Existence and multiple solutions for a variable exponent system

In this paper, we study the following variable exponent system {−div(|∇u|p(x)−2∇u)+a(x)|u|p(x)−2u=f(x,u,v,λ)in Ω,−div(|∇v|q(x)−2∇v)+b(x)|v|q(x)−2v=g(x,u,v,λ)in Ω,B(u,v)=0, under some various conditions, we present several results for the existence of multiple nontrivial solutions.