On the existence and stability of solutions for Dirichlet problem with p(x)p(x)-Laplacian

We show the existence and stability of solutions for a family of Dirichlet problems−div(a(x)|∇u(x)|p(x)−2∇u(x))+b(x)|u(x)|p(x)−2u(x)=Fuk(x,u(x)),u(x)|∂Ω=0,u∈W01,p(x)(Ω) with nonlinearity satisfying some local growth conditions. We construct a new duality theory which differs from the known ones in that it does not require a type of a Palais–Smale condition.