DOI 10.1515/anona-2014-0009 | Adv. Nonlinear Anal. 2014; 3 (S1):s37–s45 Research Article Pietro d’Avenia, Lorenzo Pisani and Gaetano Siciliano Nonautonomous Klein–Gordon–Maxwell systems in a bounded domain Abstract: This paper deals with the Klein–Gordon–Maxwell system in a bounded spatial domain with a nonuniform coupling. We discuss the existence of standing waves in equilibrium with a purely electrostatic field, assuming homogeneous Dirichlet boundary conditions on the matter field and nonhomogeneous Neu- mann boundary conditions on the electric potential. Under suitable conditions we prove existence and nonex- istence results. Since the system is variational, we use Ljusternik–Schnirelmann theory. Keywords: Klein–Gordon–Maxwell system, standing waves, electrostatic field MSC 2010: 35J50, 35J57, 35Q40 || Pietro d’Avenia: Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via E. Orabona 4, 70125 Bari, Italy, e-mail: p.davenia@poliba.it Lorenzo Pisani: Dipartimento di Matematica, Università degli Studi di Bari A. Moro, Via E. Orabona 4, 70125 Bari, Italy, e-mail: lorenzo.pisani@uniba.it Gaetano Siciliano: Departamento de Matemática, Universidade de São Paulo, Rua do Matão, 1010, 05508-090 São Paulo, SP, Brazil, e-mail: sicilian@ime.usp.br 1 Introduction In this paper we study a system of PDEs obtained in the framework of the Klein–Gordon–Maxwell (KGM for brevity) systems. Let us recall that the KGM system is given by Euler–Lagrange equations corresponding to the following total energy density: L KGM = L KG (, A, ) + L (A, ) = 1 2 (|( + )| 2 − |(∇ − A)| 2 − 2 || 2 )+ 1 8 (|∇ + A| 2 − |∇ × A| 2 ). Here denotes a scalar matter field, whose free Lagrangian density is given by L KG = 1 2 (| | 2 − |∇| 2 − 2 || 2 ), (1.1) with > 0. The field is charged and in equilibrium with its own electromagnetic field (E, B), represented by means of the gauge potentials (A, ), by E =−(∇ + A), B = ∇ × A. Abelian gauge theories provide a model for the interaction: formally we replace the ordinary derivatives ( , ∇) in (1.1) with the so-called gauge covariant derivatives ( + , ∇ − A), where is a nonzero coupling constant (see e.g. [18]). Moreover, we add to (1.1) the Lagrangian density associated with the electromagnetic field L = 1 8 (|E| 2 − |B| 2 ). Following [5], we focus on a particular class of solution, namely standing waves = () − (1.2) in equilibrium with a purely electrostatic field E =−∇(), B = 0. (1.3)