  This paper is concerned with the Klein-Gordon-Maxwell system in a bounded
spatial 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 an homogeneous Dirichlet boundary
condition on $u$ and an inhomogeneous Neumann boundary condition on $\phi$. 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.
