { 𝜕uϑ { − Wu uϑ = 0 on Γ(e) { 𝜕𝜐

(6.97)

If e, η are orthogonal directions and we define the angular derivative uϑ = uϑ(e,η) = with respect to cylindrical coordinates in the plane (e, η), as in the previous sections, it is easy to see that uϑ weakly satisfies the mixed boundary value problem 𝜕u 𝜕ϑ

Moreover, if e ∈ SN−2 is a direction of symmetry for the solution, i. e., u ≡ ue in Ω, then uϑ weakly satisfies the analogous problem in the half-domain Ω(e), namely

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6.4 Symmetry of solutions of mixed boundary value problems | 209

Proceeding exactly as in the previous sections, we get the following. Proposition 6.29. Let Ω be a cylindrically symmetric domain and u ∈ H01 (Ω ∪ Γ) ∩ C 0 (Ω) a weak solution of (6.85). Assume that for every unit vector e ∈ SN−2 we have either u(x) ≥ u(σe (x)) ∀x ∈ B(e) or

u(x) ≤ u(σe (x)) ∀x ∈ B(e)

(6.98)

Then u is sectionally foliated Schwarz symmetric. Proof. Let p󸀠 ∈ SN−2 be such that for some r, t > 0, Srt = {x = (x 󸀠 , xN ) ∈ ℝN : |x 󸀠 | = r, xN = t} ⊂ Ω and u(rp󸀠 , t) = maxSr u. We define Tp+󸀠 = {e ∈ SN−2 : e ⋅ p󸀠 > 0} and Tp− = {e ∈ SN−1 : e ⋅ p󸀠 < 0}. As in the case of rotationally symmetric domains to prove that u is foliated Schwarz symmetric with respect to p󸀠 , it is enough to show that u(x) ≥ u(σe (x)) for all x ∈ Ω(e)

(6.99)

whenever e ∈ Tp+ , and this is proved exactly as in Proposition 6.7. Arguing as in the previous sections, we get the following. Theorem 6.30 (Sufficient conditions for sectional FSS). Let Ω be a bounded rotationally symmetric domain and let u ∈ C 2 (Ω) be a solution of (6.85), where f and g are C 1 functions. Then u is sectionally foliated Schwarz symmetric provided one of the following conditions holds: (i) there exists a direction e ∈ SN−2 such that λ1 (e; Vu ; Wu ) ≥ 0 and u ≡ uσ(e) in Ω (ii) there exists a direction e ∈ SN−2 such that u < uσ(e) or u > uσ(e) in Ω(e). Using the previous theorem, we can extend the results of Section 6.3 to the case of elliptic problems in cylindrically symmetric domains. We limit ourselves to the case of convex nonlinearities, i. e., we prove the following result, analogous to Theorem 6.20. Theorem 6.31. Suppose that Ω is a cylindrically symmetric domain in ℝN , N ≥ 3 and f (|x|, xN , s), g(x󸀠 , s) are C 1 functions which are convex in the last variable. Then any solution u ∈ C 2 (Ω) of (6.85) with Morse index m(u) ≤ N − 1 is sectionally foliated Schwarz symmetric. As in the case of a rotationally symmetric domain, we first prove the following preliminary result. Lemma 6.32. Let Ω be a cylindrically symmetric domain in ℝN , N ≥ 3, and let u be a solution of problem (6.85) with Morse index m(u) ≤ N − 1. Then there exists a direction e ∈ SN−2 such that the first eigenvalue λ1e ≥ 0. Proof. The proof is immediate if m(u) ≤ 1 since in this case, for any direction e ∈ SN−2 at least one among λ1e and λ1−e must be nonnegative, otherwise taking the corresponding

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210 | 6 Morse index and symmetry for semilinear elliptic equations in bounded domains first eigenfunctions we would obtain a 2-dimensional subspace of H01 (Ω) where the quadratic form Qeu is negative definite. So let us assume that 2 ≤ j = m(u) ≤ N − 1. Recall that we denote by Φk the eigenfunctions of the problem −Δφ − f 󸀠 (|x󸀠 |, xN , u)φ = λφ { { { { φ=0 { { { { 𝜕φ − g 󸀠 (|x󸀠 |, u)φ = λφ { 𝜕𝜐

in Ω on Γ0

(6.100)

on Γ

and in particular we consider the first positive eigenfunction Φ1 . Let us denote, as in Chapter 1, V = L2 (Ω) × L2 (Γ) and for any direction e ∈ SN−2 let us consider the function (φ−e ,Φ )

1

e {( 1e ,Φ11)VV ) 2 φ1 (x) 1 ψ (x) = { (φ(φ e 1 ,Φ ) −( 1 ,Φ1 )V ) 2 φ−e 1 (x) { (φ−e 1 V 1 e

if x ∈ Ω(e) if x ∈ Ω(−e)

(in the scalar product in the space V we consider the trivial extension to Ω of the eigenfunctions φe := φe1 ). The mapping e 󳨃→ ψe is a continuous odd function from SN−2 to H01 (Ω ∪ Γ) and, by construction, (ψe , Φ1 )V = 0 The function h : SN−2 → ℝj−1 defined by h(e) = ((ψe , Φ2 )V , . . . , (ψe , Φj )V) is an odd continuous mapping, and since j − 1 < N − 1, by the Borsuk–Ulam theorem it must have a zero. This means that there exists a direction e ∈ SN−2 such that ψe is orthogonal to all the eigenfunctions Φ1 , . . . , Φj . Since m(u) = j this implies that Qu (ψe ; Ω) = Bu (ψe , ψe ) ≥ 0, which in turn implies that either Qu (φe ; Ω(e)) ≥ 0 or Qu (φ−e ; Ω(−e)) ≥ 0, i. e., either λ1e or λ1−e is nonnegative, so the assertion is proved. Proof of Theorem 6.31. Once, by Lemma 6.32, we have a direction e ∈ SN−2 such that λ1e ≥ 0 the proof is similar to the proofs of Theorem 6.14 and Theorem 6.20. Indeed let e ∈ SN−2 be such that λ1e ≥ 0. If v(x) = uσ(e) (x), by the convexity of the maps u 󳨃→ f (|x󸀠 |, xN , u), u 󳨃→ g(|x󸀠 |, u) we have that f (|x|, u(x)) − f (|x|, v(x)) ≤ f 󸀠 (|x|, u(x))(u − v) and g(|x 󸀠 |, u(x󸀠 )) − g(|x󸀠 |, v(x󸀠 )) ≤ g 󸀠 (|x|, u(x󸀠 ))(u − v) for any x such that u(x) ≥ v(x), so that 2

2

Ve ((u − v)+ ) ≤ Vu ((u − v)+ ) ,

2

We ((u − v)+ ) ≤ Wu ((u − v)+ )

2

(6.101)

(with strict inequality if f or g is strictly convex in the last variable and u(x) ≠ v(x)). + Testing (6.95) with the test function w+ = (we ) = (u − uσ(e) )+ we obtain that 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 0 = ∫ (󵄨󵄨󵄨∇w+ 󵄨󵄨󵄨 − Ve 󵄨󵄨󵄨w+ 󵄨󵄨󵄨 ) − ∫ We 󵄨󵄨󵄨w+ 󵄨󵄨󵄨 ≥ ∫ (󵄨󵄨󵄨∇w+ 󵄨󵄨󵄨 − Vu 󵄨󵄨󵄨w+ 󵄨󵄨󵄨 ) − ∫ Wu 󵄨󵄨󵄨w+ 󵄨󵄨󵄨 (6.102) Ω(e)

Γ

Ω(e)

Γ

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6.4 Symmetry of solutions of mixed boundary value problems | 211

(with strict inequality if f or g is strictly convex in the last variable and w+ ≢ 0). Since λ1e ≥ 0, we deduce that either w+ ≡ 0 or w is the first positive eigenfunction with zero eigenvalue of the associated mixed boundary value problem (6.92) (and this cannot happen if f is strictly convex in the last variable). This implies in turn that either we > 0 in Ω(e) or we ≤ 0 in Ω(e), so that, by the strong maximum principle, either we > 0 in Ω(e) or we ≡ 0 in Ω(e), or we < 0 in Ω(e). By the sufficient condition given by Theorem 6.30, u is sectionally foliated Schwarz symmetric. The same proof works if Γ = 0, and gives an analogous result for the Dirichlet problem in cylindrically symmetric domains, i. e., the following result holds. Theorem 6.33. Let Ω be a cylindrically symmetric bounded domain in ℝN , N ≥ 3 and let u ∈ H01 (Ω) ∩ C 1 (Ω) be a weak solution of the problem −Δu = f (|x|󸀠 , xN , u) { u=0

in Ω on 𝜕Ω

(6.103)

where f ∈ C 1 (Ω × ℝ). Assume that f is convex in the third variable and that u has Morse index m(u) ≤ N − 1. Then u is sectionally foliated Schwarz symmetric.

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7 Morse index and symmetry for elliptic systems in bounded domains In this chapter, we show symmetry results for solutions of elliptic systems obtained by using Morse index bounds. As for the case of scalar equations, described in Chapter 6, we get the foliated Schwarz symmetry of solutions in bounded rotationally symmetric domains, by exploiting convexity properties of the nonlinear terms. The results we present are mostly taken from [77] and [72]. However, we simplify some proofs of both papers. Let us remark that a crucial hypothesis for the symmetry results presented in this chapter is the cooperativity of the system (see Section 7.1). When this hypothesis fails, e. g., for competing systems, the same symmetry results are not expected. We refer to [206] and [215] for different symmetry properties holding in the noncooperative case. The domains considered in the next sections are either balls or annuli. Symmetry results for more general symmetric domains have been obtained in [79] assuming different bounds on the Morse index.

7.1 Morse index of solutions of elliptic systems Let us consider semilinear elliptic systems of the type −ΔU = F(x, U) in Ω

{

U=0

on 𝜕Ω

(7.1)

i. e., −Δu1 = f1 (x, u1 , . . . , um ) in Ω { { { { { {. . . ... { { −Δum = fm (x, u1 , . . . , um ) in Ω { { { { on 𝜕Ω {u1 = ⋅ ⋅ ⋅ = um = 0 where U = (u1 , . . . , um ) : Ω → ℝm , Ω is a bounded domain in ℝN and F(x, s1 , . . . , sm ) = (f1 (x, S), . . . , fm (x, S)) ∈ C 1 (Ω × ℝm ; ℝm ), S = (s1 , . . . , sm ). By solution of (7.1), we mean a weak solution as in (1.111), and inequalities will be understood in a weak sense, as in (1.112), (1.113). We refer to Section 1.5 in Chapter 1 for all the notation about systems. Let us recall the definitions of cooperativeness and full coupling. The system (7.1) is said to be 1. cooperative or weakly coupled in an open set Ω󸀠 ⊆ Ω if 𝜕fi (x, s1 , . . . , sm ) ≥ 0 𝜕sj

for every (x, s1 , . . . , sm ) ∈ Ω󸀠 × ℝm

(7.2)

and every i, j = 1, . . . , m with i ≠ j. https://doi.org/10.1515/9783110538243-007

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214 | 7 Morse index and symmetry for elliptic systems in bounded domains 2.

fully coupled in an open set Ω󸀠 ⊆ Ω along U ∈ H10 (Ω)∩C 0 (Ω; ℝm ) if it is cooperative in Ω󸀠 and in addition ∀I, J ⊂ {1, . . . , m} such that I ≠ 0, J ≠ 0, I ∩ J = 0, I ∪ J = {1, . . . , m} there exist i0 ∈ I, j0 ∈ J such that meas ({x ∈ Ω󸀠 :

𝜕fi0

𝜕sj0

(x, U(x)) > 0}) > 0

(7.3)

For what concerns semilinear systems of elliptic equations, the definition of the Morse index of a solution is analogous to the one given for equations, using the corresponding quadratic form. Definition 7.1. (i) Let U ∈ H10 (Ω)∩L∞ (Ω) be a weak solution of (7.1). We say that U is linearized stable (or has zero Morse index) if the quadratic form QU (Ψ; Ω) = ∫[|∇Ψ|2 − JF (x, U(x))(Ψ, Ψ)] dx Ω m

m

𝜕fi (x, U(x))ψi ψj ] dx ≥ 0 𝜕s j i,j=1

= ∫[∑ |∇ψi |2 − ∑ Ω

i=1

(7.4)

for any Ψ = (ψ1 , . . . , ψm ) ∈ Cc1 (Ω; ℝm ) where JF (x, U(x)) is the Jacobian matrix of F(x, S) with respect to the variables S = (s1 , . . . , sm ) computed at S = U(x). (ii) U has (linearized) Morse index equal to the integer m = m(U) ≥ 1 if m is the maximal dimension of a subspace of Cc1 (Ω; ℝm ) where the quadratic form is negative definite. (iii) U has infinite (linearized) Morse index if for any integer k there exists a k-dimensional subspace of Cc1 (Ω; ℝm ) where the quadratic form is negative definite. The crucial, simple remark that will allow us to extend some of the symmetry results known for equations to the case of systems, is that, as observed in Section 1.5, the quadratic form associated to the linearized operator at a solution U, i. e. to the linear operator LU (V) = −ΔV − JF (x, U)V

(7.5)

which in general is not selfadjoint, coincides with the quadratic form corresponding to the selfadjoint operator 1 LsU (V) = −ΔV − (JF (x, U) + JFt (x, U))V 2

(7.6)

where JFt is the transpose of the matrix JF . Therefore the symmetric eigenvalues of L as defined in Chapter 1, i. e. the eigenvalues of LsU , can be exploited to study the symmetry of the solution U, using the information on its Morse index.

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7.2 Symmetry results |

215

However, though essential in our proofs, the only consideration of these eigenvalues is not sufficient to reach the assertions of our theorems, but we also need to use the principal eigenvalue of the linearized operator (see Definition 1.63) as stressed in Remark 7.9. This can be understood by the fact that the positivity of this eigenvalue is a necessary and sufficient condition for the (weak) maximum principle to hold (see Theorem 1.64) and the maximum principle is another key-ingredient in our proofs. By the previous remark if λks = λk (−Δ + C; Ω) and W k , k ∈ ℕ+ , denote the symmetric eigenvalues and eigenfunctions of LU , i. e., W k satisfy −ΔW k + CW k = λks W k

{

k

W =0

in Ω on 𝜕Ω,

(7.7)

where C = cij (x),

𝜕fj 1 𝜕f (x, U(x))] cij (x) = − [ i (x, U(x)) + 2 𝜕sj 𝜕si

(7.8)

as in the scalar case we can prove the following. Theorem 7.2. Let Ω be a bounded domain in ℝN . Then the Morse index of a solution U to (7.1) equals the number of negative symmetric eigenvalues of the linearized operator LU . Proof. It is the same as the proof of Theorem 3.2, using Theorem 1.56(iii) and Theorem 1.57(i).

7.2 Symmetry results 7.2.1 Moving and rotating planes From now on, we will consider the case of system (7.1) in bounded rotationally symmetric domains, with radial dependence on x: −ΔU = F(|x|, U)

in Ω

U=0

on 𝜕Ω

{

(7.9)

where F(r, S) = (f1 (r, S), . . . , fm (r, S)) is a function belonging to C 1 ([0, +∞) × ℝm ; ℝm ) and Ω is a bounded rotationally symmetric domain in ℝN , m, N ≥ 2. If Ω is a ball and U is a positive solution of (7.9), the radial symmetry of U holds under some mild regularity and coupling hypotheses. Theorem 7.3. Let Ω = BR be a ball centered at the origin and suppose that U ∈ H10 (Ω) ∩ C 0 (Ω; ℝm ) is a positive weak solution of (7.9), where F = F(r, S) = (f1 (r, S), . . . , fm (r, S)) ∈

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216 | 7 Morse index and symmetry for elliptic systems in bounded domains C 1 ([0, +∞) × ℝm ; ℝm ) is nonincreasing in r ∈ [0, +∞) for any S ∈ ℝm . Assume further that the system (7.9) is cooperative. Then U is radial, i. e., U(x) = V(|x|), where V = V(r) : [0, R] → ℝm , and Vr (r) < 0. Proof. The proof is exactly the same as that of Theorem 6.1, using the corresponding comparison principles for systems, namely Theorems 1.66 and 1.67. Let us only remark that the strong comparison principle holds for fully coupled systems, while we assume here only the cooperativeness of the system. The reason is that we actually use the statement 1. of Theorem 1.67 (which only requires the weak coupling of the system) which asserts that for every component ui of the vector solution U either (ui ) < (ui )λ in Ωλ or (ui ) ≡ (ui )λ in Ωλ . Then the first alternative necessarily holds for every component because if λ < 0 then, for every component ui , there are points on the boundary of the ball where the strict inequality ui < (ui )λ holds, because they are reflected inside the ball, where all components ui are positive by hypothesis.1 Let us remark that the previous extension to systems of the analogous Gidas–Ni– Nirenberg’s theorem 6.1 has been proved by many authors (see [53, 94, 95, 153, 219] and the references therein) with different methods. As observed in Section 6.1, the proof used in Theorem 6.1 is based on the comparison principle in small domains. This makes the proof flexible and easily adaptable to different problems, as in Theorem 7.3. Moreover, as in the case of equations, this technique can be used in the analogous theorem that uses the rotating plane method (see Theorem 7.4 below). Let us recall some notation from Chapter 6. For a unit vector e ∈ SN−1 , we consider the hyperplane H(e) = {x ∈ ℝN : x ⋅ e = 0} orthogonal to the direction e and the open half-domain Ω(e) = {x ∈ Ω : x ⋅ e > 0}. We then set σe (x) = x − 2(x ⋅ e)e, x ∈ Ω, i. e., σe : Ω → Ω is the reflection with respect to the hyperplane H(e). Finally, if U : Ω → ℝm is a continuous function we define the reflected function U σ(e) : Ω → ℝm defined by U σ(e) (x) = U(σe (x)). The following result can be proved exactly as Theorem 6.3 (using the comparison principles for systems given by Theorem 1.66 and Theorem 1.67 instead of the corresponding comparison principles for scalar equations). Theorem 7.4 (Rotating planes method for systems). Let Ω be a bounded rotationally symmetric domain, F ∈ C 1 ([0, ∞) × ℝm ; ℝm ) and U ∈ H10 (Ω) ∩ C 0 (Ω; ℝm ) a weak solution of (7.9). Assume that the system (7.9) is fully coupled along U in Ω and there exists a direction eϑ0 = (cos(ϑ0 ), sin(ϑ0 ), 0, . . . , 0) such that U < U σ(eϑ0 )

in Ω(eϑ0 )

1 Note, however, that if the solution is only assumed to be nonnegative then we can deduce that it is actually positive only for fully coupled systems.

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217

7.2 Symmetry results |

Then there exists a direction eϑ1 = (cos(ϑ1 ), sin(ϑ1 ), 0, . . . , 0), with ϑ1 > ϑ0 , such that U ≡ U σ(eϑ1 )

in Ω(eϑ1 )

and U < U σ(eϑ )

in Ω(eϑ ) ∀ϑ ∈ (ϑ0 , ϑ1 )

7.2.2 Foliated Schwarz symmetry Let us now give the definition of foliated Schwarz symmetry for vector valued functions. Definition 7.5. Let Ω be a rotationally symmetric domain in ℝN , N ≥ 2. We say that a continuous vector valued function U = (u1 , . . . , um ) : Ω → ℝm is foliated Schwarz symmetric if each component ui is foliated Schwarz symmetric with respect to the same vector p ∈ ℝN . In other words, there exists a vector p ∈ ℝN , |p| = 1, such that U(x) x ⋅ p) and U is (componentwise) depends only on r = |x| and (if x ≠ 0) on θ = arccos( |x| nonincreasing in θ. Remark 7.6. Let us observe that if U is a solution of (7.9) and the system satisfies some coupling conditions, as required in Theorem 7.12, then the foliated Schwarz symmetry of U implies that either U is radial or it is strictly decreasing in the angular variable θ as in the scalar case (see Remark 6.13). The sufficient condition for the foliated Schwarz symmetry given by Proposition 6.7 can be readily extended. Proposition 7.7. Let Ω be a rotationally symmetric domain and U ∈ H10 (Ω) ∩ C 0 (Ω) a weak solution of (7.9) where F = F(r, S) ∈ C 1 ([0, ∞) × ℝm ; ℝm ). Assume that the system is fully coupled along U in Ω and that for every unit vector e ∈ SN−1 : either

U(x) ≥ U(σe (x)) ∀x ∈ Ω(e)

or

U(x) ≤ U(σe (x)) ∀x ∈ Ω(e)

(7.10)

Then U is foliated Schwarz symmetric. Proof. By Proposition 6.7, each component ui of the solution U is foliated Schwarz symmetric with respect to a vector pi ∈ ℝN , i = 1, . . . , m, so to prove that the solution U is foliated Schwarz symmetric we only need to prove that the vectors pi are all the same, for i = 1, . . . , m. To this aim, consider the vector p1 and take any hyperplane H(e) = {x ∈ ℝN : x ⋅ e = 0} passing through the axis with direction p1 . Note that also the function U σ(e) satisfies the system (7.9), hence, since u1 ≡ uσ(e) in Ω(e), by the strong comparison 1 principle (Theorem 1.67) we have that uj ≡ uσ(e) in Ω(e) for all j = 1, . . . , m, i. e., all j vectors pj coincide with p1 .

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218 | 7 Morse index and symmetry for elliptic systems in bounded domains As in the scalar case, we consider a pair of orthogonal directions η1 , η2 and the polar coordinates (ρ, ϑ) in the plane spanned by them. Then we define for U ∈ C 2 (Ω; ℝm ) the angular derivative Uϑ = Uϑ(η1 ,η2 )

(7.11)

which solves the linearized system −ΔUϑ − JF (|x|, U)Uϑ = 0 { Uϑ = 0

in Ω on 𝜕Ω

(7.12)

and, if e ∈ span (η1 , η2 ) and U ≡ U σ(e) in Ω(e), also the system −ΔUϑ − JF (|x|, U)Uϑ = 0

{

Uϑ = 0

in Ω(e) on 𝜕Ω(e)

(7.13)

Then Proposition 6.8 and Proposition 6.10 extend immediately to the vectorial case and we will not repeat them. Using the properties of the principal eigenvalue and of the corresponding eigenfunction, we deduce, as in the scalar case, the following sufficient conditions for the foliated Schwarz symmetry. Theorem 7.8 (Sufficient conditions for FSS-Systems). Let Ω be a bounded rotationally symmetric domain and U ∈ C 2 (Ω; ℝm ) a solution of (7.9), where F ∈ C 1 ([0, R] × ℝm ; ℝm ). Then U is foliated Schwarz symmetric provided one of the following conditions holds: (i) there exists a direction e ∈ SN−1 such that U ≡ U σ(e) in Ω(e) and the principal eigenvalue λ̃1 (Ω(e)) of the linearized operator LU = −Δ − JF (x, U) in Ω(e) is nonnegative; (ii) there exists a direction e ∈ SN−1 such that either U < U σ(e) or U > U σ(e) in Ω(e). Proof. If η is a direction orthogonal to e, Uϑ = Uϑ(η1 ,η2 ) satisfies (7.13) as remarked. Thus if λ̃1 (Ω(e)) > 0, since the maximum principle holds (see Theorem 1.64), we get that Uϑ ≡ 0 in Ω(e). If instead λ̃1 (Ω(e)) = 0 and Uϑ ≢ 0, by the simplicity of the principal eigenvalue we get that Uϑ is a principal eigenfunction and hence it is positive (or negative) in Ω(e). In any case, Uϑ does not change sign in Ω(e), and, arguing exactly 󸀠 󸀠 as in the scalar case, we get that for any direction e󸀠 either U ≥ U σ(e ) or U ≤ U σ(e ) . Thus, by Proposition 7.7, U is foliated Schwarz symmetric and (i) is proved. To prove (ii), we proceed as in the scalar case (see Theorem 6.12 and Remark 6.13) using Theorem 7.4. Remark 7.9. Let us observe that in Theorem 7.8 it is the nonnegativity of the principal eigenvalue the crucial hypothesis, while the information we get in the next section using the techniques introduced in Chapter 6 will concern the symmetric eigenvalues of the linearized system. Therefore, in the proofs that follow there will be an interplay and a comparison between the principal eigenvalue and the first symmetric eigenvalue in the cap Ω(e).

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7.2 Symmetry results |

219

7.2.3 Nonlinearities having convex components If U is a solution of (7.9) and the system is fully coupled along U in Ω, then the difference W = W e = U − U σ(e) = (w1 , . . . , wm ), as in the scalar case, satisfies a linear system in Ω, which is fully coupled in Ω and Ω(e). Lemma 7.10. (i) Assume that U ∈ C 1 (Ω; ℝm ) is a solution of (7.9) and that the system is fully coupled along U in Ω. Let us define for any direction e ∈ SN−1 the matrix Be (x) = (beij (x))m i,j=1 , where 1

beij (x) = − ∫ 0

𝜕fi (|x|, tU(x) + (1 − t)U σ(e) (x)) dt 𝜕sj

(7.14)

Then for any e ∈ SN−1 the function W e = U − U σ(e) satisfies in Ω(e) the linear system −ΔW e + Be (x)W e = 0 { e W =0

in Ω(e) on 𝜕Ω(e)

(7.15)

which is fully coupled in Ω(e). (ii) If Ψ = (ψ1 , . . . , ψm ) ∈ H10 (Ω(e)), let Qe (Ψ; Ω(e)) denote the quadratic form associated to the system (7.15) in Ω(e), i. e., Qe (Ψ; Ω(e)) = ∫ (|∇Ψ|2 + Be (Ψ, Ψ)) dx Ω(e) m

m

i=1

i,j=1

= ∫ (∑ |∇ψi |2 + ∑ beij ψi ψj ) dx Ω(e)

(7.16)

Then 󵄨 󵄨2 Qe (W e ; Ω(e)) = ∫ [󵄨󵄨󵄨∇(W e )󵄨󵄨󵄨 + Be (W e , W e )] dx = 0

(7.17)

Ω(e)

while for the positive and negative parts of W e the following holds: ± ± 󵄨2 ± ± 󵄨 Qe ((W e ) ; Ω(e)) = ∫ [󵄨󵄨󵄨∇(W e ) 󵄨󵄨󵄨 + Be ((W e ) , (W e ) )] dx ≤ 0

(7.18)

Ω(e)

Proof. From the equation −ΔU = F(|x|, U(x)) we deduce that the reflected function U σ(e) satisfies the equation −ΔU σ(e) = F(|x|, U σ(e) (x)), and hence the difference W e = U − U σ(e) = (w1 , . . . , wm ) satisfies the equation −ΔW e = F(|x|, U) − F(|x|, U σ(e) )

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220 | 7 Morse index and symmetry for elliptic systems in bounded domains Let us set V = U σ(e) . For any i = 1, . . . , m, we have that m 1

fi (|x|, U(x)) − fi (|x|, V(x)) = ∑ ∫ j=1 0

𝜕fi (|x|, tU(x) + (1 − t)V(x))(uj (x) − vj (x)) dt 𝜕sj

As a consequence, W e satisfies (7.15). Moreover, if i ≠ j then beij (x) ≤ 0 by (7.2), so that the linear system (7.15) is weakly coupled. If U ∈ C 1 (Ω; ℝm ) is a solution of (7.9) and the system is fully coupled along U, then the linear system associated to the matrix Be is fully coupled in Ω. Indeed, if i0 ≠ j0 and

𝜕fi0 𝜕sj0

(x, U(x)) > 0 then, since

1 𝜕f − ∫0 𝜕si [|x|, tU(x) j e

𝜕fi (y) 𝜕sj

≥ 0 for every y ∈ Ω, we get that bij (x) =

+ (1 − t)V(x)] dt < 0.

Since B is symmetric with respect to the reflection σe , (7.15) is fully coupled in Ω(e) as well and (i) is proved. To get (7.17), it is enough to multiply the ith equation of the system for wi and integrate. Instead, multiplying the ith equation of (7.15) for wi+ , we get m

m

󵄨 󵄨2 󵄨 󵄨2 0 = ∫ (󵄨󵄨󵄨∇wi+ 󵄨󵄨󵄨 + ∑ beij wj wi+ ) dx ≥ ∫ (󵄨󵄨󵄨∇wi+ 󵄨󵄨󵄨 + ∑ beij wj+ wi+ ) dx Ω(e)

j=1

j=1

Ω(e)

since wi wi+ = |wi+ |2 , while wj wi+ ≤ wj+ wi+ and bij ≤ 0 if i ≠ j. Summing on i, we get m

m

󵄨 󵄨2 0 ≥ ∫ ∑󵄨󵄨󵄨∇wi+ 󵄨󵄨󵄨 + ∑ beij wj+ wi+ dx i,j=1

Ω(e) i=1

i. e., (7.18) in the case of the positive part. For the negative part, we proceed analogously multiplying the ith equation of (7.15) for wi− and integrating. We get m

m

󵄨 󵄨2 󵄨 󵄨2 0 = − ∫ 󵄨󵄨󵄨∇wi− 󵄨󵄨󵄨 + ∑ beij wj wi− dx ≤ − ∫ 󵄨󵄨󵄨∇wi− 󵄨󵄨󵄨 + ∑ beij (−wj− )wi− dx Ω(e)

j=1

Ω(e)

m

j=1

󵄨 󵄨2 = − ∫ 󵄨󵄨󵄨∇wi− 󵄨󵄨󵄨 − ∑ beij (wj− )wi− dx Ω(e)

j=1

since wi wi− = −|wi− |2 , while wj wi− ≥ −(wj− )wi− and bij ≤ 0 if i ≠ j. Summing on i, we obtain m

m

󵄨 󵄨2 0 ≥ ∫ ∑󵄨󵄨󵄨∇wi− 󵄨󵄨󵄨 + ∑ beij wj− wi− dx Ω(e) i=1

i,j=1

i. e., (7.18) in the case of the negative part.

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7.2 Symmetry results |

221

Remark 7.11. 1. It is standard to consider the linear system (7.15) to deduce comparison principles from maximum principles. Indeed we already considered a similar system in Chapter 1 in the proofs of the comparison principles for systems in Theorem 1.66 and Theorem 1.67, where elliptic inequalities were considered. 2. Note that the inequalities in (7.18) could be strict. Indeed the products wi+ wj− could be not identically zero if i ≠ j and, therefore, Q(W e ) does not coincide in general with Q((W e )+ ) + Q((W e )− ), as it happens in the scalar case. Our first result is the counterpart of Theorem 6.20 for systems. Theorem 7.12. Let Ω be a ball or an annulus in ℝN , N ≥ 2, and let U ∈ C 2 (Ω; ℝm ) be a solution of (7.9) with Morse index m(U) ≤ N. Moreover, assume that: (i) the system is fully coupled along U in Ω (ii) for any i = 1, . . . , m the scalar function fi (|x|, S) is convex in the variable S = (s1 , . . . , sm ) ∈ ℝm . Then U is foliated Schwarz symmetric and if U is not radial then it is strictly decreasing in the angular variable. We observe that, with respect to the corresponding result in [77], in Theorem 7.12 only the convexity of the functions fi , i = 1, . . . , m, is required, as in [79], where more general domains are considered. Remark 7.13. Requiring that the system is fully coupled along a solution U in Ω is an assumption easily satisfied by most of the systems encountered in applications (see Section 7.3). It is essentially needed to ensure the validity of the strong maximum and comparison principles. Obviously, the previous theorem holds in particular for stable solutions. However, in this case, as for scalar equations, it is not difficult to see that the solution is radial, even without the assumption (ii). Therefore, we have the following Remark 7.14. If the system (7.9) is fully coupled along a stable solution U in Ω, then U is radial. We will deduce from the proof of Theorem 7.12 that for nonradial Morse index one solutions the following condition holds. Theorem 7.15. Under the assumptions of Theorem 7.12 if a solution U has Morse index one and is not radial then necessarily m

∑ j=1

m 𝜕f 𝜕uj 𝜕uj 𝜕fi j (r, U(r, ϑ)) (r, ϑ) = ∑ (r, U(r, ϑ)) (r, ϑ) 𝜕sj 𝜕ϑ 𝜕s 𝜕ϑ i j=1

(7.19)

for any i = 1, . . . , m, with (r, ϑ) as in Definition 7.5.

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222 | 7 Morse index and symmetry for elliptic systems in bounded domains In particular, if m = 2 then (7.19) implies that 𝜕f1 𝜕f (|x|, U(x)) = 2 (|x|, U(x)), 𝜕s2 𝜕s1

∀x ∈ Ω

(7.20)

Remark 7.16. The result of Theorem 7.15 is somewhat surprising because it asserts that under the hypotheses of Theorem 7.12, for any nonradial Morse index one solution another coupling condition, namely (7.19) (and in particular (7.20) if m = 2), must hold along the solution. We will prove Theorem 7.12 by several auxiliary results. Lemma 7.17. Assume that U is a solution of (7.9) and that the hypotheses (i)–(ii) of Theorem 7.12 hold. Then for any direction e ∈ SN−1 QU ((W e ) ; Ω(e)) ≤ 0 +

where QU is the quadratic form defined in (7.4) and W e is as in Lemma 7.10. Proof. For any i = 1, . . . , m we have −Δwi = fi (|x|, U) − fi (|x|, U σ(e) )

in Ω(e)

Testing the equation with wi+ , we obtain 󵄨 󵄨2 ∫ 󵄨󵄨󵄨∇(wi )+ 󵄨󵄨󵄨 dx = ∫ (fi (|x|, U) − fi (|x|, U σ(e) ))wi+ dx

Ω(e)

(7.21)

Ω(e)

Observe that fi (|x|, S) is convex in S, so that (fi (|x|, U(x)) − fi (|x|, U σ(e) (x)))wi+ ≤ (∇fi (|x|, U(x)) ⋅ (U(x) − U σ(e) (x)))wi+ m

= (∇fi (|x|, U(x)) ⋅ W e )wi+ = ∑ j=1

𝜕fi (|x|, U(x))wj wi+ 𝜕uj

where ∇ stands for the gradient of fi with respect to the variables S = (s1 , . . . , sm ). Moreover, 𝜕fi 𝜕f 󵄨 󵄨2 w w+ = i 󵄨󵄨󵄨wi+ 󵄨󵄨󵄨 , 𝜕si i i 𝜕si because

𝜕fi 𝜕sj

while

𝜕fi 𝜕f w w+ ≤ i wj+ wi+ 𝜕sj j i 𝜕sj

if i ≠ j

≥ 0 by the weak coupling assumption.

By (7.21), taking into account the previous inequalities, we get m 𝜕f 󵄨 󵄨2 ∫ 󵄨󵄨󵄨∇(wi )+ 󵄨󵄨󵄨 dx ≤ ∫ ∑ i (|x|, U(x))wj+ wi+ dx 𝜕s j j=1

Ω(e)

Ω(e)

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7.2 Symmetry results | 223

Thus, summing on i = 1, . . . , m, we obtain m m 𝜕f 󵄨 󵄨2 ∫ (∑󵄨󵄨󵄨∇(wi )+ 󵄨󵄨󵄨 − ∑ i (|x|, U(x))wi+ wj+ ) dx ≤ 0 𝜕sj i,j=1 i=1

(7.22)

Ω(e) e +

i. e., QU ((W ) ; Ω(e)) ≤ 0. Lemma 7.18. Suppose that U is a solution of (7.9) with Morse index m(U) ≤ N and assume that the hypothesis (i) of Theorem 7.12 holds. Then there exists a direction e ∈ SN−1 such that QU (Ψ; Ω(e)) ≥ 0 for any Ψ ∈ Cc1 (Ω(e); ℝm ). Equivalently, the first symmetric eigenvalue λ1s (LU , Ω(e)) of the linearized operator LU = −Δ − JF (x, U) in Ω(e) is nonnegative (and hence also the principal eigenvalue λ̃1 (LU , Ω(e)) is nonnegative, by Theorem 1.64). Proof. It is analogous to the proof of Proposition 6.21. Proof of Theorem 7.12. By Lemma 7.18, there exists a direction e such that the first symmetric eigenvalue λ1s (LU , Ω(e)) of the linearized operator is nonnegative, so that the principal eigenvalue λ̃1 (Ω(e)) is nonnegative as well. Moreover, by Lemma 7.17 we have that QU ((W e )+ ) ≤ 0, so that either (W e )+ ≡ 0, or λ1s (LU , Ω(e)) = 0 and (W e )+ is the positive first symmetric eigenfunction in Ω(e). In any case, either U ≤ U σ(e) or U ≥ U σ(e) in Ω(e) holds. Thus, by the strong maximum principle, either U ≡ U σ(e) in Ω(e), and the principal eigenvalue λ̃1 (Ω(e)) is nonnegative, or U < U σ(e) in Ω(e) or U > U σ(e) in Ω(e). Hence, by Theorem 7.8, U is foliated Schwarz symmetric. Remark 7.19. 1. In the previous proof when (U − U σ(e) )+ ≡ 0, we also have by construction that λ1s (Lu , Ω(e)) ≥ 0 and, therefore, the principal eigenvalue satisfies λ̃1 (Ω(e)) = λ̃1 (Ω(−e)) ≥ 0. In the case when U < U σ(e) in Ω(e) or U > U σ(e) in Ω(e), by rotating the planes 󸀠 we find a different direction e󸀠 such that U ≡ U σ(e ) in Ω(e󸀠 ) and it could happen that λ1s (Ω(e󸀠 )) < 0. However, let us observe explicitly that the sign of the principal eigenvalue is preserved in the rotation, i. e., λ̃1 (Ω(e󸀠 )) = λ̃1 (Ω(−e󸀠 )) ≥ 0, and actually λ̃1 (Ω(e󸀠 )) = λ̃1 (Ω(−e󸀠 )) = 0. Indeed since U < U σ(g) for any direction g between e and e󸀠 , we have that 0 is the principal eigenvalue of the system satisfied by U − U σ(g) , namely (7.15), with coefficients bgij (x)

1

= −∫ 0

𝜕fi [|x|, tU(x) + (1 − t)U σ(g) (x)] dt 𝜕sj

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224 | 7 Morse index and symmetry for elliptic systems in bounded domains As g → e󸀠 , where e󸀠 is the symmetry position, the coefficients bij approach the 𝜕f coefficients of the linearized system, namely c = − i , so by continuity λ̃ (Ω(e󸀠 )) = ij

2.

𝜕sj

1

λ̃1 (Ω(−e󸀠 )) = 0. We can deduce, exactly as in Remark 6.13, point 2, that the solution U is either x radial or strictly decreasing with respect to θ, where θ = arccos( |x| ⋅p) is the angular variable that appears in Definition 7.5.

Proof of Theorem 7.15. By the proof of Theorem 7.12 and Remark 7.19, we can find a direction e such that U is symmetric with respect to the hyperplane H(e) and the principal eigenvalue λ̃1 (Ω(e)) = λ̃1 (Ω(−e)) ≥ 0. As in the proof of Proposition 6.21, it is easy to see that if U is a Morse index one solution then for any direction e either λ1s (LU , Ω(e)) or λ1s (LU , Ω(−e)) must be nonnegative. On the other hand, by symmetry, λ1s (Ω(e)) = λ1s (Ω(−e)), so that λ̃1 (Ω(e)) = λ̃1 (Ω(−e)) ≥ λ1s (Ω(e)) = λ1s (Ω(−e)) ≥ 0. As observed in the previous remark, for any couple of directions η1 , η2 , where η2 is the direction of the axis of symmetry and η1 is any direction orthogonal to it (η1 could be the direction e), the derivative with respect to the angular variable in the plane (η1 , η2 ) coincides in Ω(η1 ), with the opposite of the angular derivative Uθ , where θ is as in Definition 7.5. Then, if λ̃1 (Ω(e)) > 0, the angular derivative Uθ must vanish (since it satisfies (7.13) and the maximum principle holds in Ω(e)). Hence U is radial. So if Uθ ≢ 0 necessarily λ̃1 (Ω(e)) = λ1s (Ω(e)) = 0 and by (iv) of Proposition 1.64 the derivative Uθ is a negative first eigenfunction of the symmetrized system in Ω(e), as well as a solution of (7.13). Thus we get that 1 JF (|x|, U)Uθ = (JF (|x|, U) + JFt (|x|, U))Uθ 2 i. e., (7.19) and if m = 2 we get (7.20), since Uθ is strictly negative. 7.2.4 Nonlinearities with convex derivatives The second symmetry result we get is the counterpart of Theorem 6.22 for systems. Theorem 7.20. Let Ω be a ball or an annulus in ℝN , N ≥ 2, and let U ∈ C 2 (Ω; ℝm ) be a solution of (7.9) with Morse index m(U) ≤ N − 1. Moreover, assume that: (i) the system is fully coupled along U in Ω 𝜕f (ii) for any i, j = 1, . . . m the function 𝜕si (|x|, S) is convex in S = (s1 , . . . , sm ): j

𝜕fi 𝜕f 𝜕f (|x|, tS󸀠 + (1 − t)S󸀠󸀠 ) ≤ t i (|x|, S󸀠 ) + (1 − t) i (|x|, S󸀠󸀠 ) 𝜕sj 𝜕sj 𝜕sj for any t ∈ [0, 1], S󸀠 , S󸀠󸀠 ∈ ℝm and x ∈ Ω.

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7.2 Symmetry results |

225

Then U is foliated Schwarz symmetric and if U is not radial then it is strictly decreasing in the angular variable. Remark 7.21. As for the scalar case the assumption (ii) of Theorem 7.20 allows to get the symmetry of solutions in cases not covered by Theorem 7.12. However, the assumption on the Morse index is more restrictive since we require m(U) ≤ N − 1, while in the analogous result for the scalar case (Theorem 6.22) the Morse index was assumed to be less than or equal to N. For systems, technical difficulties arise in the proof when m(U) = N. Theorem 7.22. Under the assumptions of Theorem 7.20 assume that U is a nonradial solution of (7.9) and either a) U has Morse index one or 𝜕fi b) there exist i0 , j0 ∈ {1, . . . , m} such that the function 𝜕s 0 (|x|, S) satisfies the following j0

strict convexity assumption: 𝜕fi0

𝜕sj0

(|x|, tS󸀠 + (1 − t)S󸀠󸀠 ) < t

𝜕fi0

𝜕sj0

(|x|, S󸀠 ) + (1 − t)

𝜕fi0

𝜕sj0

(|x|, S󸀠󸀠 )

(7.23)

for any t ∈ (0, 1), whenever x ∈ Ω and S󸀠 , S󸀠󸀠 ∈ ℝm satisfy s󸀠k ≠ s󸀠󸀠 k for any k ∈ {1, . . . , m}. Then m

∑ j=1

m 𝜕f 𝜕uj 𝜕uj 𝜕fi j (r, U(r, θ)) (r, θ) = ∑ (r, U(r, θ)) (r, θ) 𝜕sj 𝜕θ 𝜕si 𝜕θ j=1

(7.24)

for any i = 1, . . . , m, with (r, θ) as in Definition 7.5. In particular, if m = 2 then (7.24) implies that 𝜕f1 𝜕f (|x|, U(x)) = 2 (|x|, U(x)), 𝜕s2 𝜕s1

∀x ∈ Ω.

(7.25)

Remark 7.23. Note that (7.24) and (7.25) were also deduced under the assumptions of Theorem 7.12, but only for Morse index one solutions. Remark 7.24 (Radial symmetry of stable solutions). The symmetry result of Theorem 7.20 holds in particular for stable solutions of (7.9). However, in this case it is easy to get that the solution is radial without any assumption on the nonlinearity (see also Remark 7.14). The proof of Theorem 7.20 follows the scheme of the proof of Theorem 7.12, and it is based upon the following results.

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226 | 7 Morse index and symmetry for elliptic systems in bounded domains Lemma 7.25. Assume that U is a solution of (7.9) and the hypotheses of Theorem 7.20 hold. Let Be (x) = (beij (x))m i,j=1 be the matrix associated to the fully coupled system (7.15) defined by (7.14), i. e., 1

beij (x) = − ∫ 0

𝜕fi [|x|, tU(x) + (1 − t)U σ(e) (x)] dt 𝜕sj

and let us define the matrix Be,s (x) = (be,s (x))m i,j=1 , where ij 𝜕f 1 𝜕fi be,s (|x|, U(x)) + i (|x|, U σ(e) (x))) ij (x) = − ( 2 𝜕sj 𝜕sj

(7.26)

Then the linear system with matrix Be,s is fully coupled in Ω and Ω(e) for any e ∈ SN−1 . Moreover, for any i, j = 1, . . . , m and x ∈ Ω it holds beij (x) ≥ be,s ij (x) with strict inequality for any i0 , j0 such that uσ(e) (x) k

𝜕fi0 𝜕sj0

(7.27)

satisfies the strict convexity assumption

(7.23) if uk (x) ≠ for any k ∈ {1, . . . , m}. Finally, for the quadratic forms Qe and Qe,s associated to the matrices Be and Be,s we have that ± ± 󵄨2 ± ± 󵄨 0 ≥ Qe ((W e ) ; Ω(e)) = ∫ [󵄨󵄨󵄨∇(W e ) 󵄨󵄨󵄨 + Be ((W e ) , (W e ) )] dx Ω(e)

± 󵄨2 ± ± ± 󵄨 ≥ ∫ [󵄨󵄨󵄨∇(W e ) 󵄨󵄨󵄨 + Be,s ((W e ) , (W e ) )] dx = Qe,s ((W e ) ; Ω(e)) Ω(e)

for W e = U − U σ(e) , with the strict inequality Qe,s ((W e ) ; Ω(e)) = Qe,s ((W e ); Ω(e)) < 0 +

if F satisfies the hypothesis b) of Theorem 7.22 and W e > 0 in Ω(e), while Qe,s ((W e ) ; Ω(e)) = Qe,s ((W e ); Ω(e)) < 0 −

if F satisfies the hypothesis b) of Theorem 7.22 and W e < 0 in Ω(e). Proof. By hypothesis (ii) of Theorem 7.20, we get 1

−beij (x) = ∫ 0

𝜕fi [|x|, tU(x) + (1 − t)U σ(e) (x)] dt 𝜕sj

1

≤ ∫(t 0

𝜕fi 𝜕f [|x|, U(x)] + (1 − t) i [|x|, U σ(e) (x)]) dt 𝜕sj 𝜕sj

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(7.28)

7.2 Symmetry results | 227

𝜕f 1 𝜕f = ( i (|x|, U(x)) + i (|x|, U σ(e) (x))) = −be,s ij (x) 2 𝜕sj 𝜕sj This implies (7.27) and the inequality is strict for any i0 , j0 such that

𝜕fi0 𝜕sj0

(7.29) satisfies the

strict convexity assumption (7.23) if uk (x) ≠ uσ(e) (x) for any k ∈ {1, . . . , m}. k This in turn implies the full coupling of the system with matrix Be,s , since by Lemma 7.10 the system with matrix Be is fully coupled. From (7.18) and (7.27), since wk± ≥ 0, we get m

m

i=1

i,j=1

m

m

i=1

i,j=1

󵄨 󵄨2 0 ≥ ∫ (∑󵄨󵄨󵄨∇wi+ 󵄨󵄨󵄨 + ∑ beij wj+ wi+ ) dx Ω(e)

󵄨 󵄨2 + + ≥ ∫ (∑󵄨󵄨󵄨∇wi+ 󵄨󵄨󵄨 + ∑ be,s ij wj wi ) dx Ω(e)

i. e., (7.28) in the case of the positive part of W e , with strict inequality if F satisfies the hypothesis b) of Theorem 7.22 and W e > 0. Analogously, we get the corresponding inequality for the negative part of W e . Lemma 7.26. Suppose that U is a solution of (7.9) with Morse index m(U) ≤ N − 1 and assume that the hypothesis (i) of Theorem 7.20 holds. Let Qe,s be the quadratic form associated to the operator Le,s (V) = −ΔV + Be,s V, Be,s being the matrix defined in (7.26): Qe,s (Ψ; Ω󸀠 ) = ∫ [|∇Ψ|2 + Be,s (Ψ, Ψ)] dx Ω󸀠 m

m

𝜕f 1 𝜕fi ( (|x|, U(x)) + i (|x|, U σ(e) (x)))ψi ψj ] dx (7.30) 2 𝜕sj 𝜕sj i,j=1

= ∫[∑ |∇ψi |2 − ∑ Ω

i=1

Then there exists a direction e ∈ SN−1 such that Qe,s (Ψ; Ω(e)) ≥ 0 ∀Ψ ∈ Cc1 (Ω(e); ℝm ) Equivalently, the first symmetric eigenvalue λ1s (Le,s , Ω(e)) of the operator Le,s (V) = −ΔV + Be,s V in Ω(e) is nonnegative (and hence also the principal eigenvalue λ̃1 (Le,s , Ω(e)) is nonnegative). Proof. Let us assume that 1 ≤ j = m(U) ≤ N − 1 and let Φ1 , . . . , Φj be mutually orthogonal eigenfunctions corresponding to the negative symmetric eigenvalues λ1s (LU , Ω), . . . , λjs (LU , Ω) of the linearized operator LU (V) = −ΔV − JF (x, U)V in Ω. For any e ∈ SN−1 , let ϕe,s be the first positive L2 -normalized eigenfunction of the symmetric system associated to the linear operator Le,s in Ω(e). We observe that ϕe,s is uniquely determined since the corresponding system is fully coupled in Ω(e). Let Φe,s

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228 | 7 Morse index and symmetry for elliptic systems in bounded domains be the odd extension of ϕe,s to Ω, and let us observe that Φ−e,s = −Φe,s , because Be,s is symmetric with respect to the reflection σe . The mapping e 󳨃→ Φe,s is a continuous odd function from SN−1 to H01 (Ω ∪ Γ), therefore, the mapping h : SN−1 → ℝj defined by h(e) = ((Φe,s , Φ1 )L2 (Ω) , . . . , (Φe,s , Φj )L2 (Ω) ) is an odd continuous mapping, and since j ≤ N − 1, by the Borsuk–Ulam theorem it must have a zero. This means that there exists a direction e ∈ SN−1 such that Φe,s is orthogonal to all the eigenfunctions Φ1 , . . . , Φj . This implies that QU (Φe,s ; Ω) ≥ 0, because m(U) = j, and since Φe,s is an odd function, we obtain that 0 ≤ QU (Φe,s ; Ω) = Qe,s (Φe,s , Ω) = 2Qe,s (ϕe,s , Ω(e)) = 2λ1s (Le,s , Ω(e)) Proof of Theorem 7.20. By Lemma 7.26, there exists a direction e such that the first symmetric eigenvalue λ1s (Le,s , Ω(e)) of the operator Le,s (V) = −ΔV + Be,s V in Ω(e) is nonnegative, and hence also the principal eigenvalue λ̃1 (Le,s , Ω(e)) is nonnegative. Since Qe,s ((W e )± ; Ω(e)) ≤ 0 by Lemma 7.25, we have two alternatives. The first one is that (W e )+ and (W e )− both vanish, in which case W e ≡ 0 in Ω(e), and this implies in turn that Le,s = LU . Then U is symmetric and the principal eigenvalue λ̃1 (LU , Ω(e)) is nonnegative, so that the hypothesis (i) of Theorem 7.8 holds and we get that U is foliated Schwarz symmetric. The second alternative is that one among (W e )+ and (W e )− does not vanish and λ1s (Le,s , Ω(e)) = 0. Then either (W e )+ or (W e )− is a first symmetric eigenfunction of the operator Le,s (V) in Ω(e). If (W e )+ is a first symmetric eigenfunction of the operator Le,s (V) = −ΔV + Be,s V in Ω(e), then it is positive in Ω(e), i. e., U > U σe in Ω(e). In the case when (W e )− is the first symmetric eigenfunction, we get the reversed inequality. Then, by the sufficient condition (ii) given by Theorem 7.8, u is foliated Schwarz symmetric. Remark 7.27 (Alternative proof of Theorem 7.20). By Lemma 7.25, we have the comparison (7.27) between the entries of the matrices Be and Be,s . However, this does not imply that the quadratic forms are ordered (this happens, by Theorem 1.57(iv) a), if (Be,s − Be ) is positive semidefinite). This is the reason why we considered in Lemma 7.25 and in the previous proof of Theorem 7.20 the positive and negative parts of W e . Nevertheless, the comparison (7.27) between the single entries of the matrices Be and Be,s implies, by Theorem 1.57(iv) b), the inequality λ1s (−Δ + Be,s ; Ω(e)) ≤ λ1s (−Δ + Be ; Ω(e))

(7.31)

for the first symmetric eigenvalues. Then part of the proof of Theorem 7.20 can be made using this property. Indeed we can start with a direction e such that the first symmetric eigenvalue λ1s (Le,s , Ω(e)) of the operator Le,s (V) = −ΔV + Be,s V in Ω(e) is nonnegative, and hence also the principal eigenvalue λ̃1 (Le,s , Ω(e)) is nonnegative.

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7.3 Examples | 229

Then by (7.31) we get that the first symmetric eigenvalue λ1s (−Δ + Be , Ω(e)) of the operator −Δ + Be in Ω(e) (Be being defined in (7.26)), is also nonnegative. If λ1s (−Δ + Be , Ω(e)) > 0, then necessarily the difference W e = U − U σ(e) must vanish since Qe (W e ; Ω(e)) = 0 by (7.17). This implies that Be = Be,s = JF (|x|, U), so that we find a direction e with U ≡ U σ(e) in Ω(e) and since, by symmetry, Le,s = LU , we also have λ1s (Le,s , Ω(e)) = λ1s (LU , Ω(e)) > 0. If instead λ1s (−Δ + Be , Ω(e)) = 0, then necessarily λ1s (−Δ + Be,s ; Ω(e)) = λ1s (−Δ + Be , Ω(e)) = 0. Since Qe (W e ; Ω(e)) = 0 either W e vanishes or W e is a first symmetric eigenfunction of the system (7.15), which is fully coupled. This implies that it does not change sign in Ω(e), and assuming that, e. g., U ≥ U σ(e) then U > U σ(e) , by the strong comparison principle, and again by Theorem 7.8 the solution U is foliated Schwarz symmetric. Proof of Theorem 7.22. Let us remark that in the proof of Theorem 7.20 we found a direction e such that U is symmetric with respect to H(e) and not only the principal eigenvalue λ̃1 (LU , Ω(e󸀠 )) of the linearized operator in Ω(e󸀠 ) is nonnegative, but also the first symmetric eigenvalue λ1s (LU , Ω(e)) = λ1s (LU , Ω(−e)) ≥ 0. If hypothesis b) of Theorem 7.22 holds, then necessarily W e ≡ 0, since Qe,s ((W e )+ ; Ω(e)) < 0 if (W e )+ is positive, analogously for the negative part. The same happens if U is a Morse index one solution since in this case for any direction e either λ1s (LU , Ω(e)) or λ1s (LU , Ω(−e)) must be nonnegative, so that the proof goes on as the one of Theorem 7.15.

7.3 Examples A first type of elliptic systems that could be considered are those of “gradient type” 𝜕g (see [95]), i. e., systems of type (7.9) where fj (|x|, S) = 𝜕s (|x|, S) for some scalar function j

g ∈ C 2,α ([0, +∞) × ℝm ). In this case, the solutions correspond to critical points of the functional Φ(u) =

1 ∫ |∇U|2 dx − ∫ g(|x|, U) dx 2 Ω

Ω

H10 (Ω)

in and the linearized operator (7.5) coincides with the second derivative of Φ. Thus standard variational methods apply which often give solutions of finite (linearized) Morse index, as, for example, in the case when the mountain pass theorem can be used (see Section 3.2). So, if the hypotheses of Theorem 7.12 are satisfied, the symmetry results of Section 7.2 can be applied. Many systems of this type have been studied (see [95] and the references therein). An example is the nonlinear Schrödinger system −Δu1 + u1 = |u1 |2q−2 u1 + b|u2 |q |u1 |q−2 u1

{

2

−Δu2 + ω u2 = |u2 |

2q−2

q

q−2

u2 + b|u1 | |u2 |

in Ω u1

in Ω

(7.32)

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230 | 7 Morse index and symmetry for elliptic systems in bounded domains where b > 0, Ω is either a bounded domain or the whole ℝN , N ≥ 2, q > 1 if N = 2 and N 1 < q < N−2 if N ≥ 3. If Ω ≠ ℝN the Dirichlet boundary conditions are imposed u1 = u2 = 0

on 𝜕Ω

(7.33)

The system (7.32) has been considered in several papers (see [13, 170] and the references therein). It is easy to see that the system is of gradient type and the solutions of (7.32), (7.33) are critical points of the functional I(U) = I(u1 , u2 ) =

1 1 b ∫ |∇U|2 dx − ∫(|u1 |2q + |u2 |2q ) dx − ∫ |u1 u2 |q dx 2 2q q Ω

Ω

(7.34)

Ω

in the space H10 (Ω). Thus the linearized operator (7.5) at a solution U corresponds to the second derivative of I in H10 (Ω), and hence the corresponding linear system is symmetric. Moreover, as observed in [170], the system (7.32) is cooperative, and fully coupled in Ω(e), for any e ∈ SN−1 , along every purely vector solution U = (u1 , u2 ), i. e., such that u1 and u2 are both not identically zero. In particular, the system is fully coupled along positive solutions in any Ω(e). Then Theorem 7.12 could be applied to any purely vector solution with Morse index less than or equal to N, in an annulus or in a ball. In particular, we can consider solutions obtained by the mountain pass theorem which have a Morse index not bigger than one which are purely vector positive solutions for suitable values of b and q ≥ 2, as could be obtained by the same proof of [170] for the case when Ω = ℝN . A second type of interesting systems of two equations are the so-called “Hamiltonian type” systems (see [95] and the references therein). More precisely, we consider the system −Δu1 = f1 (|x|, u1 , u2 ) { { { {−Δu2 = f2 (|x|, u1 , u2 ) { { {u1 = u2 = 0

in Ω

(7.35)

in Ω on 𝜕Ω

with f1 (|x|, u1 , u2 ) =

𝜕H (|x|, u1 , u2 ), 𝜕u2

f2 (|x|, u1 , u2 ) =

𝜕H (|x|, u1 , u2 ) 𝜕u1

(7.36)

for some scalar function H ∈ C 2,α ([0, +∞) × ℝ2 ). These systems can be studied by considering the associated functional J(U) = I(u1 , u2 ) =

1 ∫ ∇u1 ⋅ ∇u2 dx − ∫ H(|x|, u1 , u2 ) dx 2 Ω

Ω

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(7.37)

7.3 Examples | 231

either in H10 (Ω) or in other suitable Sobolev spaces (see [95]). It is easy to see that the linearized operator defined in (7.5) does not correspond to the second derivative of the functional J, which is strongly indefinite. Nevertheless, solutions of (7.35) can have finite linearized Morse index as we show with a few simple examples. Let us consider the following system: −Δu1 = λeu2 { { { −Δu2 = μeu1 { { { {u1 = u2 = 0

in Ω in Ω

(7.38)

on 𝜕Ω

where λ, μ ∈ ℝ. If λ, μ > 0, then the system is fully coupled along every solution in any Ω(e), e ∈ SN−1 , and the hypotheses of Theorem 7.12 are satisfied. Let us consider the function G : ℝ2 × (C 2,α (Ω))2 → (C 0,α (Ω))2 defined by G((λ, μ), (u1 , u2 )) = (−Δu1 − λeu2 , −Δu2 − λeu1 ). We have that G((0, 0), (0, 0)) = (0, 0), 𝜕(u𝜕G,u ) ((0, 0), 1 2 (0, 0))(ϕ, ψ) = (−Δϕ, −Δψ), and the first (symmetric) eigenvalue of this operator is strictly positive, so that the operator is invertible. Thus, by the implicit function theorem, for small λ, μ there is a unique nontrivial solution (u1 (λ, μ), u2 (λ, μ)) close to the trivial solution of the system (7.38) corresponding to λ = μ = 0. Moreover, the solution is positive by the maximum principle and it is linearized stable, so that it is radial. Indeed the first (symmetric) eigenvalue of the linearized operator corresponding to λ = μ = 0, u = v = 0 is strictly positive, and by continuity (see Theorem 1.57) it is positive for small λ, μ. The same happens substituting the exponential with other nonlinearities f (u2 ), g(u1 ) with nonnegative derivative in a neighborhood of 0 and such that f (0), g(0) > 0. We now consider, as another example, the system p

−Δu1 = u2 { { { { { {−Δu2 = uq1 { { {u1 , u2 > 0 { { { {u1 = u2 = 0

in Ω in Ω in Ω

(7.39)

on 𝜕Ω

N+2 where 1 < p, q < N−2 if N ≥ 3, p, q > 1 if N = 2. The idea is to proceed as in the previous example starting from the case p = q and the solution u1 = u2 = z, where z is a scalar solution of the equation

−Δz = z p { { { z>0 { { { {z = 0

in Ω in Ω

(7.40)

on 𝜕Ω

which is nondegenerate.

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232 | 7 Morse index and symmetry for elliptic systems in bounded domains Let us observe that if p = q and z has Morse index equal to the integer m(z), then m(z) is also the Morse index of the solution U = (u1 , u2 ) = (z, z) of the system (7.39). Indeed the linearized equation at z for the equation (7.40) and the linearized system at (z, z) for the system (7.39) are respectively −Δϕ − pz p−1 ϕ = 0

in Ω

ϕ=0

on 𝜕Ω

{

(7.41)

and −Δϕ1 − pz p−1 ϕ2 = 0 { { { −Δϕ2 − pz p−1 ϕ1 = 0 { { { {ϕ1 = ϕ2 = 0

in Ω in Ω

(7.42)

on 𝜕Ω

and the eigenvalues of the two operators are the same. Indeed, if ϕ is an eigenfunction for (7.41) corresponding to the eigenvalue λk , then taking ϕ1 = ϕ2 = ϕ we obtain an eigenfunction (ϕ1 , ϕ2 ) for (7.42) corresponding to the same eigenvalue. Vice versa, if (ϕ1 , ϕ2 ) is an eigenfunction for (7.42) corresponding to the eigenvalue λk , then ϕ = ϕ1 + ϕ2 is an eigenfunction for (7.41) corresponding to the same eigenvalue. So proceeding as in the previous example we can start from a nondegenerate solution of (7.40) with ) (or p ∈ (1, +∞) if N = 2) and find a branch of solutions a fixed exponent p ∈ (1, N+2 N−2 of (7.39) corresponding to (possibly different) exponents p, q close to p. For example, if we start with a mountain pass (positive) solution z in the ball of equation (7.40) with the exponent p, knowing that its Morse index is one (see Chapter 3) and it is nondegenerate (see [74]), we get a branch of Morse index one radial solutions for p, q close to p. Thus, starting from an exponent p for which the mountain pass solution z of (7.40) is not radial in the case when Ω is an annulus, we can construct solutions U = (u1 , u2 ) of (7.39) in correspondence of exponents p, q close to p, with Morse index one. Then Theorem 7.12 applies and, in particular, we get that the coupling condition (7.20) holds, which, in this case, can be written as pup−1 = quq−1 1 2

in Ω

(7.43)

Note that more generally, by Corollary 7.15, the equality (7.43) must hold for every solution U = (u1 , u2 ) of (7.39) with Morse index one giving so a sharp condition to be satisfied by the components of a solution of this type. We point out that we must be careful when z is a nonradial foliated Schwarz symmetric function, since in this case, the solution is automatically degenerate because of the rotation invariance of the equation. To bypass this difficulty, we could work in subspaces of symmetric functions. To be more precise, let us start with a solution z of equation (7.40), with exponent p, which is nonradial but foliated Schwarz symmetric with respect to a fixed vector p and with low Morse index. This is the case, for example, of a positive minimal energy solution

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7.3 Examples | 233

in the annulus for some values of the exponent and of the radii of the annulus, which has Morse index one. In these cases, 0 is an eigenvalue of the linearized operator and there are N − 1 corresponding orthogonal eigenfunctions which are odd with respect to N − 1 orthogonal symmetry axes passing through p. Roughly speaking, they are the ones which induce the degeneration. So we consider the subspace H e of H01 (Ω) consisting of the functions that are even with respect to the symmetry hyperplanes. If the solution is nondegenerate in this space, we can use the implicit function theorem and the continuation method as before and find a branch of foliated Schwarz symmetric solutions to the system (7.39) for values of p, q close to the initial exponent p. Then, in the case of a positive Morse index one nonradial foliated Schwarz symmetric solution z in the annulus corresponding to the exponent p, we obtain from Corollary 7.15 a branch of foliated Schwarz symmetric solutions (u, v) to the system (7.39) corresponding to the values p, q close to p, which satisfy the coupling relation (7.20), which, in this case, are p|v|p−1 = q|u|q−1 The same considerations can be made for sign changing solutions, when the hypotheses of Theorem 7.20 are satisfied. In particular, we can consider the system −Δu1 = |u2 |p−1 u2 { { { −Δu2 = |u1 |q−1 u1 { { { {u1 = u2 = 0

in Ω in Ω

(7.44)

on 𝜕Ω

N+2 . Then we start from the case p = q and the solution u1 = u2 = z, where 1 < p, q < N−2 where z is a scalar solution of the equation

−Δz = |z|p−1 z { z=0

in Ω on 𝜕Ω

(7.45)

Let us observe that if p = q and z has Morse index equal to the integer m(z), then as before m(z) is also the Morse index of the solution U = (u1 = u2 ) = (z, z) of the system (7.44). So if we start from a nondegenerate solution of (7.45) with a fixed exponent p ∈ N+2 (1, N−2 ) (or p ∈ (1, +∞) if N = 2), using the implicit function theorem, we find a branch of solutions of (7.44) corresponding to (possibly different) exponents p, q close to p. For example, if we start with a least energy nodal solution z in the ball of equation (7.45) with the exponent p, knowing that its Morse index is two (see Section 3.3.1) we get a branch of Morse index two solutions for p, q close to p. Note that, as proved in Chapter 3 (see also [7]), the least energy nodal solution of (7.45) is not radial but foliated Schwarz symmetric. So it is obviously degenerate, but working in the space of

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234 | 7 Morse index and symmetry for elliptic systems in bounded domains axially symmetric functions we could remove the degeneracy and apply the continuation method described above, if there are not other degeneracies. Then Theorem 7.20 applies if p, q ≥ 2 and, in particular, we get that the coupling condition (7.25) holds, which in this case can be written as p|u2 |p−1 = q|u1 |q−1

in Ω

(7.46)

Note that more generally, by Theorem 7.22, the equality (7.46) must hold for every solution U = (u1 , u2 ) of (7.44) with Morse index m(U) ≤ N − 1, giving so a sharp condition to be satisfied by the components of a solution of this type. Let us remark that the symmetry results of Section 7.2 also apply when the nonlinearity depends on |x| (in any way). Arguing as before, it is not difficult to construct systems having solutions with low Morse index, in particular with Morse index one or two. An example could be the “Henon system” −Δu1 = |x|α |u2 |p−1 u2 { { { { { {−Δu2 = |x|β |u1 |q−1 u1 { { {u 1 , u 2 > 0 { { { {u 1 = u 2 = 0

in Ω in Ω in Ω

(7.47)

on 𝜕Ω

with α, β > 0, p, q ≥ 1. Starting again from a solution of the scalar equation with Morse index one or two, it is possible to construct solutions of (7.47) with the same Morse index, for p, q and α, β close to each other. Note that for some values of p, q, α, β such solutions would not be radial, even if Ω is a ball, but rather foliated Schwarz symmetric as for the scalar case (see [38]). Finally, as in the scalar case, we can have examples of nonhomogeneous systems with convex nonlinearities for which there exist solutions of Morse index one which change sign and to which our results apply.

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8 Some results in unbounded domains In this chapter, we briefly indicate some extensions to unbounded domains of the symmetry theorems considered in Chapter 6 and Chapter 7. We will focus on results obtained by moving planes and Morse index bounds, emphasizing, in both cases, the importance of assuming the solution in a suitable function space. The results about foliated Schwarz symmetry in unbounded domains are based on the same ideas developed in the case of bounded domains but the proofs are rather technical and long and we refer to [131] for them. In the last section, we recall several Liouville-type nonexistence results for solution with finite Morse index.

8.1 Moving planes and symmetry in unbounded domains Soon after the famous result by Gidas, Ni and Nirenberg [122] about the radial symmetry of positive solutions of semilinear elliptic equations in balls, the same authors considered in [123] the following problem in ℝN : −Δu = f (u), u > 0 in ℝN , { u → 0 when |x| → ∞

(8.1)

Using again the Alexandrov–Serrin moving planes method, they proved that the C 2 solutions of (8.1) are radial provided f ∈ C 1+α [0, ∞), f (0) = 0, f 󸀠 (0) < 0. They also obtain some symmetry results in the case f 󸀠 (0) = 0 under appropriate assumptions on the growth of f near 0 and on the decay of u at ∞. Later Y. Li and W. Ni [156] and C. Li [155] extended the previous symmetry results to fully nonlinear strictly elliptic equations proving, in particular, the symmetry of solutions of (8.1) when one of the following hypotheses holds: (i) ∃s0 > 0 : f 󸀠 (s) ≤ 0 ∀s ∈ (0, s0 ) (ii) ∃α > 0, m ∈ ℝ : f 󸀠 (s) = O(sα )(s → 0), u = O( |x|1m )(|x| → ∞) and mα > 2. N

Note that if (ii) holds then u ∈ Lα 2 (ℝN ). As a matter of fact, it is possible to obtain the symmetry result under the only N assumption that u belongs to the space Lα 2 (ℝN ) using a technique introduced in [216, 217], which is based on Sobolev inequalities together with the moving planes method (see also [81] for a similar technique using Poincaré’s and Hardy’s inequalities). We will sketch below the proof given in [9] (where this technique is exploited to obtain symmetry results for solutions of elliptic equations on unbounded manifolds) to illustrate the main difficulties that arise when considering problems in unbounded domains and the importance of assuming that the solutions belong to suitable function spaces. https://doi.org/10.1515/9783110538243-008

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236 | 8 Some results in unbounded domains More precisely, we will prove the following result. Theorem 8.1. Let u ∈ C 1 (ℝN ) be a (weak) solution of the problem −Δu = f (u) { u>0

in ℝN

(8.2)

in ℝN

Suppose that one of the following set of hypotheses holds: (H1) (i) u(x) → 0 as |x| → ∞ (ii) ∃s0 > 0 : f is nonincreasing in (0, s0 ). (H2) (i) u(x) → 0 as |x| → ∞ (a) ≤ C(a + b)α when 0 < a < b < s0 and u ∈ (ii) ∃s0 , α > 0: such that f (b)−f b−a αN/2 N L (ℝ ). (H3) ∃α > 0, p ∈ [2, N) such that (a) | ≤ C(a + b)α for all a, b > 0 (i) | f (b)−f b−a n

(ii) u ∈ 𝒟1,p (ℝN ) ∩ Lα 2 (ℝN ) ∗ where 𝒟1,p (ℝN ) := {u ∈ Lp (ℝN ) : |∇u| ∈ Lp (ℝN )}, p∗ =

Np N−p

Then, u is radially symmetric around some point x0 ∈ ℝN , i. e., u(x) = u(r), where r := |x − x0 |. Moreover, u󸀠 (r) < 0 for all r > 0. Proof. For simplicity, we assume N ≥ 3, though it is possible to obtain analogous results when N = 2, using the corresponding Sobolev embeddings. As usual in radial symmetry results in ℝN , it is enough to fix an arbitrarily chosen direction and to prove symmetry w. r. t. that direction. Then it is easy to see that all the symmetry hyperplanes meet in a single point. For simplicity of notation, we fix the e1 = (1, 0, . . . , 0) direction and denote a point in ℝN as x = (x1 , x󸀠 ) with x󸀠 ∈ ℝN−1 . Given λ ∈ ℝ, we set Σλ := {x = (x1 , x󸀠 ) ∈ ℝN : x1 < λ}, xλ := σλ (x) := (2λ − x1 , x󸀠 ),

Hλ := {x ∈ ℝN : x1 = λ},

uλ (x) := u(xλ ),

Σλ = σλ (Σλ )

Here, xλ := σλ (x) := (2λ − x1 , x󸀠 ) is the image of the point x = (x1 , x 󸀠 ) under the reflection through the hyperplane Hλ . The first step of the proof will consist in showing that the set Λ := {λ ∈ ℝ : ∀μ > λ, u ≥ uμ in Σμ } is nonempty and bounded from below. The second step will then be to show that if λ0 := inf Λ, then u ≡ uλ0 in Σλ0 . Step 1: Λ ≠ 0 and is bounded from below. Case of hypotheses (H2). First, we see that Λ is bounded from below, since u → 0 when |x| → ∞.

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8.1 Moving planes and symmetry in unbounded domains | 237

We write v = uλ and suppose q ≥ 1 (to be chosen below). For ε > 0, we let wε = wε,q (x) := [(v − u − ε)+ ]q . Using wε as test function (it has compact support in Σλ since u → 0 as |x| → ∞ and v ≡ u on Hλ ), we obtain, once we subtract the equation for u from the equation for v, 󵄨󵄨D[(v − u − ε)+ ]󵄨󵄨󵄨2 = ∫[f (v) − f (u)][(v − u − ε)+ ]q 󵄨 󵄨

q−1 󵄨

q ∫[(v − u − ε)+ ]

(8.3)

Σλ

Σλ q+1

Define zε := [(v − u − ε)+ ] 2 . If λ is sufficiently large, then v < s0 . Hence, since we integrate in a set where v > u + ε > u > 0 and (H2) holds, we get that 4q f (v) − f (u) q q (v − u)[(v − u − ε)+ ] ≤ C ∫ vα (v − u)[(v − u − ε)+ ] . ∫ |Dzε |2 = ∫ 2 v−u (q + 1) Σλ

Σλ

Σλ

Since u ∈ L∞ (ℝN )∩LαN/2 (ℝN ), we can fix a sufficiently large q (say q such that α+q+1 ≥ α N2 ) so that q

∫ vα (v − u)[(v − u − ε)+ ] ≤ ∫ vα+q+1 ≤ ∫ uα+q+1 < +∞. Σλ

Σλ

(8.4)

ℝN

Passing to the limit as ε → 0, and denoting z = [(v − u)+ ] dominated convergence theorem)

q+1 2

, we obtain (using the

∫ |Dz|2 ≤ C ∫ vα z 2 . Σλ

Σλ

By Hölder and Sobolev inequalities, it follows that 2

∫ |Dz| ≤ C(∫ v Σλ

αN 2

) (∫ z

Σλ

= C(

2 N

2N N−2

αN 2

)

Σλ

∫

αN 2

2 N

u ) (∫ z 2 ) ∗

2 2∗

Σλ

∈ L1 (ℝN ), we have that limλ→∞ ∫Σλ u αN 2

αN 2

αN 2

(8.5)

2 N

≤ C1 (∫ u ) ∫ |Dz|2 .

Σλ

Σλ =σλ (Σλ )

Since u

N−2 N

Σλ

= 0, and thus, for sufficiently large λ,

2 N

C1 (∫ u ) < 1. Σλ

This, together with (8.5), yields that ∫Σ |Dz|2 = 0, and thus |Dz| = 0 in Σλ , and hence z λ is constant in Σλ . Since z = 0 on Hλ , this implies that z = 0 in Σλ , which proves step 1.

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238 | 8 Some results in unbounded domains Case of hypotheses (H1). In this case, the proof is much easier: we can simply take q = 1 and, since f is decreasing near zero, for sufficiently large λ the r. h. s. in (8.3) is nonpositive and, therefore, is zero. Passing to the limit as ε → 0, again we obtain 󵄨 󵄨2 ∫󵄨󵄨󵄨D(v − u)+ 󵄨󵄨󵄨 = 0, Σλ

for λ sufficiently large. We conclude as in the previous case. Case of hypotheses (H3). Let us set q = N(p−1)−p , so that p = N−p

N(q+1) N+q−1

and p∗ =

N (q + 1) N−2 . By the summability assumptions on the solution, it is possible to take directly the function [(v − u)+ ]q , v = uλ , as a test function in the equations for u and uλ in Σλ . More precisely, since u, v belong to 𝒟1,p (ℝN ) and they coincide on the hyperplane Hλ, there ∗ exists a sequence φj of functions in Cc∞ (Σλ ) such that φj → [(v − u)+ ] in Lp (Σλ ) and Dφj → D(v − u)+ in Lp (Σλ ). Moreover, passing to a subsequence and substituting if necessary φj with φ+j , we can assume that 0 ≤ φj → [(v − u)+ ], Dφj → D(v − u)+ a. e.

in Σλ , and that there exist functions ψ0 ∈ Lp , ψ1 ∈ Lp , such that |φj | ≤ ψ0 , |Dφj | ≤ ψ1 a. e. in Σλ . Taking the functions φqj as test functions in the equations for v and u in Σλ and subtracting, we get ∗

q ∫ φq−1 D(v − u) ⋅ Dφj = ∫[f (v) − f (u)]φqj j Σλ

Σλ

If we can pass to the limit for j → ∞, getting q−1 󵄨

q ∫[(v − u)+ ]

󵄨󵄨D[(v − u)+ ]󵄨󵄨󵄨2 = ∫[f (v) − f (u)][(v − u)+ ]q < ∞, 󵄨 󵄨

Σλ

Σλ

then the proof is exactly the same as in the previous case. So it is enough to justify the passage to the limit, which easily follows from the dominated convergence theorem. Indeed we have 󵄨󵄨 q󵄨 α q α q 󵄨󵄨[f (v) − f (u)]φj 󵄨󵄨󵄨 ≤ C(u + v) |v − u||φj | ≤ C(u + v) |v − u||ψ0 | and (u + v)α |v − u||ψ0 |q belongs to L1 , since (u + v)α ∈ Lr1 , |v − u| ∈ Lr2 , |ψ0 |q ∈ Lr3 , where ∗ N N , r3 = pq = q+1 with r1 + r1 + r1 = 1. Analogously, we r1 = N2 , r2 = p∗ = (q + 1) N−2 q N−2 1 2 3 have that 󵄨󵄨 q−1 󵄨 󵄨 q−1 󵄨 󵄨󵄨φj D(v − u) ⋅ Dφj 󵄨󵄨󵄨 ≤ |ψ0 | 󵄨󵄨󵄨D(v − u)󵄨󵄨󵄨|ψ1 | and |ψ0 |q−1 |D(v − u)||ψ1 | belongs to L1 , since ψq−1 ∈ Ls1 , |D(v − u)|, |ψ1 | ∈ Ls2 , where 0 s1 =

p∗ q−1

=

q+1 N ,s q−1 N−2 2

=p=

N(q+1) N+q−1

and

1 s1

+ 2 s1 = 1. 2

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8.2 Foliated Schwarz symmetry | 239

Step 2: u ≡ uλ0 . Let us first consider the case of hypothesis (H2) or (H3). Since λ0 is the infimum, by continuity we have that u ≥ uλ0 in Σλ0 . Thus, if we suppose u ≢ uλ0 in Σλ0 , the strong comparison principle yields that u > uλ0 in Σλ0 and ux1 < 0 on Hλ0 . αN

Since u ∈ L 2 (ℝN ), we can choose a compact set K ⊂ Σλ0 and a number δ > 0 such that ∀λ ∈ (λ0 − δ, λ0 ) we have K ⊂ Σλ and C1 (

∫

Rλ (Σλ \K)

2 N

1 u ) < , 2 αN 2

(8.6)

where C1 is as in (8.5). On the other hand, since u − uλ0 is positive in Σλ0 , there exists 0 < δ1 < δ, such that u > uλ ,

in K

∀λ ∈ (λ0 − δ1 , λ0 ).

(8.7)

Using (8.6) and proceeding as in Step 1, since the integrals are on Σλ \ K, we see that (uλ − u)+ ≡ 0 in Σλ \ K. By (8.7) we get u ≥ uλ in Σλ for all λ ∈ (t0 − δ1 , λ0 ), contradicting the definition of λ0 . In the case of hypothesis (H1), since u > uλ0 in Σλ0 and ux1 < 0 on Hλ0 , proceeding as in Remark 6.2 it is easy to show that, for any R > 0, there exists δ = δR > 0 such that u > uλ in Σλ ∩ BR (0), if λ0 − δ < λ < λ0 . If R > 0 is sufficiently large, then u(xλ ) < s0 for any x ∈ Σλ \ BR0 and, proceeding as in Step 1, it is easy to show that u > uλ in Σλ \ BR (0), if λ0 − δ < λ < λ0 , contradicting the definition of λ0 . Remark 8.2. Note that in the case of critical problems, i. e., when α = 4/(N −2) = 2∗ −2, it follows that αN/2 = 2N/(N − 2) = 2∗ , and the radial symmetry holds for solutions ∗ in L2 vanishing at infinity, or solutions belonging to the space D1,p (ℝN ) for some p ∈ [2, N) without assuming a priori that it converges to zero at infinity, if the nonlinearity f satisfies the growth condition both at zero and at infinity. Note also that there could be solutions with infinite energy, i. e., whose gradient does not belong to L2 (and this happens, e. g., in the supercritical case). This is why we have to take q ≥ 1 in the proof of Theorem 8.1. Remark 8.3. Symmetry and monotonicity properties of solutions of elliptic equations in unbounded domains other than ℝN , like half-spaces or cylinders, have been largely studied, e. g., in the series of papers [28–30] by Berestycki, Caffarelli and Nirenberg.

8.2 Foliated Schwarz symmetry Using the Morse index of a solution, as defined in Chapter 3 for bounded or unbounded domains, it is possible to extend the symmetry results of Chapter 6 and Chapter 7 to

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240 | 8 Some results in unbounded domains the case of rotationally symmetric unbounded domains. The symmetry we consider is the foliated Schwarz symmetry, as defined in Chapter 6 and Chapter 7. We observe that the sufficient condition given by Proposition 6.7 holds also in unbounded domain. Let us consider the problem − Δu = f (|x|, u) in Σ

(8.8)

where Σ is either ℝN or the exterior of a ball, i. e., Σ = ℝN \ B where B is a ball centered at the origin, N ≥ 2, and f : Σ × ℝ → ℝ is locally a C 1 -function. When Σ = ℝN \ B, we also require the boundary condition u=0

on 𝜕Σ.

(8.9)

In the sequel by a solution of (8.8), (8.9) we mean a classical C 2 (Σ) solution. The following results, which extend the result of Chapter 6, have been proved in [131]. Theorem 8.4. Suppose that f (|x|, s) has a convex derivative f 󸀠 (|x|, s) = 𝜕f (|x|, s) for every 𝜕s x ∈ Σ. Then every solution u of (8.8) and (8.9) with |∇u| ∈ L2 (Σ) and Morse index j ≤ N is foliated Schwarz symmetric. Theorem 8.5. Suppose that f (|x|, s) is convex in the s-variable for every x ∈ Σ. Then every solution u of (8.8) and (8.9) with |∇u| ∈ L2 (Σ) and Morse index j ≤ N is foliated Schwarz symmetric. We point out that the previous convexity assumptions are not satisfied for the Allen–Cahn nonlinearity f (u) = u − u3 and its generalizations, for which, in contrast to our results, there exist in ℝN rather complicated solutions with low Morse index (see the remarks in [90] and the references therein for a discussion on this topic). On the other hand, in the previous theorems, nonlinearities depending explicitly on the radial space variable |x| are allowed. We will not give the proofs of these theorems, for which we refer to the paper [131], but we only indicate the main steps and point out that in each of them a crucial role is played by the assumption that |∇u| ∈ L2 (Σ). As in the bounded domain case, an important step is to prove the following counterpart of Theorem 6.12. Proposition 8.6. Let u be a solution of (8.8) and (8.9) such that |∇u| ∈ L2 (Σ). Assume that there exists e ∈ 𝒮 such that u is symmetric with respect to the hyperplane H(e) and inf

Qu (ψ) ≥ 0.

(8.10)

𝜕f 󵄨 󵄨2 (|x|, u(x))󵄨󵄨󵄨ψ(x)󵄨󵄨󵄨 ]dx 𝜕s

(8.11)

ψ∈Cc1 (Σ(e))

where Qu (ψ) = ∫[|∇ψ|2 − Σ

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8.2 Foliated Schwarz symmetry | 241

Then u is foliated Schwarz symmetric. Another important ingredient in the proof of the symmetry results is the rotating plane method introduced in Chapter 6. We need the following definition. Definition 8.7. The solution u of (8.8) and (8.9) is said to be stable outside a compact set 𝒦 ⊂ Σ if Qu (ψ) ≥ 0 for all ψ ∈ Cc1 (Σ \ 𝒦) where Qu is the quadratic form defined in (8.11). It is easy to see that the following holds (see [89, 90, 117]). Remark 8.8. If u has finite Morse index, then u is stable outside a compact set 𝒦 ⊂ Σ. Denoting, as in Chapter 6, by we the difference between a function u and its reflection with respect to the hyperplane H(e), it is possible to prove the following result by means of a rotating plane argument. Proposition 8.9. Let u be a solution of (8.8) and (8.9) stable outside a compact set 𝒦 ⊂ Σ and such that |∇u| ∈ L2 (Σ). Suppose that either f or f 󸀠 is convex in the second variable, and that there exists a direction e ∈ 𝒮 such that we (x) < 0

∀x ∈ Σ(e) or

we (x) > 0

∀x ∈ Σ(e).

(8.12)

Then there exists another direction e󸀠 ∈ 𝒮 such that we󸀠 ≡ 0, (i. e., u is symmetric with respect to the hyperplane H(e󸀠 )) and inf

ψ∈Cc1 (Σ(e󸀠 ))

Qu (ψ) ≥ 0.

(8.13)

Another result, which is the counterpart of Proposition 6.21, and which does not depend on the convexity of the nonlinearities, is the following. Proposition 8.10. Suppose that u has Morse index j ≤ N. Then there exists e ∈ 𝒮 such that inf

ψ∈Cc1 (Σ(e))

Qu (ψ, ψ) ≥ 0.

(8.14)

Let us finally remark that although the characterization of the Morse index by means of the number of negative eigenvalues (Theorem 3.2) is not available for unbounded domains, in many proofs, this characterization is exploited in large balls, assuming that the solution has a Morse index not exceeding the dimension N. Remark 8.11. In the paper [73], the results for systems of Chapter 7 have also been extended to rotationally symmetric unbounded domains, with suitable hypotheses on the solution, the main one being that the gradient of the solution belongs to L2 . Besides the technical difficulties that arise when considering solutions in unbounded domains, also the various type of eigenvalues that can be considered for systems play a role, as in Chapter 7. We refer to [73] for the statements and the proofs.

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242 | 8 Some results in unbounded domains

8.3 Nonexistence and classification results Information on the Morse index of a solution can also be used to deduce some Liouville-type nonexistence results. Indeed as a consequence of the foliated Schwarz symmetry the nonexistence of nontrivial solutions have been proved in some cases (see [131]). We state some of them below. Theorem 8.12. Every stable solution u of (8.8) and (8.9) such that |∇u| ∈ L2 (Σ) is radial. If in addition Σ = ℝN and f does not depend on |x|, then u is constant. This theorem generalizes and complements results on stable solutions obtained in [55] and [90]. We stress that in Theorems 8.4–8.12 we do not assume boundedness of the solution u. In the case when f does not depend on |x|, some other nonexistence results can be deduced from Theorem 8.4 and Theorem 8.5. Theorem 8.13. Assume that Σ = RN and f = f (s), i. e., f does not depend on x and that either f is convex or f 󸀠 is convex. Then there are no sign changing solutions u of (8.8) with |∇u| ∈ L2 (ℝN ),

u(x) → 0

as |x| → ∞

and Morse index j ≤ N. Theorem 8.14. Assume that Σ = ℝN \ B and f = f (s), i. e., f does not depend on x and that either f is convex or f 󸀠 is convex. Then there are no solutions u (neither positive nor sign changing) of (8.8) and (8.9) with |∇u| ∈ L2 (Σ),

u(x) → 0

as |x| → ∞

and Morse index j ≤ N. These results are quite easy corollaries of Theorems 8.4 and 8.5 except in the case when nodal solutions in ℝN \ B have to be excluded. This part is more difficult and requires additional ideas (see [131]). The previous results apply to many nonlinear problems. In particular, to semilinear elliptic equations of the type 4

− Δu = |u| N−2 u in ℝN , N ≥ 3

(8.15)

and − Δu + u = |u|σ u, where σ > 0 and σ
3.

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(8.16)

8.3 Nonexistence and classification results | 243

In particular, the nonexistence result of Theorem 8.13 applies to the case of the critical power nonlinearity (8.15) which satisfies the hypothesis that f 󸀠 is convex for 3 ≤ N ≤ 6. In this case, the assumptions |∇u| ∈ L2 (Σ),

u(x) → 0

as |x| → ∞

(8.17)

are automatically satisfied since it is proved in [117] that every classical solution u with finite Morse index belongs to the space D1,2 (ℝN ) ∩ L∞ (ℝN ) where D1,2 (ℝN ) is defined as the completion of Cc∞ (ℝN ) in the norm ‖v‖D1,2 (ℝN ) = ‖∇v‖L2 (ℝN ) . We believe that the properties (8.17) should hold for finite Morse index solutions corresponding to a more general class of nonlinearities, although they are certainly not satisfied in the case of the Allen–Cahn nonlinearity f (u) = u − u3 . We end by mentioning that many nonexistence and classification results have been proved by using the Morse index of a solution but with other methods. In particular, we state below some very interesting results proved by Farina in [117] for the solutions of the Lane–Emden equation − Δu = |u|p−1 u in Ω

(8.18)

where Ω is an unbounded domains. They are based on a method which exploits suitable test functions (we refer to the paper and the references therein for a detailed discussion of the results). Theorem 8.15. Let Ω = ℝN and let u ∈ C 2 (ℝN ) be a stable solution of (8.18) with 1