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

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.