On the existence and stability of solutions for Dirichlet problem with 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)−2 u(x) = Fk u x,u(x) , u(x)|∂Ω = 0, u∈W 1,p(x) 0 (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. © 2006 Elsevier Inc. All rights reserved.