In this paper we are dealing with a Schroedinger–Maxwell system in a bounded domain of R^3; the unknowns are the charged standing waves in equilibrium with a purely electrostatic potential. The system is not autonomous, in the sense that the coupling depends on a function q = q(x). The non-homogeneous Neumann boundary condition on φ prescribes the flux of the electric field F and gives rise to a necessary condition. On the other hand we consider the usual normalizing condition for u. Under mild assumptions involving F and the function q, we prove that this problem has a variational framework: its solutions can be characterized as constrained critical points. Then, by means of the Ljusternick–Schnirelmann theory, we get the existence of infinitely many solutions.

We study the class of nonlinear Klein–Gordon–Maxwell systems describing a standing wave (charged matter field) in equilibrium with a purely electrostatic field. We improve some previous existence results in the case of an homogeneous nonlinearity. Moreover, we deal with a limit case, namely when the frequency of the standing wave is equal to the mass of the charged field; this case shows analogous features of the well-known ‘zero-mass case’ for scalar field equations.

The paper is concerned with the Klein-Gordon-Maxwell system in a bounded space domain. We discuss the existence of standing waves $\psi = u(x) exp (-i\omega t)$ in equilibrium with a purely electrostatic field $-\nabla phi$. 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.

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 Neumann boundary conditions on the electric potential. Under suitable conditionswe prove existence and nonexistence results. Since the system is variational, we use Ljusternik–Schnirelmann theory.

We study the sum of weighted Lebesgue spaces, by considering an abstract measure space and the Nemytskii operator defined on it. Then we apply our general results to prove existence and multiplicity of solutions to a class of nonlinear p-Laplacian equations in R^n; with a nonnegative measurable potential, possibly singular and vanishing at infinity, and Carathéodory functions satisfying a double-power growth condition in u.