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

In this paper we study the existence of positive solutions for the following Schrödinger–Maxwell system of singular elliptic equations 1 $\{\begin{array}{cc}-div(A(x)\mathrm{\nabla }u)+\psi u^{r-1}=\frac{f(x)}{u^{\theta }}\hfill & \text{ in }\mathrm{\Omega },\hfill \\ -div(M(x)\psi )=u^r\hfill & \text{ in }\mathrm{\Omega },\hfill \\ u,\psi >0\hfill & \text{ in }\mathrm{\Omega },\hfill \\ u=\psi =0\hfill & \text{ on }\partial \mathrm{\Omega },\hfill \end{array}$ \( \textstyle\begin{cases} -\operatorname{div}(A(x) \nabla u)+\psi u^{r-1}= \frac{f(x)}{u^{\theta }} & \text{ in } \Omega , \\ -\operatorname{div}(M(x) \psi )=u^{r} & \text{ in } \Omega , \\ u, \psi >0 & \text{ in } \Omega , \\ u=\psi =0 & \text{ on } \partial \Omega ,\end{cases} \) where $\mathrm{\Omega }$ $\Omega $ is a bounded open set of ${\mathbb{R}}^N,N>2$ $\mathbb{R}^{N}, N>2$ , $r>1$ $r>1$ , $0<\theta <1$ $0 < \theta <1$ and $f$ $f$ 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.