Regularizing Effect for a Nonlocal Maxwell-Schr\"odinger System

Abstract

In this paper we prove existence and regularity of weak solutions for the following system \begin{align*} \begin{cases} &-\mbox{div}\Bigg(\bigg(\|\nabla u\|^{p}_{L^{p}}+\|\nabla v\|^{p}_{L^{p}}\bigg)|\nabla u|^{p-2}\nabla u\Bigg) + g(x,u,v)=f \ \ \ \mbox{in} \ \Omega; &-\mbox{div}\Bigg(\bigg(\|\nabla u\|^{p}_{L^{p}}+\|\nabla v\|^{p}_{L^{p}}\bigg)|\nabla v|^{p-2}\nabla v\Bigg) = h(x,u,v) \ \ \ \ \mbox{in} \ \Omega; &u=v=0 \ \mbox{on} \ \partial\Omega. \end{cases} \end{align*} where $\Omega$ is an open bounded subset of $\mathbb{R}^N$, $N>2$, $f\in L^m(\Omega)$, where $m>1$ and $g$, $h$ are two Carath\'eodory functions, which may be non monotone. We prove that under appropriate conditions on $g$ and $h$, there is gain of Sobolev and Lebesgue regularity for the solutions of this system.