Klein-Gordon-Maxwell systems in a bounded domain

This paper is concerned with the Klein-Gordon-Maxwell system in a bounded space domain. We discuss the existence of standing waves $\psi = u(x)e^{-i\omega t}$ in equilibrium with a purely electrostatic field $\mathbf{E} = −\nabla \phi(x)$. 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.