Low energy solutions for singularly perturbed coupled nonlinear systems on a Riemannian manifold with boundary

TL;DR

This paper investigates low energy solutions for a singularly perturbed nonlinear Klein-Gordon-Maxwell system on a Riemannian manifold with boundary, showing their existence depends on boundary topology and they concentrate near boundary points.

Contribution

It establishes a link between the number of low energy solutions and the topological complexity of the boundary using Lusternik Schnirelmann theory.

Findings

Number of solutions depends on boundary's topological properties.

Solutions have a unique maximum point on the boundary.

Existence of solutions for small perturbation parameter.

Abstract

Let (M,g) be asmooth, compact Riemannian manifold with smooth boundary, with n= dim M= 2,3. We suppose the boundary of M to be a smooth submanifold of M with dimension n-1. We consider a singularly perturbed nonlinear system, namely Klein-Gordon-Maxwell-Proca system, or Klein-Gordon-Maxwell system of Scrhoedinger-Maxwell system on M. We prove that the number of low energy solutions, when the perturbation parameter is small, depends on the topological properties of the boundary of M, by means of the Lusternik Schnirelmann category. Also, these solutions have a unique maximum point that lies on the boundary.