In this paper, we study the existence of weak solutions for Dirichlet boundary-value problems given in the following quasilinear elliptic system −div σ(x, u, Du) + b(x, u, Du) = f (x, u, Du) in Ω, u = 0 on ∂Ω. We prove the needed result, relying on the theory of Young measures, Galerkin's approximation and weak monotonicity assumptions on σ, in reflexive Orlicz-Sobolev spaces.