Existence for an elliptic system with nonlinear boundary conditions

In this paper we prove the existence of a weak solution to the following system Delta_p u = Delta_q v = 0  in Omega |nabla u|^{p-2}partial_{nu}u = f(x,u) - (alpha+1)K(x) |u|^{alpha-1}u |v|^{beta+1} + f_1  on partial Omega |nabla v|^{q-2}partial_{nu}v = g(x,u) - (beta+1)K(x) |v|^{beta-1}v |u|^{alpha+1}+ g_1  on partial Omega where  Omega is a bounded domain in R^N (N ≥ 2), f_1, g_1, f, g and K are functions that satisfy some conditions.


(g c,d
): There exist c > 0 and a function d ∈ L r (∂Ω) such that, x ∈ ∂Ω and for all s ∈ R.
Existence results for nonlinear elliptic systems when the nonlinear term appears as a source in the equation complemented with Dirichlet boundary conditions have been studied by various authors; we cite the works [1,3,4].For the nonlinear boundary condition, the authors in [2] proved the existence of nontrivial solutions to the system where (F u ; F v ) is the gradient of some positive potential The proofs are done under suitable assumptions on the potential F , and based on variational arguments.Our purpose in the present paper is to show that the problem (1.1) admits at least a solution (u, v) ∈ W 1,p (Ω) × W 1,q (Ω), we also give a special case of the problem (1.1) (see Corollary 3.3A).Our proofs are based on variational arguments.

Preliminaries
In this section, we collect some results relative to the eigenvalue problem where the weight m is assumed to lie in M p := {m ∈ L p(∂Ω); m + ≡ 0 and ∂Ω mdσ < 0}.
O. Torné in [5] showed, by using infinite dimensional Ljusternik-Schnirelman theory, that the problem (2.1) admits a sequence of eigenvalues where is the Krasnoselski genus of C, and let ) is a nondecreasing and unbounded sequence of positive eigenvalues of the problem (2.1).Moreover λ 1 (m, p) > 0 is the first positive eigenvalue of (2.1).Moreover λ 1 (m, p) is simple, isolated and it is the only nonzero eigenvalue associated to an eigenfunction of definite sign.
Remark 2.2 This theorem is proved in [5] by applying infinite dimensional Ljusternik-Schnirelman theory for existence of the sequence λ k (m, p) and Picone's identity for simplicity of the first eigenvalue.

Existence of solution for a system Steklov problem
In the whole continuation, we note by λ 1 (m, p) (resp.λ 1 (n, q)) the first eigenvalue of the problem (2.1) for the integer p and the weight m (resp.the integer q and the weight n).We also note Theorem 3.1 If m ∈ M p and n ∈ M q , then the problem (1.1) admits at least a solution for λ 1 (m, p) > 1 and λ 1 (n, q) > 1.
Remark 3.2 We can have λ 1 (m, p) > 1 , since λ 1 (m, p) is homogeneous respect to the weight in the sense where Consider the space W = W 1,p (Ω) × W 1,q (Ω) equipped with the norm where .
Let the energy functional Φ : Remark 3.3 The conditions lim sup |s|→+∞ pF (x,s) |s| p := m(x) imply that for all ε > 0, there exists d ε ∈ L r (∂Ω) such that a.e.x ∈ ∂Ω and for all s ∈ R, we have lim sup |s|→+∞ qG(x,s) |s| q := n(x) imply that for all ε > 0, there exists d ′ ε ∈ L r (∂Ω) such that a.e.x ∈ ∂Ω and for all s ∈ R, we have p ∂Ω m|u| p dσ is continuous in its two arguments (m, u), we deduce for k sufficiently large, Existence for an elliptic system 53 As ε is arbitrary, lim sup We distinguish two cases.First case: , for a subsequence still denoted by Dividing by ||u k || p 1,p and passing to the limit, we have , passing to limit, we obtain 1 p |cst| p ∂Ω mdσ = 0, this contradicts ∂Ω mdσ < 0 and cst = 0. Seconde case: (u k ) is bounded, for a subsequence still denoted (u k ), there exists Passing to the limit, we obtain On the other hand It then follows the (S + ) property that u k → u strongly in W 1,p (Ω).In addition u = 0 (since 1 p ∂Ω m|u| p dσ = 1).Thus λ is an eigenvalue of the problem (1.1).Since ∂Ω mdσ < 0, we deduce 0 < λ.Consequently, we have 0 < λ < λ 1 (m, p), this contradicts the Theorem 2.1.Finally, we have Lemma 3.1 If m ∈ M p and n ∈ M q , then the functional Φ is coercive for λ 1 (m, p) > 1 and λ 1 (n, q) > 1.
Proof: Suppose by contradiction that there exist a sequence w n ∈ W and c ≥ 0 ) and and (3.7) Existence for an elliptic system 55 Dividing (3.6) and (3.7) respectively by k p n and k q n , we obtain (3.9) Since ũn is a bounded, for a further subsequence still denoted by ũn ⇀ ũ weakly in W 1,p (Ω) and ũn → ũ strongly in L p (Ω), on the other hand we have ∂Ω n(x)dσ ≥ 0.
Proof: It suffices to see that the trace mapping W → L pp p−1 (∂Ω) × L qq q−1 (∂Ω) is compact. 2 Proof: [Proof of Theorem 2.1.]By Lemma 3.2, Φ is weakly lower semicontinuous and by Lemma 3.1, Φ is coercive.Φ is continuously differentiable.The proof is complete.2 Now we can give a special case of Theorem 2.1.