Nonlinear Klein-Gordon-Maxwell systems with Neumann boundary conditions on a Riemannian manifold with boundary

Abstract

Let (M, g) be a smooth compact, n dimensional Riemannian manifold, n = 3, 4 with smooth n − 1 dimensional boundary ∂ M. We search for the positive solutions of the singularly perturbed Klein Gordon Maxwell Proca system with homogeneous Neumann boundary conditions or for the singularly perturbed Klein Gordon Maxwell system with mixed Dirichlet Neumann homogeneous boundary conditions. We prove that C1 stable critical points of the mean curvature of the boundary generates H1(M) solutions when the perturbation parameter ɛ is sufficiently small.