Klein-Gordon-Maxwell System in a bounded domain

TL;DR

This paper investigates the existence and multiplicity of solutions for the Klein-Gordon-Maxwell system within a bounded domain, focusing on standing waves and electrostatic fields under specific boundary conditions.

Contribution

It provides new results on the existence of solutions for both linear and nonlinear Klein-Gordon-Maxwell systems with particular boundary conditions.

Findings

Existence of nontrivial solutions in the linear case for small boundary data.

Infinitely many solutions in the nonlinear perturbed case.

Characterization of solutions under Dirichlet and Neumann boundary conditions.

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 $\mathbf{E}=-\nabla\phi(x)$. 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.