Number and profile of low energy solutions for singularly perturbed Klein Gordon Maxwell systems on a Riemannian manifold

TL;DR

This paper studies the existence and quantity of low-energy solutions for Klein-Gordon-Maxwell and Schrödinger-Maxwell systems on 3D Riemannian manifolds, linking solutions to the manifold's topology.

Contribution

It establishes a relationship between the number of solutions and the topological complexity of the manifold using Lusternik-Schnirelmann theory.

Findings

Number of solutions depends on the Lusternik-Schnirelmann category of the manifold.

Existence of positive solutions for the systems under subcritical nonlinearity.

Solutions exhibit a single peak profile related to the manifold's topology.

Abstract

Given a 3-dimensional Riemannian manifold (M,g), we investigate the existence of positive solutions of the nonlinear Klein-Gordon-Maxwell system and nonlinear Schroedinger-Maxwell system with subcritical nonlinearity. We prove that the number of one peak solutions depends on the topological properties of the manifold M, by means of the Lusternik Schnirelmann category.