Existence and Regularity of Positive Solutions for Schrödinger-Maxwell System with Singularity

Abstract

In this paper we study the existence of positive solutions for the following Schrödinger–Maxwell system of singular elliptic equations

$$ \textstyle\begin{cases} -\operatorname{div}(A(x) \nabla u)+\psi u^{r-1}= \frac{f(x)}{u^{\theta }} & \text{ in } \Omega , \\ -\operatorname{div}(M(x) \psi )=u^{r} & \text{ in } \Omega , \\ u, \psi >0 & \text{ in } \Omega , \\ u=\psi =0 & \text{ on } \partial \Omega ,\end{cases} $$

(1)

where \(\Omega \) is a bounded open set of \(\mathbb{R}^{N}, N>2\), \(r>1\), \(0 < \theta <1\) and \(f\) is nonnegative function belongs to a suitable Lebesgue space. In particular, we take advantage of the coupling between the two equations of the system by proving how the structure of the system gives rise to a regularizing effect on the summability of the solutions.