Existence results for a class of ( p , q ) Laplacian systems

where 1 < p, q < N and Ω is a regular set of R , N ≥ 3, α > 0, β > 0, λ and μ are positive parameters, functions a(x), b(x) and d(x) ∈ C(Ω) are smooth functions with change sign on Ω, we assume here that 1 < γ < min(p, q), γ < α + β, α + β > max(p, q) and α/p + β/q = 1. For p ≥ 1 ∆pu is the p-Laplacian defined by ∆pu = div(|∇u|p−2∇u) and W 1,p 0 (Ω) is the closer of C∞ 0 (Ω) equipped by the norm ‖u‖1,p := ‖∇u‖p, where ‖.‖p represent the norm of Lebesgue space L(Ω). The Lebesgue integral in Ω will be denote by the symbol ∫ whenever the integration is carried out over all Ω. Let p′ be the conjugate to p,W ′ 0 (Ω) is the dual space toW 1,p 0 (Ω) and we denote by ‖.‖−1,p′ its norm. We denote by 〈x∗, x〉X∗,X the natural duality paring betweenX and


Introduction
In this paper we deal with the nonlinear elliptic system where 1 < p, q < N and Ω is a regular set of R N , N ≥ 3, α > 0, β > 0, λ and µ are positive parameters, functions a(x), b(x) and d(x) ∈ C(Ω) are smooth functions with change sign on Ω, we assume here that 1 < γ < min(p, q), γ < α + β, α + β > max(p, q) and α/p + β/q = 1.For p ≥ 1 ∆ p u is the p-Laplacian defined by ∆ p u = div(|∇u| p−2 ∇u) and W 1,p 0 (Ω) is the closer of C ∞ 0 (Ω) equipped by the norm u 1,p := ∇u p , where .p represent the norm of Lebesgue space L p (Ω).The Lebesgue integral in Ω will be denote by the symbol whenever the integration is carried out over all Ω.
Let p be the conjugate to p, W −1,p 0 (Ω) is the dual space to W 1,p 0 (Ω) and we denote by .−1,p its norm.We denote by x * , x X * ,X the natural duality paring between X and where (Ω) has been studied in [1] for p = q and in a recent paper [2] for p = q on arbitrary domains with lack of compactness.
It is clear that problem (1) has a variational structure.
It is well known if the Euler function φ is bounded below and φ has a minimizer on X, then this minimizer is a critical point of φ.However, the Euler function φ(u, v), associated with the problem (1), is not bounded below on the whole space X, but is bounded on an appropriate subset, and has a minimizer on this set (if it exists) which gives rise to solution to (1).Clearly, the critical points of φ are the weak solutions of problem (1).
We introduce the operators J 1 , J 2 , D 1 , D 2 , B 1 , B 2 : X → X * in the following way

Main results
Our main result is the following: (Ω), non of the functions f and g is identically to zero on Ω and: Then, there exists a pair (u * , v * ) ∈ Λ such that the sequence (u n , v n ) converges strongly to (u * , v * ) in X, Moreover, (u * , v * ) is a solution of system (1) satisfies the property φ(u * , v * ) < 0. Definition 2. We say that φ satisfies the Palais-Smale condition (P S) c if every sequence Proof.This lemma is proved in [3].
Proof.Since the sequence (u n , v n ) is bounded in X, we may consider that there is a subsequence (denote again by (u n , v n )), which is weakly convergent in X.
Lemma 3. Let c ∈ R. Then the functional φ(u, v) satisfies the (P S) c condition.
Proof.According to Lemma 2, it sufficient to prove that the sequence Using successively the Holder's inequality and the Young inequality on the terms f, u n and g, v n , we can write Since the real numbers θ and ν being arbitrary, a suitable choose of θ and ν assure the boundedness of the sequence (u n , v n ).
Lemma 4. The critical value of φ on Λ, m 1 = inf (u,v)∈Λ φ(u, v), has the following property: Proof.Let u f be the unique solution of the Dirichlet problem and let v g be the unique solution of the problem It is clear that (u f , 0), (0, v g ) are two elements of Λ and we have Similarly to proof of J. Velin [13, 4.2], we can show that Thus, the Lemma is proved.