Existence of solutions for p(x)-Laplacian problems on a bounded domain

In this paper we study the following p(x)-Laplacian problem: −div a(x)|∇u|p(x)−2∇u +b(x)|u|p(x)−2u = f (x,u), x ∈ Ω, u = 0, on ∂Ω, where 1 < p1 p(x) p2 < n, Ω ⊂ Rn is a bounded domain and applying the mountain pass theorem we obtain the existence of solutions in W 1,p(x) 0 (Ω) for the p(x)-Laplacian problems in the superlinear and sublinear cases.  2004 Elsevier Inc. All rights reserved