An Asymmetric Steklov Problem With Weights: the singular case

abstract: We prove the existence of a first nonprincipal eigenvalue for an asymmetric Steklov problem with weights. We are interested in the singular case (in where one of the weights has meanvalue zero), this case requires some special attention in connection with the Palais Smale (PS) conditions and with the mountain pass geometry.


Introduction
Let Ω ⊂ R N (N ≥ 2) be a bounded domain with a Lipschitz continuous boundary. Let 1 < p < ∞ and let N −1 p−1 < q < ∞ if p < N and q ≥ 1 if p ≥ N . m, n ∈ L q (∂Ω) with m + = 0 and n + = 0. The asymmetric Steklov problem is defined by where λ ∈ R + is the eigenvalue, u ∈ W 1,p (Ω) is an associated eigenvalue and ν is the unit exterior normal. The solutions of (1) or of related equations are always understood in the weak sense, i.e., u ∈ W 1,p (Ω) with where dσ is the N − 1 dimensional Hausdorff measure. In a previous work (see [1]), we proved the existence of a first nonprincipal eigenvalue for (1) are ∂Ω mdσ = 0 and ∂Ω ndσ = 0 by applying a version of the mountain pass theorem to the functional f (u) = 1 p Ω |∇u| p dx restricted to the manifold in this case (P S) condition is satisfied and the geometry of the mountain pass was derived from observation that ϕ m and ϕ n where strict local minima (ϕ m denotes the positive first eigenvalue of (1) with m = n). Our purpose in this work is to prove the existence of a first nonprincipal eigenvalue for (1) where ∂Ω mdσ = 0 or ∂Ω ndσ = 0. In this case the Palais Smale condition is not satisfied any more at level 0 and at least one of the two naturals candidates for local minimum fails to belong to the manifold M m,n . To by pass this difficulty we apply a version of the mountain pass theorem for a local C 1 functional restricted to a C 1 manifold and which satisfies the Palais-Smale condition of Cerami (P SC) at certain levels(see [2]).

Preliminaries
Our main purpose in this preliminaries section, is to collect some results relative to the following eigenvalue problem Clearly 0 is a principal eigenvalue of (3) with the constants as eigenfunctions. The search for another principal eigenvalue involves the following quantity we have λ * 1 (m) < ∞ since m + = 0 in Ω. Proposition 2. 1 1. If Ω mdσ < 0. Then λ * 1 (m) > 0 is the first positive Steklov eigenvalue. Moreover λ * 1 (m) is simple and isolated and it is the only nonzero Steklov eigenvalue associated to an eigenfunction of definite sign .
3. If Ω mdσ = 0. Then λ * 1 (m) = 0 and 0 is the unique principal eigenvalue. Proposition 2.1 is proved in [4] (see also [1]). In case 1 or 2 of Proposition 2.1, the infimum is achieved at ϕ m ∈ M m,n the positive eigenfunction associated to λ * 1 (m) with 1 p ∂Ω mϕ p m = 1. In the case 3 the fact that λ * 1 (m) = 0 is easily verified by considering the sequence where ψ is any fixed smooth function with ψ ≥ 0 and ∂Ω mψ > 0. Note that in case 3 of Proposition 2.1, the infimum in (4) is not achieved (since no constant satisfies the constraint in that case). To get a first nonprincipal eigenvalue for an asymmetric Steklov problem with weights, we will use a version of the mountain pass theorem on C 1 manifold, which we now recall. Let E be a real Banach space and let M := {u ∈ E; g(u) = 1}, where g ∈ C 1 (E, R) and 1 is a regular value of g.
Let f ∈ C 1 (E, R) and consider the restrictionf of f to M .

Proposition 2.2 ( [3]) Assumef bounded from below and let
Thenf satisfies (P SC) c if and only iff satisfies (P S) c .
Remark 2.1 Going back to case 3 of Proposition (2.1), one can see that the functional f (u) = 1 p Ω |∇u| p dx restricted to the manifold M m,n does not satisfy the (P S) 0 . Indeed the sequence v k from (5) provides an unbounded (P S) sequence. That the (P SC) 0 condition does not hold neither will follow from Proposition 2.2.
Assume thatf satisfies (P SC) c for c given in (6). Then c is a critical value off .

A first nontrivial eigenvalue
The assumptions on m, n in this section are m, n ∈ L q (∂Ω) with ∂Ω m = 0 or ∂Ω n = 0 and m + = 0, n + = 0. We look for nonnegative eigenvalues λ of (1). Clearly the only nonnegative principal eigenvalues of (1) are 0, λ * (m) and λ * (n). Moreover multiplying by u + or u − one easily sees that if (1) with λ ≥ 0 has a solution which changes sign then λ > max(λ * (m), λ * (n)). Proving the existence of such a solution which changes sign and which in addition corresponds to a minimum value of λ is our purpose in this section. We will use a variational approach and consider the functional f (u) = 1 p Ω |∇u| p dx on E = W 1,p (Ω), the manifold M m,n defined in introduction and the restrictionf of f to M m,n . To state our main result let us introduce the following family of paths in M m,n Γ = {γ ∈ C([0, 1], M m,n ) : γ(0) ≤ 0 and γ(1) ≥ 0}, which is nonempty (see [1]), and the finite minimax value Theorem 3.1 Assume ∂Ω m = 0 or ∂Ω n = 0. Then c(m, n) is an eigenvalue of (1) which satisfies max{λ * 1 (m), λ * 1 (n)} < c(m, n). Moreover there is no eigenvalue of (1) between max{λ * 1 (m), λ * 1 (n)} and c(m, n).
The rest of this section is devoted to the proof of Theorem 3.1.
Proposition 3.1f satisfies (P SC) c for all c > 0.
Proof. Let u k ∈ M m,n be a (P SC) c sequence forf , with c > 0.
where ǫ k → 0. We will show that u k remains bounded and concludes that u k admits a convergent subsequence. Let us assume by contradiction that for a subsequence, Since Ω |∇u k | p dx remains bounded, one has Ω |∇v k | p dx → 0 and it follows easily that v 0 ≡ cst = 0 and that v k → v 0 strongly in W 1,p (Ω). On the other hand, taking ξ = a k (ω) : where ω ∈ W 1,p (Ω) and dividing by u k p−1 , one gets By passing to the limit, we implies that v 0 is a solution of where c is the level appearing in the (P SC) c sequence. Since v 0 ≡ cst, the righthand side of (9) is ≡ 0, and since c > 0, one gets m(v + 0 ) p−1 − n(v − 0 ) p−1 ≡ 0. This relation with a nonzero constant v 0 implies m ≡ 0 or n ≡ 0, which contradicts m + ≡ 0 and n + ≡ 0. Thus u k remains bounded, for a subsequence, u k → u 0 weakly in W 1,p (Ω). Taking ξ = a k (ω) in (8), one deduces for some constant D; taking now ω = u k − u 0 in the above, one obtains Since Ω |u k | p−2 u k (u k − u 0 ) → 0, it then follows from the (S + ) property that u k → u 0 strongly in W 1,p (Ω), which yields the conclusion. 2 We now turn to the geometry off . The situation here is again simpler in the non singular case (see Preliminaries). We start by giving an important proposition, which is proved in [1]. Proposition 3.2 If ∂Ω mdσ = 0, then ϕ m ∈ M m,n is a strict local minimum of f , with in addition for some ǫ 0 > 0 and all 0 < ǫ < ǫ 0 , where B(ϕ m , ǫ) denotes the ball in W 1,p (Ω) of center ϕ m and radius ǫ. Similar conclusion for −ϕ n if ∂Ω ndσ = 0.