MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS ON COMPACT RIEMANNIAN MANIFOLDS

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Niveau: Supérieur, Licence, Bac+2
MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS ON COMPACT RIEMANNIAN MANIFOLDS JEROME VETOIS Abstract. Let (M, g) be a smooth, compact Riemannian n-manifold, and h be a Holder continuous function on M . We prove the existence of multiple changing sign solutions for equations like ∆gu + hu = |u| 2??2 u, where ∆g is the Laplace–Beltrami operator and the exponent 2? = 2n/ (n? 2) is critical from the Sobolev viewpoint. 1. Introduction 1.1. Statement of the results Let (M, g) be a smooth, compact Riemannian manifold of dimension n ≥ 3 and h be a Holder continuous function on M , namely a function which belongs to C0,? (M) for some real number ? in (0, 1). We consider equations like ∆gu+ hu = |u| 2??2 u , (1.1) where ∆g = ? divg? is the Laplace–Beltrami operator, and 2? = 2n/ (n? 2). If H21 (M) stands for the Sobolev space of all functions in L2 (M) with one derivative in L2 (M), then 2? is the critical exponent for the embeddings of H21 (M) into Lebesgue spaces. We provide H21 (M) with the scalar product ?u, v?H21 (M) = ∫ M ??u,?v?g dvg + ? ∫ M uvdvg , (1.2) where ? is a positive constant to be chosen large later on.

• like ∆g


• multiple solutions


• positive solution


• elliptic equations


• critical elliptic


• unique positive


• solution u∞


• changing sign


• nodal solutions


• euclidean space

Sujets

• Euclidean space

Informations

• Publié par: profil-vieg-2012
• Nombre de lectures: 20
• Langue: English

Extrait

MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS COMPACT RIEMANNIAN MANIFOLDS ´ ˆ ´ JEROME VETOIS

Abstract.Let (M, g) be a smooth, compact Riemanniann-manifold, andhH¨olderbea continuous function onM prove the existence of multiple changing sign solutions for. We 2 ∗ 2 heΔ g  operator and theis ami equations likeΔ g u+hu=|u| − u re, w Laplace–Beltr the exponent 2 ∗ = 2n/(n−2) is critical from the Sobolev viewpoint.

1.1.Statement of the results

1.Introduction

ON

Let (M, g) be a smooth, compact Riemannian manifold of dimensionn≥3 andhbe a Ho¨ldercontinuousfunctiononM, namely a function which belongs toC 0,θ (M) for some real numberθin (0,1). We consider equations like

Δ g u+hu=|u| 2 ∗ −2 u ,(1.1) whereΔ g =−div g ris the Laplace–Beltrami operator, and 2 ∗ = 2n/(n−2). IfH 12 (M) stands for the Sobolev space of all functions inL 2 (M) with one derivative inL 2 (M), then 2 ∗ is the critical exponent for the embeddings ofH 12 (M) into Lebesgue spaces. We provide H 12 (M) with the scalar product hu, vi H 12 (M) = Z M hru,rvi g dv g +Λ Z M uvdv g ,(1.2) whereΛtyofniiuoctndlreHeo¨Thn.rotelagearnlesohcebottnatsnosipasotivicehprovides the regularity of weak solutions of equation (1.1). In case there holdsh≡ 4( n n − − 2 1) Scal g , where Scal g is the scalar curvature of the manifold (M, g), equation (1.1) is the intensively studied Yamabe equation whose positive solutionsuare such that the scalar curvature of the conformal metricu 2 ∗ −2 g this InSchoen [49], Trudinger [58], and Yamabe [59]).is constant (see Aubin [3], paper, we deal with multiplicity of solutions for equation (1.1) when the functionhis locally n−2 less than 4(n−1) Scal g in Theorem 1.1, and globally less than 4( n n − − 2 )1 Scal g in Theorems 1.2 and 1.3. We define the energy of a solutionuof equation (1.1) to be the real numberE(u) given by E(u) = Z M |u| 2 ∗ dv g ,(1.3) wheredv g is the volume element of the manifold (M, g). We say that an operator likeΔ g +h is coercive onH 12 (M) if the energy associated to this operator controls theH 12 -norm. We Date: July 3, 2006. Published inInternational Journal of Mathematics18 1071–1111. 9,(2007), no. 1

MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS 2 letD 1,2 (R n ) be the homogeneous Sobolev space defined as the completion of the space of all smooth functions onR n with compact support with respect to the scalar product hu, vi D 1,2 (R n ) = Z R n hru,rvidx . We let alsoK n the sharp constant for the embedding ofbe D 1,2 (R n ) intoL 2 ∗ (R n ), namely K n = s n(n−) 4 2ω 2n/n , whereω n is the volume of the unitn associate each solution of equation (1.1) with-sphere. We its opposite one, and call that a pair of solutions. We state our first result as follows.

Theorem 1.1.Let(M, g)be a smooth, compact Riemannian manifold of dimensionn≥4 andhcnitsuufniouoctnlderaH¨obeononMsuch that the operatorΔ g +his coercive on H 12 (M). If there exists a pointx 0 inMsuch thath(x 0 )< 4( n n − − 2 1) Scal g (x 0 ), then equation (1.1)admits at least(n+ 2)/2pairs of nontrivial solutions with energy less than2K n −n . More precisely, we prove that either we do have infinitely many solutions of equation (1.1) or the (n+ 2)/we get in Theorem 1.1 have distinct energies.2 pairs of nontrivial solutions In the particular case where the manifold is locally conformally flat,n≥7, andhis aC 1 - function less than 4(n n − − )1 2 Scal g  Inmanifold, the above result can be improved.on the whole such a setting, we establish two results. We first consider families of equations like Δ g u+hu=|u| p α −2 u ,(1.4) where (p α ) α is a sequence in [2,2 ∗ ] converging to 2 ∗ . A sequence (u α ) α is said to be a sequence of solutions for the family (1.4) if for anyα,u α  weis a solution of equation (1.4). First, prove a compactness result for the family of equations (1.4) similar to the one proved by Devillanova–Solimini [17] in the case of smooth, bounded domains of the Euclidean space. Our compactness result is as follows.

Theorem 1.2.Let(M, g)be a smooth, compact, locally conformally flat Riemannian manifold of dimensionn≥7andhbe aC 1 -function onM. We let(p α ) α be a sequence in[2,2 ∗ ] converging to2 ∗ , and we consider the family of equations(1.4) there holds. Ifh < 4( n n − − 2 )1 Scal g inM, then any bounded sequence inH 12 (M)of solutions for this family of equations remains bounded inC 0 (M).

An equivalent conclusion of Theorem 1.2 is that any bounded sequence inH 12 (M) of solu- tions for the family of equations (1.4) is compact inH 12 (M). In particular, such a sequence converges up to a subsequence inH 12 (M This) to a solution of the critical equation (1.1). easily follows from standard elliptic estimates (see, for instance, Gilbarg–Trudinger [28] Theo- rem 9.11) and the compactness of the embedding ofH 2 p (M) intoH 12 (M) for all real numbers p >2n/(n−2). As a remark, note thatp α →2 ∗ is the only interesting difficult case for com- pactness since the embeddings ofH 12 (M) intoL p (M) are compact forp <2 ∗ . Theorem 1.2 is the key argument in the proof of our last result which states as follows.

Theorem 1.3.Let(M, g)be a smooth, compact, locally conformally flat Riemannian manifold of dimensionn≥7andhbe aC 1 -function onM. If there holdsh < 4( n n − − 2 )1 Scal g inM, then equation(1.1)infinitely many solutions with unbounded energies.admits

MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS

3

There are several situations where we do know that the solutions we get in Theorems 1.1 and 1.3 truly change sign. Such changing sign solutions are referred to as nodal solutions. Let us assume, for instance, that the Ricci curvature Ric g of the manifold (M, g) satisfies Ric g ≥ 4 n ( (n n − − )2 )1 λg(1.5) for some positive real numberλof bilinear forms, the inequality being strict, in the sense when the manifold is conformally diffeomorphic to the sphere. Then, as proved by Bidaut- Ve´ron–V´eron[6],equation(1.1)withh≡λhas a unique positive solution, which turns out to beu=λ (n−2)/4 all but one pairs of solutions we get in particular, in such a situation, . In Theorem 1.1 are nodal. Concerning Theorem 1.3, it has been proved by Druet [21] that there is ana prioribound on the energy of positive solutions of equation (1.1) whenh < 4(n n − − )1 2 Scal g inM precisely, for any smooth, compact Riemannian manifold (. MoreM, g) of dimension n≥3, there exists a real numberE 0 such that ifuis a positive solution of equation (1.1), thenE(u)≤E 0 whereE(uas a direct consequence of the particular,  In) is as in (1.3). existence of thisa prioribound for positive solutions, Theorem 1.3 provides infinitely many nodal solutions for equation (1.1). Summarizing, the following corollary holds true.

Corollary 1.4.Let(M, g)be a smooth, compact Riemannian manifold of dimensionnandh be aC 1 -function onMsuch thath < 4( n n − − 2 )1 Scal g inM. Ifn≥7and the manifold is locally conformally flat, then equation(1.1) Ifadmits infinitely many nodal solutions.n≥4, the manifold is arbitrary,h≡λfor someλ >0, and(1.5)holds true, the inequality being strict when the manifold is conformally diffeomorphic to the sphere, then equation(1.1)admits at leastn/2pairs of nodal solutions.

Compactness of positive solutions of equations like (1.1) have been intensively studied in recent years. Possible references on this topic, in the case of manifolds, are Druet [21, 22], Li– Zhang [40–42], Li–Zhu [43], Marques [45], and Schoen [50–52]. A survey reference on the sub- ject is Druet–Hebey [23]. We refer also to Hebey [31, 32] for compactness of positive solutions of critical elliptic systems in potential form and to Hebey–Robert–Wen [33] for compactness of positive solutions of critical fourth order equations. Compactness of changing sign solutions of equations like (1.1), in the case of smooth, bounded domains of the Euclidean space, have been studied in Devillanova–Solimini [17]. We follow this reference by Devillanova–Solimini [17] in several places in Section 3, as well as we follow the reference Clapp–Weth [15] in several places in Section 2. Possible other references on the existence of multiple nodal solutions for equations like (1.1) are Atkinson–Brezis–Peletier [2], Bahri–Lions [4], Capozzi–Fortunato–Palmieri [8], Castro–Cossio–Neuberger [9], Cerami–Fortunato–Struwe [10], Cerami–Solimini–Struwe [11], Devillanova–Solimini [18], Ding [19], Djadli–Jourdain [20], Fortunato–Jannelli, [26], Hebey– Vaugon [34], Holcman [35], Jourdain [36], Solimini [53], Tarantello [57], and Zhang [60]. Need- less to say, the above list does not pretend to exhaustivity. We refer also to the recent very nice paper by Ammann–Humbert [1] where the question of the existence of at least one changing sign solution to the Yamabe equation is addressed. A final remark in this introduction concerns the condition h < 4( n n− − 1) 2 Scal g in Theorem 1.2. Let (S n ,std) be the unitn holds Scal-sphere. There std ≡n(n−1), and the Yamabe equation on the unitn-sphere reads as

n n Δ std u+ ( 4 −2) u=u 2 ∗ −1 .

(1.6)