KLEIN-GORDON-MAXWELL SYSTEMS IN A BOUNDED DOMAIN

This paper is concerned with the Klein-Gordon-Maxwell system in a hounded space domain. We discuss the existence of standing waves psi = u(x)e(-iwt) in equilibrium with a purely electrostatic field E = -del phi(x). We assume homogeneous Dirichlet boundary conditions on u and non homogeneous Neumann boundary conditions on phi. In the linear case we prove the existence of a nontrivial solution when the coupling constant is sufficiently small. Moreover we show that a suitable nonlinear perturbation in the matter equation gives rise to infinitely many solutions. These problems have a variational structure so that we can apply global variational methods.