Abstract

Using an Orlicz-Sobolev Space setting, we consider an eigenvalue problem for a system of the form {Δ φ1u = λ(a1(x,u)+b(x) γ1(u)Γ2(v)) in Ω, -Δφ2 v = λ(a2(x, v) + b(x)Γ1(u)γ2(u)) in Ω, u = v =0 on ∂Ω. We prove that the solution to a suitable minimizing problem, with a restriction, yields a solution to this problem for a certain λ. The differential operators involved lack homogeneity and in addition the Orlicz-Sobolev spaces needed may not be reflexive and the corresponding functional in the minimization problem is in general neither everywhere defined nor a fortiori C 1. © Springer Science+Business Media, LLC 2006.