Klein-Gordon-Maxwell System in a bounded domain

Abstract

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 $\mathbf{E}=-\nabla \varphi (x)$. We assume an homogeneous Dirichlet boundary condition on $u$ and an inhomogeneous Neumann boundary condition on $\varphi$. 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.

17 pages

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