A singular Schr\"odinger-Maxwell system

TL;DR

This paper investigates the existence of positive solutions to a Schrödinger-Maxwell system with singular terms, using approximation schemes and variational methods to handle singularities and establish solution properties.

Contribution

It introduces a novel approach to handle strongly singular terms in Schrödinger-Maxwell systems via approximation and barrier techniques, and characterizes energy solutions as saddle points.

Findings

Established existence of positive solutions for singular Schrödinger-Maxwell systems.

Developed an approximation scheme to manage singularities effectively.

Identified conditions under which energy solutions are saddle points.

Abstract

In this paper we are concerned with existence of positive solutions for a Schr\"odinger-Maxwell system with singular or strongly-singular terms. We overcome the difficulty given by the singular terms through an approximation scheme and controlling the approximated sequences of solutions with suitable barriers from above and from below. Besides this, in some particular case, we show that the unique energy solution of the singular system is a saddle point of a suitable functional.