In this article, we study the existence of solution for the following elliptic system of variable exponents with perturbation terms − div |∇u| p(x)−2∇u) + |u| p(x)−2u = λa(x)|u| γ(x)−2u + Fu(x, u, v) in R N , − div |∇v| q(x)−2∇v) + |v| q(x)−2 v = λb(x)|v| δ(x)−2 v + Fv(x, u, v) in R N , u ∈ W1,p(·) (R N ), v ∈ W1,q(·) (R N ), where the corresponding functional does not satisfy PS conditions. We obtain a sufficient condition for the existence of solution and also present a result on asymptotic behavior of solutions at infinity.