Existence and regularity of positive solutions for Schr\"odinger-Maxwell system with singularity

TL;DR

This paper proves the existence of positive solutions for a coupled Schr"odinger-Maxwell system with singularities, highlighting how the system's structure induces regularizing effects on solutions.

Contribution

It establishes the existence of positive solutions for a singular elliptic Schr"odinger-Maxwell system, leveraging the coupling to enhance solution regularity.

Findings

Existence of positive solutions under specified conditions.

Coupling induces regularizing effects on solution regularity.

Solutions are shown to belong to certain Lebesgue spaces.

Abstract

In this paper we are going to prove existence for positive solutions of the following Schr\"odinger-Maxwell system of singular elliptic equations: begin{equation} \left\{\begin{array}{l} u \in W_{0}^{1,2}(\Omega):-\operatorname{div}\left(a(x) \nabla u\right)+\psi|u|^{r-2} u=\frac{f(x)}{u^{\theta}}, \psi \in W_{0}^{1,2}(\Omega):-\operatorname{div}(M(x) \nabla \psi)=|u|^{r} \end{array}\right. \end{equation} where $\Omega$ is a bounded open set of $\mathbb{R}^{N}, N>2,$ $r>,1,$ $u>0,$ $\psi>0,$ $0 < \theta<1$ and $f$ belongs to a suitable Lebesgue space. In particular, we take advantage of the coupling between the two equations of the system by demonstrating how the structure of the system gives rise to a regularizing effect on the summability of the solutions.