The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds

TL;DR

This paper explores how the scalar curvature of a 3D Riemannian manifold influences the existence and concentration of positive solutions to singularly perturbed Klein-Gordon-Maxwell and Schrödinger-Maxwell systems, revealing geometric effects on solutions.

Contribution

It establishes a link between stable critical points of scalar curvature and the existence of solutions concentrating at those points for small perturbations.

Findings

Solutions concentrate at critical points of scalar curvature as epsilon approaches zero.

Stable critical points of scalar curvature generate positive solutions.

The results apply to both Klein-Gordon-Maxwell and Schrödinger-Maxwell systems.

Abstract

Given a 3-dimensional Riemannian manifold (M,g), we investigate the existence of positive solutions of singularly perturbed Klein-Gordon-Maxwell systems and Schroedinger-Maxwell systems on M, with a subcritical nonlinearity. We prove that when the perturbation parameter epsilon is small enough, any stable critical point x_0 of the scalar curvature of the manifold (M,g) generates a positive solution (u_eps,v_eps) to both the systems such that u_eps concentrates at xi_0 as epsilon goes to zero.