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.