Existence of positive solutions for a class of p(x)p(x)-Laplacian systems

We mainly consider the system{−Δp(x)u=λf(x,v)inΩ,−Δp(x)v=λg(x,u)inΩ,u=v=0on∂Ω, where Ω⊂RNΩ⊂RN is a bounded domain, p(x)p(x) is a function which satisfies some conditions, −Δp(x)u=−div(|∇u|p(x)−2∇u)−Δp(x)u=−div(|∇u|p(x)−2∇u) is called p(x)p(x)-Laplacian. We give the existence of positive solutions under some conditions. In particular, we do not assume radial symmetric conditions on the system.