In this paper, we consider the system {{{{{{{{{{{{{{{{{{{ −M1(∫ Ω |∇u|p(x) + |u|p(x) p(x) dx)(Δp(x)u − |u|p(x)−2u) = λa(x)|u|r1(x)−2u − μb(x)|u|α(x)−2u in Ω, −M2(∫ Ω |∇v|q(x) + |v|q(x) q(x) dx)(Δq(x)v − |v|q(x)−2v) = λc(x)|v|r2(x)−2v − μd(x)|v|β(x)−2v in Ω, u = v = 0 on ∂Ω, where Ω is a bounded domain in ℝN (N ≥ 2) with a smooth boundary ∂Ω, M1(t), M2(t) are continuous functions and λ, μ > 0. We prove that for any μ > 0 there exists λ∗ sufficiently small such that for any λ ∈ (0, λ∗) the above system has a nontrivial weak solution. The proof relies on some variational arguments based on Ekeland’s variational principle, and some adequate variational methods.
