A NONLINEAR EIGENVALUE PROBLEM WITH INDEFINITE WEIGHTS RELATED TO THE SOBOLEV TRACE EMBEDDING

In this paper we study the Sobolev trace embedding W 1,p(Ω) ↪→ LpV (∂Ω), where V is 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.


Introduction
Let Ω be a bounded smooth domain in R N and V : ∂Ω → R an indefinite weight.In this paper we consider the Sobolev trace embedding W 1,p (Ω) → L p V (∂Ω), where We only require mild integrability hypotheses on the weight V (x).More precisely, we assume V + ≡ 0 on ∂Ω and V ∈ L s (∂Ω), (1.1) where s > (N − 1)/(p − 1) if 1 < p ≤ N and s ≥ 1 if p > N.
Under these hypotheses on the weight V , this embedding is compact and therefore there exists a constant S p = S p (Ω, V ) such that the following inequality holds, Here and in what follows, we use the following norm in W 1,p (Ω): By the compactness of the embedding, we can prove (see Theorem 1.1) that there exists functions, usually called extremals, where the constant S p is attained.In fact, the extremals are weak solutions of (1.2) Here ∆ p u = div(|∇u| p−2 ∇u) is the p-Laplacian and ∂ ∂ν is the outer unit normal derivative.
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] so we 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.
The study of the eigenvalue problem −∆ p u = λ|u| p−2 u complemented 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 The proof of this theorem relies on the Ljusternik-Schnirelman critical point theory on C 1 manifolds using the genus, γ.We find the following variational characterization of a sequence of eigenvalues , where Regarding the first eigenvalue λ 1 , 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 λ 1 are in fact the extremals for the embedding W 1,p (Ω) → L p 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 = λ 1 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.
Related to λ 1 we have a nonexistence result.In fact, if we consider the equation where λ 1 is 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 Ω and g ≡ 0 on ∂Ω.In this case, u = ku 1 where u 1 is an eigenfunction associated to λ 1 .
Since λ 1 is isolated in the spectrum and there exists eigenvalues different from λ 1 , it make sense to define the second eigenvalue of (1.2) as λ2 := inf{λ ∈ R : λ is an eigenvalue and λ > λ 1 }.
We denote by K q the best constant in the Sobolev trace embedding W 1,p (Ω) → L q (∂Ω) and set p * = p(N − 1)/(N − p) the critical Sobolev exponent.Observe that Concerning the second eigenvalue, we have the following result Theorem 1.5.The eigenvalue λ 2 found in Theorem 1.1 coincides with λ2 .In particular, λ2 is an eigenvalue for (1.2).Moreover, it holds the following variational characterization of λ2 = λ 2 , where 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.

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 Over this class we define the genus, γ : Σ → N ∪ {∞}, as For the properties of the genus and some applications we refer to [16].
Let us consider We are looking for critical points of ϕ restricted to the manifold M using a minimax technique.First we observe that ϕ satisfies the Palais-Smale condition on M .Recall that ϕ satisfies the Palais-Smale condition on M , means that if (u j ) ⊂ M is a Palais-Smale sequence (i.e.ϕ(u j ) → C and ϕ (u j ) → 0) then there exists a convergent subsequence (u j k ).Our functional ϕ verifies the Palais-Smale condition on M for Palais-Smale sequences above a positive value.We state this as a lemma for future reference.
Lemma 2.1.Let β > 0 and (u j ) ⊂ M be a Palais-Smale sequence on M above level β.Then there exists a subsequence that converges strongly in W 1,p (Ω).
Proof: See [13].Now we seek for critical values of ϕ.

symmetric and γ(C) ≤ k} and let
Proof: First, let us see that β k > 0. It is immediate that γ(M ) = +∞, hence β k is well defined in the sense that for every k, C k = ∅.As we can choose a set C ∈ C k with the property ∂Ω |u| p V (x) dσ = 0 if u ∈ C, we conclude that β k = sup C∈C k min u∈C ϕ(u) > 0. Now, for a fixed k let us prove the existence of the solution u k .By a standard deformation argument we can assume that there exists a sequence (u j ) ∈ M such that ϕ(u j ) → β k and ϕ (u j ) → 0, see [13] for the details.Now, from Lemma 2.1 we can extract a converging subsequence u j → u k that gives us the desired solution that must verify, by the continuity of ϕ, ϕ βk is well defined and βk ≥ β k > 0. Let us prove that lim k βk = 0. Assume, by contradiction, that there exists a constant κ > 0 such that βk > κ > 0 for all k.Then for every k there exists C k such that βk > min Hence there exists As M is bounded, we can assume, taking a subsequence if necessary, that u k u weakly in W 1,p (Ω) and u k → u strongly in L p (∂Ω).Hence ϕ(u) ≥ κ > 0 but this is a contradiction with the fact that u ≡ 0 because

Simplicity, isolation and monotonicity of λ 1
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 C 1,α loc (Ω) (see [18], [11], etc.) but, as far as we know, with V under the hypotheses (1.1) this regularity is not known to hold up to the boundary.
First we prove that eigenfunctions associated to λ 1 must have definite sign.Proof: Let u be an eigenfunction associated to λ 1 .Since u W 1,p (Ω) = |u| W 1,p (Ω) and u L p V (∂Ω) = |u| L p V (∂Ω) , from the variational characterization of λ 1 given by (2.1), it follows that |u| is also an eigenfunction associated to λ 1 .By the strong maximum principle, see [19], or using Harnack inequality, see [17], it follows that |u| > 0 in Ω, therefore either u > 0 or u < 0 in Ω and so u ≥ 0 or u ≤ 0 in Ω.
For the proof of the simplicity of λ 1 we use the following "Piccone's identity" proved in [1].
Proof: Assume by contradiction that u ≥ 0 (if u ≤ 0 the argument is analogous).Arguing as in Lemma 3.1, it follows that u > 0 in Ω.Let ϕ > 0 be an eigenfunction associated to λ 1 and ε > 0. We apply Piccone's identity to the pair ϕ, u + ε.We have For the second part, in the case 1 < p < N, we consider w(x) = u(x) if x ∈ N and 0 otherwise, then w ∈ W 1,p (Ω) and if we use w in the weak formulation of u, we get by Hölder inequality, where p * = p(N −1) N −p is the critical exponent in the Sobolev trace imbedding Theorem.Now, by the Sobolev trace embedding Theorem, there exists a constant K p * = K p * (N, p, Ω) such that Hence, and the proposition follows.The cases where p ≥ N and s > 1 can be handled in a similar fashion.
For the case p > N, s = 1, we proceed as before to obtain 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 u is finite.
Next, we make use of Piccone's identity to prove the simplicity of λ 1 .
Proof: We argue similarly as in Proposition 3.1.Let u, v be two eigenfunctions associated to λ 1 .We can assume that u and v are both positive in Ω.We apply Piccone's identity to the pair u, v + ε and obtain

Since the function
but then, L(u, v) = 0 and by Theorem 3.2, there exists k ∈ R such that u = kv.
To end the proof of Theorem 1.2, we need a lemma from [13].
Proof: The result is a consequence of estimate (3.4) or (3.5).Suppose that the statement of the proposition is false.Then there exists a sequence of eigenvalues λ n λ 1 .Let u n be an eigenfunction associated to λ n and we can assume that ∂Ω |u n | p V (x) dσ = 1.Now, by (1.2), it is easy to see that u n is bounded in W 1,p (Ω), so there exists a subsequence (that we still denote u n ) and a function u ∈ W 1,p (Ω) such that u n → u in L p (∂Ω) and weakly in W 1,p (Ω).Moreover, if we define φ n , φ as for every v ∈ W 1,p (Ω), we get that, by Hölder inequality, φ n , φ ∈ W 1,p (Ω) and φ n → φ in W 1,p (Ω) .By the continuity of the operator A p given by Lemma 3.3, we get that the sequence u n converges strongly in W 1,p (Ω), therefore, passing to the limit in the weak formulation of u n we deduce that u is an eigenfunction of (1.2) associated to λ 1 .We can assume, by Lemma 3.1, that u > 0 (the case u < 0 is analogous).Then, from the fact that u n → u in L p (∂Ω), it follows that u n → u in measure, so |∂Ω − n | → 0, but this contradicts either estimate (3.4) or estimate (3.5).
Next we prove Theorem 1.3, that shows the monotonicity of the first eigenvalue with respect to the weight.
Proof of Theorem 1.3: Let u 1 be an eigenfunction associated to the first eigenvalue of the weight V 1 .Then , and the proof is finished.
We end this section with the proof of Theorem 1.4, a nonexistence result for problem (1.3).
Proof of Theorem 1.4:The 'if' part is immediate.Let u be a solution of (1.3), therefore If we choose φ = u − , we have This implies that either u − = 0 or In the latter case, we have that u − is a multiple of the principal eigenfunction, so u − > 0 in Ω and so u = −u − and the proof is complete.
Suppose now that u − = 0 then u = u + and then by the maximum principle (see [17], [19]), u > 0 in Ω. Applying now Piccone's identity to the pair u 1 , u + ε we get which is a contradiction unless f ≡ 0 in Ω, g ≡ 0 on ∂Ω.

Variational characterization of the second eigenvalue
This section is concerned with the study of the second eigenvalue.Let us recall that, as λ 1 is isolated, it make sense to define λ2 := inf{λ ∈ R : λ is an eigenvalue of (1.2) and λ > λ 1 }.
It can also be checked, arguing exactly as in the first part of the proof of Proposition 3.3 that λ2 is an eigenvalue of (1.2), i.e. the infimum is achieved.
The goal of this section is to show that this eigenvalue λ2 coincides with the second variational eigenvalue λ 2 found in Theorem 1.1.For this purpose, first we give a variational characterization of λ2 .
Proof of Theorem 1.5: We deal with the case s > 1 or 1 < p ≤ N .The arguments for the case s = 1 with p > N are analogous and we leave the details to the reader.Let us call It is easy to check that µ ≤ λ2 , because if we take u 2 an eigenfunction of (1.2) associated with λ2 such that ∂Ω |u| p V (x) dσ = 1, by Proposition 3.1, we have that u 2 is admissible in the variational characterization of µ and the claim follows.Hence, the proof will follows if we show that µ ≥ λ 2 .It is easy to check that This last equation (4.9) together with (4.8) imply the desired result.

Proposition 3 . 1 .
) ≥ 0 a.e. and (iii) L(u, v) = 0 a.e. in Ω if and only if u = kv for some k ∈ R. Now, we define a nodal domain N of a function u as 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.Any eigenfunction u associated to a positive eigenvalue 0 < λ = λ 1 changes sign on the boundary.Moreover, if N is a nodal domain of u then