Existence of solutions for quasilinear elliptic systems in divergence form with variable growth

This paper is concerned with the following Dirichlet problem for a quasilinear elliptic system with variable growth: -div sigma(x, u(x), Du(x)) = f in Omega, u(x) = 0 on partial derivative Omega, where Omega subset of R-n is a bounded domain. By means of the Young measure and the theory of variable exponent Sobolev spaces, we obtain the existence of solutions in W-0(1,p(x)) (Omega, R-m) for each f is an element of (W-0(1,p(x)) (Omega, R-m))*.