ar X iv : 0 80 2 . 43 52 v 1 [ m at h . A P ] 2 9 Fe b 20 08 Klein - Gordon - Maxwell System in a bounded domain ∗

This paper is concerned with the Klein-Gordon-Maxwell system in a bounded spatial domain. We discuss the existence of standing waves ψ = u(x)e in equilibrium with a purely electrostatic field E = −∇φ(x). We assume an homogeneous Dirichlet boundary condition on u and an inhomogeneous Neumann boundary condition on φ. In the “linear” case we characterize the existence of nontrivial solutions for small boundary data. With a suitable nonlinear perturbation in the matter equation, we get the existence of infinitely many solutions. Mathematics Subject Classification 2000: 35J50, 35J55, 35Q60