Klein-Gordon-Maxwell system in a bounded domain

The paper is concerned with the Klein-Gordon-Maxwell system in a bounded space domain. We discuss the existence of standing waves $\psi = u(x) exp (-i\omega t)$ in equilibrium with a purely electrostatic field $-\nabla phi$. 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