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

TL;DR

This paper investigates positive solutions to singularly perturbed Klein-Gordon-Maxwell systems on Riemannian manifolds with boundary, linking solutions to boundary mean curvature in the small perturbation limit.

Contribution

It establishes a connection between stable critical points of boundary mean curvature and solutions of the Klein-Gordon-Maxwell systems under Neumann and mixed boundary conditions.

Findings

Solutions exist when boundary mean curvature has stable critical points.

Solutions are generated for sufficiently small perturbation parameters.

The work extends understanding of boundary effects in nonlinear field equations.

Abstract

Let (M,g) be a smooth compact, n dimensional Riemannian manifold, n=3,4 with smooth n-1 dimensional boundary. We search 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 stable critical points of the mean curvature of the boundary generates solutions when the perturbation parameter is sufficiently small.