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