Neumann condition in the Schrodinger-Maxwell system

We study a system of (nonlinear) Schrodinger and Maxwell equation in a bounded domain, with a Dirichelet boundary condition for the wave function psi and a nonhomogeneous Neumann datum for the electric potential phi. Under a suitable compatibility condition, we establish the existence of infinitely many static solutions psi = u(x) in equilibrium with a purely electrostatic field E = -del phi. Due to the Neumann condition, the same electric field is in equilibrium with stationary solutions psi = e(-iwt)u(x) of arbitrary frequency omega.