Existence of minimizers for a quasilinear elliptic system of gradient type

The aim of this paper is to investigate the existence of weak solutions of the coupled quasilinear elliptic system of gradient type -div(a(x,u,∇u))+A_{t}(x,u,∇u)= g₁(x,u,v) in Ω, -div(b(x,v,∇v))+B_{t}(x,v,∇v) = g₂(x,u,v) in Ω u=v=0 on ∂Ω where Ω is an open bounded domain of R^N, N ≥2 and A(x,t,ξ), B(x, t,ξ) are C^1- Carathéodory functions on Ωx Rx R^N with partial derivatives A_{t} = ∂A/∂t, a = ∇_{ξ} A, respectively B_{t} = ∂B/∂t , b = ∇_{ξ} B, while g₁(x, t, s), g₂(x, t, s) are given Carathéodory maps defined on Ω x R x R which are partial derivatives with respect to t and s of a function G(x, t, s). We prove that, even if the general form of the terms A and B makes the variational approach more difficult, under suitable hypotheses, the functional related to the problem is bounded from below and attains its minimum in a "right" Banach space X. The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami-Palais-Smale condition and a suitable generalization of the Weierstrass Theorem.