Klein-Gordon-Maxwell systems in a bounded domain
Pietro d’Avenia Lorenzo Pisani Gaetano Siciliano
Discrete & Continuous Dynamical Systems - A 2010, 26(1): 135-149 doi: 10.3934/dcds.2010.26.135
This paper is concerned with the Klein-Gordon-Maxwell system in a bounded space 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 homogeneous Dirichlet boundary conditions on $u$ and non homogeneous Neumann boundary conditions on $\phi$. In the "linear" case we prove the existence of a nontrivial solution when the coupling constant is sufficiently small. Moreover we show that a suitable nonlinear perturbation in the matter equation gives rise to infinitely many solutions. These problems have a variational structure so that we can apply global variational methods.
keywords: standing waves electrostatic field. Klein-Gordon-Maxwell system

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