On a Dirichlet problem with p(x)p(x)-Laplacian

We show the existence and stability of solutions for a family of Dirichlet problems−div(Vz1(x,∇u),…,VzN(x,∇u))+Lu(x,u)=Fuk(x,u),u∈W01,p(x)(Ω) in a bounded domain and with nonconvex nonlinearity satisfying some local growth conditions. The conditions upon V and L   allow for considering the p(x)p(x)-Laplacian equation. We use the relations between critical points and critical values to the primal and a suitable dual action functional to get the existence, stability and some properties of the solutions.