A Nonlinear eigenvalue problem with indefinite weights related to the Sobolev trace embedding

Publ. Mat. 46 (2002), 221–235 A NONLINEAR EIGENVALUE PROBLEM WITH INDEFINITE WEIGHTS RELATED TO THE SOBOLEV TRACE EMBEDDING Juli´ an Fern´ andez Bonder and Julio D. Rossi Abstract In this paper we study the Sobolev trace embedding W1,p(Ω) → Lp V(∂Ω), where Vis an indefinite weight. This embedding leads to a nonlinear eigenvalue problem where the eigenvalue appears at the (nonlinear) boundary condition. We prove that there exists a sequence of variational eigenvalues λk+∞and then show that the first eigenvalue is isolated, simple and monotone with respect to the weight. Then we prove a nonexistence result related to the first eigenvalue and we end this article with the study of the second eigenvalue proving that it coincides with the second variational eigenvalue. 1. Introduction Let Ω be a bounded smooth domain in RNand V:∂Ω→Ran indefinite weight. In this paper we consider the Sobolev trace embedding W1,p(Ω) →Lp V(∂Ω), where Lp V(∂Ω) = u:∂Ω→R;∂Ω |u|pV(x)dσ < +∞. We only require mild integrability hypotheses on the weight V(x). More precisely, we assume V+≡ 0on∂Ω and V∈Ls(∂Ω),(1.1) where s>(N−1)/(p−1) if 1 <p≤Nand s≥1ifp>N. 2000 Mathematics Subject Classification. 35P30, 35J70, 35J20. Key words. p-Laplacian, eigenvalue problems, nonlinear boundary conditions. Supported by ANPCyT PICT No. 03-05009. J. D. Rossi is a member of CONICET. 222 J. Fern´ andez Bonder, J. D. Rossi Under these hypotheses on the weight V, this embedding is compact and therefore there exists a constant Sp=Sp(Ω,V) such that the following inequality holds, S1/p puLp V(∂Ω) ≤uW1,p(Ω),where up Lp V(∂Ω) =∂Ω |u|pV(x)dσ. Here and in what follows, we use the following norm in W1,p(Ω): up W1,p(Ω) =Ω |∇u|p+|u|pdx. By the compactness of the embedding, we can prove (see Theorem 1.1) that there exists functions, usually called extremals, where the constant Spis attained. In fact, the extremals are weak solutions of    ∆pu=|u|p−2u, in Ω, |∇u|p−2∂u ∂ν =λV (x)|u|p−2u, on ∂Ω. (1.2) Here ∆pu= div(|∇u|p−2∇u) is the p-Laplacian and ∂ ∂ν is the outer unit normal derivative. Problems of the form (1.2) appears in several branches of pure and applied mathematics, such as the theory of quasiregular and quasiconformal mappings in Riemannian manifolds with boundary (see [12], [18], etc.), non-Newtonian fluids, reaction diffusion problems, flow through porus media, nonlinear elasticity, glaciology, etc. (see [4], [5], [6], [10], etc.). Observe that in (1.2), we are dealing with a nonlinear eigenvalue problem. In the case p= 2, this eigenvalue problem becomes linear and it is known as the Steklov problem, [7]. Our main concern here is the study of eigenvalues for problem (1.2). First we extend the results in [13] and [15] to our more general setting and study the dependence of the first eigenvalue with respect to the weight. Some of these results are adaptations of the proofs in [13]sowe only sketch them in order to make the paper self contained. The main difference in proving these results comes in the proof of the isolation and simplicity of the first eigenvalue were the arguments of [15] cannot be applied. This difficulty is overcome by the use of a “Piccone’s identity” in the same spirit of [1], [8]. Once we have proved that the first eigenvalue is isolated it make sense to define the second eigenvalue. Then we characterize this second eigenvalue and prove that coincides with the second variational eigenvalue found before. This last result is new even in the case V≡1 and is the main result in this paper. Nonlinear Eigenvalue Problem 223 The study of the eigenvalue problem −∆pu=λ|u|p−2ucomplemented with Dirichlet boundary conditions have received considerable attention in recent years. See for example [1], [2], [3], [8], [9], [14]. However, problem (1.2) is less covered in the literature. With V≡1, problem (1.2) has been studied in [13] and in [15]. In those papers it is proved that there exists an unbounded sequence of eigenvalues and that the first eigenvalue is isolated and simple. Next, we state the precise results of the paper. We prove, Theorem 1.1. Let V(x)satisfy (1.1), then there exists a sequence of eigenvalues λkof (1.2) such that λk→+∞as k→+∞. The proof of this theorem relies on the Ljusternik-Schnirelman critical point theory on C1manifolds using the genus,γ. We find the following variational characterization of a sequence of eigenvalues 1 λk = sup C∈Ck min u∈C u||p Lp V(∂Ω) up W1,p(Ω) , where Ck={C⊂W1,p(Ω); Cis compact, symmetric and γ(C)≥k}. Regarding the first eigenvalue λ1, following ideas from [8] and [15], we prove Theorem 1.2. The first eigenvalue of (1.2) is simple and isolated. Moreover, any associated eigenfunction does not change sign in Ω. The eigenfunctions associated to λ1are in fact the extremals for the embedding W1,p(Ω) →Lp V(∂Ω). Hence our result says that the extremal is unique up to a multiplicative constant. We observe that any eigenfunction associated to an eigenvalue λ=λ1 changes sign in ∂Ω. Also the number of nodal domains is finite. See Section 3. Moreover, we prove that the first eigenvalue is monotone with respect to the weight. Theorem 1.3. Let V1,V2be two weight functions satisfying hypotheses (1.1).IfV1≤V2then λ1(V1)≥λ1(V2). Related to λ1we have a nonexistence result. In fact, if we consider the equation    ∆pu=|u|p−2u−f(x),in Ω, |∇u|p−2∂u ∂ν =λ1V(x)|u|p−2u+g(x),on ∂Ω, (1.3) 224 J. Fern´ andez Bonder, J. D. Rossi where λ1is the principal eigenvalue and f,g ≥0 are bounded and locally smooth, we have Theorem 1.4. The problem (1.3) has a solution if and only if f≡0 on Ωand g≡0on ∂Ω. In this case, u=ku1where u1is an eigenfunction associated to λ1. Since λ1is isolated in the spectrum and there exists eigenvalues different from λ1, it make sense to define the second eigenvalue of (1.2) as ¯ λ2:= inf{λ∈R:λis an eigenvalue and λ>λ 1}. We denote by Kqthe best constant in the Sobolev trace embedding W1,p(Ω) →Lq(∂Ω) and set p∗=p(N−1)/(N−p) the critical Sobolev exponent. Observe that Kp=Spif V≡1. Concerning the second eigenvalue, we have the following result Theorem 1.5. The eigenvalue λ2found in Theorem 1.1 coincides with ¯ λ2. In particular, ¯ λ2is an eigenvalue for (1.2). Moreover, it holds the following variational characterization of ¯ λ2=λ2, ¯ λ2=λ2= inf u∈AΩ |∇u|p+|u|pdx, where A={u∈W1,p(Ω); uLp V(∂Ω) =1and |∂Ω±|≥c},ifs>1or 1<p≤Nand A={u∈W1,p(Ω); uLp V(∂Ω) =1and ∂Ω±V(x)dσ ≥ c},ifs=1with p>N. Here ∂Ω+=∂Ω∩{u>0},∂Ω−=∂Ω∩{u<0} and c=(K−1 p∗λ1VLs(∂Ω))−γor c=K∞/λ1respectively. We want to remark that this last result was not known to hold in the case V≡1 and is the main result in this paper. The rest of the paper is organized as follows. In Section 2, we deal with the existence of a sequence of eigenvalues and prove Theorem 1.1. Next, in Section 3, we study the first eigenvalue and prove Theorems 1.2, 1.3 and the nonexistence result, Theorem 1.4. Finally, in Section 4 we prove Theorem 1.5. 2. Existence of {λ k } The proof is a rather straightforward adaptation of Theorem 1.3 in [13] where problem (1.2) with V≡1 is considered, so we only make a sketch in order to make the paper self contained. We introduce a topological tool, the genus. Given a Banach space X, we consider the class Σ={A⊂X:Ais closed, A=−A}. Nonlinear Eigenvalue Problem 225 Over this class we define the genus, γ:Σ→N∪ {∞},as γ(A) = min{k∈N: there exists ϕ∈C(A, Rk−{0}),ϕ(x)=−ϕ(−x)}. For the properties of the genus and some applications we refer to [16]. Let us consider M={u∈W1,p(Ω) : up W1,p(Ω) =p}and ϕ(u)=1 p∂Ω |u|pV(x)dσ. We are looking for critical points of ϕrestricted to the manifold Musing a minimax technique. First we observe that ϕsatisfies the Palais-Smale condition on M. Recall that ϕsatisfies the Palais-Smale condition on M, means that if (uj)⊂Mis a Palais-Smale sequence (i.e. ϕ(uj)→Cand ϕ(uj)→0) then there exists a convergent subsequence (ujk). Our functional ϕverifies the Palais-Smale condition on Mfor Palais-Smale sequences above a positive value. We state this as a lemma for future reference. Lemma 2.1. Let β>0and (uj)⊂Mbe a Palais-Smale sequence on M above level β. Then there exists a subsequence that converges strongly in W1,p(Ω). Proof: See [13]. Now we seek for critical values of ϕ. Theorem 2.1. Let Ck={C⊂M:Cis compact, symmetric and γ(C)≤k}and let βk= sup C∈Ck min u∈Cϕ(u).(2.1) Then βk>0, there exists uk∈Msuch that ϕ(uk)=βkand ukis a weak solution of (1.2) with λk=1/βk. Moreover limkβk=0and hence limkλk=+∞. Proof: First, let us see that βk>0. It is immediate that γ(M)=+∞, hence βkis well defined in the sense that for every k,Ck=∅. As we can choose a set C∈Ckwith the property ∂Ω|u|pV(x)dσ =0ifu∈C, we conclude that βk= supC∈Ckminu∈Cϕ(u)>0. Now, for a fixed k let us prove the existence of the solution uk. By a standard deformation argument we can assume that there exists a sequence (uj)∈Msuch that ϕ(uj)→βkand ϕ(uj)→0, see [13] for the details. Now, from Lemma 2.1 we can extract a converging subsequence uj→ukthat gives us the desired solution that must verify, by the continuity of ϕ,ϕ(uk)= βk. 226 J. Fern´ andez Bonder, J. D. Rossi Let us see that limkβk= 0. Let Ejbe a sequence of subspaces of W1,p(Ω), such that Ei⊂Ei+1,∪Ei=W1,p(Ω) and dim(Ei)=i. Let Ec ia topological complementary of Ei. Let ˜ βk= sup C∈Ck min u∈C∩Ec k−1 ϕ(u). ˜ βkis well defined and ˜ βk≥βk>0. Let us prove that limk˜ βk= 0. Assume, by contradiction, that there exists a constant κ>0 such that ˜ βk>κ>0 for all k. Then for every kthere exists Cksuch that ˜ βk>min u∈Ck∩Ec k−1 ϕ(u)>κ. Hence there exists uk∈Ck∩Ec k−1with ˜ βk>ϕ(uk)>κ.AsMis bounded, we can assume, taking a subsequence if necessary, that uk$u weakly in W1,p(Ω) and uk→ustrongly in Lp(∂Ω). Hence ϕ(u)≥ κ>0 but this is a contradiction with the fact that u≡0 because uk∈Ec k−1. 3. Simplicity, isolation and monotonicity of λ1 In this section we prove Theorems 1.2, 1.3 and 1.4. First we deal with Theorem 1.2 and we divide the proof in a series of lemmas in order to clarify the exposition. Then we deal with Theorem 1.3 and we end this section with the proof of the nonexistence result, Theorem 1.4. Observe that solutions of (1.2), by a well known fact, belong to C1,α loc (Ω) (see [18], [11], etc.) but, as far as we know, with Vunder the hypotheses (1.1) this regularity is not known to hold up to the boundary. First we prove that eigenfunctions associated to λ1must have definite sign. Lemma 3.1. Eigenfunctions associated to λ1are either positive or negative in Ω. Moreover if u∈C1,α(Ω) then uhas definite sign in Ω. Proof: Let ube an eigenfunction associated to λ1. Since uW1,p(Ω) = |u|W1,p(Ω) and uLp V(∂Ω) =|u|Lp V(∂Ω), from the variational characterization of λ1given by (2.1), it follows that |u|is also an eigenfunction associated to λ1. By the strong maximum principle, see [19], or using Harnack inequality, see [17], it follows that |u|>0 in Ω, therefore either u>0oru<0 in Ω and so u≥0oru≤0inΩ. If u∈C1,α(Ω), assume that there exists x0∈∂Ω such that |u(x0)|= 0. By Hopf’s Lemma, [19], we have that ∂|u| ∂ν (x0)<0, but the boundary condition impose ∂|u| ∂ν (x0) = 0, a contradiction. So |u|>0inΩ and the result follows. Nonlinear Eigenvalue Problem 227 For the proof of the simplicity of λ1we use the following “Piccone’s identity” proved in [1]. Lemma 3.2 ([1, Theorem 1.1]).Let v>0,u≥0be two continuous functions in Ωdifferentiable a.e. Denote L(u, v)=|∇u|p+(p−1)up vp|∇v|p−pup−1 vp−1|∇v|p−2∇v∇u, R(u, v)=|∇u|p−|∇v|p−2∇up vp−1∇v. Then (i) L(u, v)=R(u, v),(ii) L(u, v)≥0a.e. and (iii) L(u, v)=0 a.e. in Ωif and only if u=kv for some k∈R. Now, we define a nodal domain Nof a function uas the closure of a connected component of Ω \{u=0}. In the next result we give an estimate on the measure of N∩∂Ω for an eigenfunction u. Recall that p∗=p(N−1)/(N−p) is the critical Sobolev exponent. Proposition 3.1. Any eigenfunction uassociated to a positive eigenvalue 0<λ=λ1changes sign on the boundary. Moreover, if Nis a nodal domain of uthen |N ∩ ∂Ω|≥(K−1 p∗λVLs(∂Ω))−γ,(3.2) where γ=s(N−1) sp−Nif 1<p≤Nand γ=2sif p>N,s>1.Ifp>N and s=1we get N∩∂Ω |V(x)|dσ ≥K∞ λ.(3.3) In particular, if ∂Ω+={x∈∂Ω:u(x)>0}and ∂Ω−={x∈∂Ω: u(x)<0}then |∂Ω+|≥cλ,|∂Ω−|≥cλ,(3.4) or ∂Ω+ |V(x)|dσ ≥cλ,∂Ω− |V(x)|dσ ≥cλ,(3.5) where cλ=(K−1 p∗λVLs(∂Ω))−γor cλ=K∞ λrespectively. Here Kqis the best constant in the Sobolev trace embedding W1,p(Ω)→ Lq(∂Ω) and |A|denotes the (N−1)-dimensional measure of a subset A⊂∂Ω. 228 J. Fern´ andez Bonder, J. D. Rossi Proof: Assume by contradiction that u≥0 (if u≤0 the argument is analogous). Arguing as in Lemma 3.1, it follows that u>0inΩ. Let ϕ>0 be an eigenfunction associated to λ1and ε>0. We apply Piccone’s identity to the pair ϕ,u+ε. We have 0≤Ω L(ϕ, u +ε)dx =Ω R(ϕ, u +ε)dx ≤λ1∂Ω ϕpV(x)dσ−Ω ϕpdx−Ω |∇u|p−2∇u∇ϕp (u+ε)p−1dx. (3.6) As ϕp (u+ε)p−1∈W1,p(Ω), it is admissible in the weak formulation of u. Then from (3.6) it follows that 0≤∂Ωλ1−λup−1 (u+ε)p−1ϕpV(x)dσ. Letting ε→0weget 0≤∂Ω (λ1−λ)ϕpV(x)dσ, which is impossible, as λ>λ 1and ∂ΩϕpV(x)dσ =ϕp W1,p(Ω)/λ1>0. Therefore, uchanges sign. For the second part, in the case 1 <p<N, we consider w(x)=u(x) if x∈N and 0 otherwise, then w∈W1,p(Ω) and if we use win the weak formulation of u, we get N |∇u|p+|u|pdx =λN∩∂Ω |u|pV(x)dσ ≤λVLs(∂Ω)up Lsp(N∩∂Ω) ≤λVLs(∂Ω)up Lp∗(N∩∂Ω)|N ∩ ∂Ω|p∗−sp sp∗ by H¨older inequality, where p∗=p(N−1) N−pis the critical exponent in the Sobolev trace imbedding Theorem. Now, by the Sobolev trace embedding Theorem, there exists a constant Kp∗=Kp∗(N,p,Ω) such that Kp∗up Lp∗(N∩∂Ω) =Kp∗wp Lp∗(∂Ω) ≤Ω |∇w|p+|w|pdx =N |∇u|p+|u|pdx. Hence, Kp∗≤λVLs(∂Ω)|N ∩ ∂Ω| p∗−s(p−1) sp∗ and the proposition follows. The cases where p≥Nand s>1 can be handled in a similar fashion. Nonlinear Eigenvalue Problem 229 For the case p>N,s= 1, we proceed as before to obtain up W1,p(N)≤λup L∞(N∩∂Ω) N∩∂Ω |V(x)|dσ. Therefore (3.3) follows. The proof is complete. As an easy consequence of Proposition 3.1 we get the following Corollary 3.1. Let (λ, u)be an eigenpair of (1.2) with λ>λ 1. Then the number of nodal domains of uis finite. Next, we make use of Piccone’s identity to prove the simplicity of λ1. Proposition 3.2. λ1is simple. Proof: We argue similarly as in Proposition 3.1. Let u,vbe two eigenfunctions associated to λ1. We can assume that uand vare both positive in Ω. We apply Piccone’s identity to the pair u,v+εand obtain 0≤Ω L(u, v +ε)dx =Ω R(u, v +ε)dx =λ1∂Ω upV(x)dσ −Ω updx −Ω |∇v|p−2∇v∇up (v+ε)p−1dx. Since the function up (v+ε)p−1∈W1,p(Ω), it is admissible in the weak formulation of v. It follows, arguing as in Proposition 3.1, that 0≤Ω L(u, v +ε)dx ≤λ1∂Ω up(1 −vp−1 (v+ε)p−1)V(x)dσ. Letting ε→0, we obtain Ω L(u, v)dx =0, but then, L(u, v) = 0 and by Theorem 3.2, there exists k∈Rsuch that u=kv. To end the proof of Theorem 1.2, we need a lemma from [13]. Lemma 3.3 ([13, Lemma 2.1]).Let φ∈W1,p(Ω), where W1,p(Ω)denotes the dual space of W1,p(Ω). Then there exists a unique weak solution u∈W1,p(Ω) of −∆pu+|u|p−2u=φ. Moreover, the operator Ap:φ→ uis continuous. Now we can prove, Proposition 3.3. λ1is isolated, that is, there exists δ>0such that there is no other eigenvalue of (1.2) in the interval (λ1,λ 1+δ).