Existence of solutions for a resonant Steklov Problem

abstract: In this paper, we prove the existence of weak solutions to the problem △pu = 0 in Ω and |∇u| ∂u ∂ν = λ1m(x)|u|u + f(x, u) − h on ∂Ω, where Ω is a bounded domain in R (N ≥ 2), m ∈ L(∂Ω) is a weight, λ1 is the first positive eigenvalue for the eigenvalue Steklov problem △pu = 0 in Ω and |∇u| ∂u ∂ν = λm(x)|u|u on ∂Ω. f and h are functions that satisfy some conditions.

Ω will be a bounded domain in R N (N ≥ 2), with a Lipschitz continuous boundary, 1 < p < ∞, m ∈ L q (∂Ω) where N −1 p−1 < q < ∞ if p < N and q ≥ 1 if p ≥ N .We assume that m + = max(m, 0) ≡ 0 and ∂Ω mdσ < 0. f : ∂Ω × R →R is a Carathéodory function satisfying the growth condition for all s ∈ R and a.e.x ∈ ∂Ω.Here a = cst > 0, b ∈ L r ′ (∂Ω) and h ∈ L r ′ (∂Ω), where r ′ is the conjugate of r = pq q−1 .λ 1 design the first positive eigenvalue of the following Steklov problem It is well-known that Recall that λ 1 is simple (see [7]).Moreover, there exists a unique positive eigenfunction ϕ 1 whose norm in W 1,p (Ω) equals to one.We say that u ∈ W 1,p (Ω) is a weak solution of (1) if , where dσ is the N − 1 dimensional Hausdorff measure.Classical Dirichlet problems involving the p-Laplacian have been studied by various authors, we cite the works [1], [2], [3], [4], [5], [6] and [8] .Our purpose of this paper is to extend some of the results known in the Dirichlet p-Laplacian problem.We prove the existence of solutions for a resonant Steklov problem under Landesman-Lazer conditions.

Existence of solutions for a resonant Steklov problem
In this section, we study the solvability of the Steklov problem (1) under Landesman-Lazer conditions and by using the minimum principle.The following theorem is our main ingredient.
Theorem 2.1 (Minimum principle) Let X be a Banach space and Φ ∈ C 1 (X, R).Assume that Φ satisfies the Palais-Smale condition and bounded from below.Then c = inf X Φ is a critical point.

Suppose that f satisfies the hypotheses below
where ϕ 1 is the normalized positive eigenfunction associated to λ 1 .
The following theorem is main result in this paper.
The following lemmas will be used in the proof of Theorem 2.2, it guarantees the existence of a critical point.The functional energy associated to the problem (1) is where Resonant Steklov Problem 87 Lemma 2.1 Let m ∈ L q (∂Ω), m + = 0 and ∂Ω mdσ < 0. Assume (4) and (5) are fulfilled.Then Φ satisfies the Palais-Smale condition (PS) on W 1,p (Ω).
Proof: Let (u n ) be a sequences in W 1,p (Ω) and c be a real number such that |Φ(u n )| ≤ c for all n and Φ ′ (u n ) → 0. We prove that (u n ) is bounded in W 1,p (Ω), we assume by contradiction that ||u n || → +∞ as n → +∞.Let v n = un ||un|| , thus v n is bounded, for a subsequence still denoted by Since, by hypotheses on p, h, u n and using ( 4) Using the weak lower semi-continuity of norm and the definition of λ 1 , we get Thus, v n → v strongly in W 1,p (Ω) and This implies, by the definition of ϕ 1 , that v = ±ϕ 1 (since ∂Ω mdσ < 0).Letting Therefore, the Lebesgue theorem implies that (6) and Φ ′ (u n ) → 0 implies that for all ε > 0 there exists n 0 ∈ N such that for all n ≥ n 0 , we have By summing up ( 6) and ( 7), we get Passing to the limit, we obtain which contradicts (5).Case 2: Suppose that v n → −ϕ 1 , then we have u n (x) → −∞ and f (x, u n (x)) → l(x) a.e.x ∈ ∂Ω, g(x, u n (x)) → l(x) a.e.x ∈ ∂Ω.
By summing up ( 6) and ( 7), we get Passing to the limits, we get which contradicts (5).Finally, (u n ) is bounded in W 1,p (Ω), for a subsequences still denoted by (u n ), there exists u ∈ W 1,p (Ω) such that u n ⇀ u weakly in W 1,p (Ω) and u n → u strongly in L pq q−1 (∂Ω).By the hypotheses on m, h, u n and using (4), we deduce that On the other hand, we have it then follows from the (S + ) property that u n → u strongly in W 1,p (Ω). 2 Lemma 2.2 Let m ∈ L q (∂Ω), m + = 0 and ∂Ω mdσ < 0. Assume (4) and (5) are fulfilled.Then Φ is bounded from below.
Proof: It suffices to show that Φ is coercive.Suppose by contradiction that there exists a sequence (u n ) such that ||u n || → +∞ and Φ(u n ) ≤ c.As in proof of Lemma 2.1, we can show that v n = un ||un|| → ±ϕ 1 .By the definition of λ 1 , we have