arXiv:0902.2507v1 [math.AP] 15 Feb 2009 EXISTENCE OF WEAK SOLUTIONS FOR NONLINEAR ELLIPTIC SYSTEMS INVOLVING THE (p(x),q(x))-LAPLACIAN MOUNIR HSINI Institut Pr´ eparatoire aux Etudes d’Ing´ enieurs de Tunis mounir.hsini@ipeit.rnu.tn Abstract. In this paper, we prove the existence of weak solutions for the following nonlinear elliptic system −Δ p(x) u = a(x)|u| p(x)-2 u − b(x)|u| α(x) |v| β(x) v + f (x) in Ω, −Δ q(x) v = c(x)|v| q(x)-2 v − d(x)|v| β(x) |u| α(x) u + g(x) in Ω, u = v =0 on ∂Ω, where Ω is an open bounded domains of R N with a smooth boundary ∂Ω and Δ p(x) denotes the p(x)-Laplacian.The existence of weak solutions is proved using the theory of monotone operators. Similar result will be proved when Ω= R N . 1. Introduction The purpose of this paper is to study the existence of weak solutions for the following nonlinear elliptic system involving the p(x)-Laplacian.    −Δ p(x) u = a(x)|u| p(x)−2 u − b(x)|u| α(x) |v| β(x) v + f (x) in Ω, −Δ q(x) v = c(x)|v| q(x)−2 v − d(x)|v| β(x) |u| α(x) u + g(x) in Ω, u = v =0 on ∂ Ω, (1) where Ω is an open bounded domains in R N with a smooth boundary ∂ Ω. The operator Δ p(x) u = div ( |∇u | p(x)−2 ∇u ) is called p(x)-Laplacian, which will be reduced to the p-Laplacian when p(x)= p a constant. The study of various mathematical problems with variable exponent has been re- ceived considerable attention in recent years, for examples we cite works of X-L Fan and V. Radulescu [20], [27]. The operator p(x)-Laplacian turns up in many math- ematical settings, e.g., Non-Newtonian fluids, reaction-diffusion problems, porous media, astronomy, quasi-conformal mappings..etc. see [2, 3, 9]. Problems including this operator for bounded domains have been studied in [20, 27] and for unbounded domains in [10, 21, 14]. Many authors have studied semilinear and non linear elliptic systems, as a reference we cite [7, 10, 28, 22, 29]. 2000 Mathematics Subject Classification. 35B45; 35J55. Key words and phrases. Weak solutions; nonlinear elliptic systems; p(x)-Laplacian; monotone operators; generalized Lebesbgue-Soboles spaces. 1