Klein-Gordon-Maxwell systems in a bounded domain

Pietro d’Avenia; Lorenzo Pisani; Gaetano Siciliano

doi:10.3934/dcds.2010.26.135

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2010, Volume 26, Issue 1: 135-149. Doi: 10.3934/dcds.2010.26.135

This issue Previous Article Singularly perturbed ODEs and profiles for stationary symmetric Euler and Navier-Stokes shocks Next Article The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy

Klein-Gordon-Maxwell systems in a bounded domain

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

Received: November 2008

Revised: July 2009

Published: March 2010

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J50, 35J55, 35Q60; Secondary: 35Q40, 35Q60.

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