# American Institute of Mathematical Sciences

January  2010, 26(1): 135-149. doi: 10.3934/dcds.2010.26.135

## Klein-Gordon-Maxwell systems in a bounded domain

 1 Dipartimento di Matematica, Politecnico di Bari, Via Orabona, 4, I-70125 Bari, Italy 2 Dipartimento di Matematica, Università degli Studi di Bari, Via Orabona, 4, I-70125 Bari, Italy

Received  November 2008 Revised  July 2009 Published  October 2009

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.
Citation: Pietro d’Avenia, Lorenzo Pisani, Gaetano Siciliano. Klein-Gordon-Maxwell systems in a bounded domain. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 135-149. doi: 10.3934/dcds.2010.26.135
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