Schr\"odinger-Maxwell systems on Hadamard manifolds

TL;DR

This paper investigates the existence and multiplicity of isometry-invariant solutions to nonlinear Schr"odinger-Maxwell systems on Hadamard manifolds, overcoming compactness issues via variational methods and symmetry considerations.

Contribution

It introduces a novel approach to handle non-compact Hadamard manifolds by exploiting isometric actions to establish solution existence and multiplicity.

Findings

Existence of solutions under certain nonlinear conditions

Uniqueness of solutions in specific cases

Multiple solutions depending on nonlinear term behavior

Abstract

In this paper we study nonlinear Schr\"odinger-Maxwell systems on $n-$dimensional Hadamard manifolds, $3\leq n\leq 5.$ The main difficulty resides in the lack of compactness of such manifolds which is recovered by exploring suitable isometric actions. By combining variational arguments, some existence, uniqueness and multiplicity of isometry-invariant weak solutions are established for the Schr\"odinger-Maxwell system depending on the behavior of the nonlinear term.