Neumann condition in the Schrödinger-Maxwell system

Abstract

We study a system of (nonlinear) Schrödinger 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 ${\bold E}=-\nabla\phi$. Due to the Neumann
condition, the same electric field is in equilibrium with stationary
solutions $\psi=e^{-i\omega t}u(x)$ of arbitrary frequency
$\omega$.