On a Dirichlet problem with 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) = Fk u (x, u), u ∈W 1,p(x) 0 (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)-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. © 2007 Elsevier Inc. All rights reserved