Semi linear elliptic system at resonance

In this work, we investigate the existence of weak solutions for the following semi-linear elliptic system −∆u + p(x)u = αu + ϕ (x, v) in Ω, −∆v + q(x)v = βv + ψ (x, u) in Ω, with Dirichlet boundary condition, where Ω is a bounded open set of R N (N ≥ 2), α, β two real parameters, (p(x), q(x)) ∈ (L∞ (Ω))2 and p(x), q(x) ≥ 0. using the Leray-Schauder’s topological degree and under some suitable conditions for the non linearities ϕ and ψ, we show the existence of nontrivial solutions.