Regularizing Effect for a Class of Maxwell-Schr\"odinger Systems

Abstract

In this paper we prove the existence and regularity of weak solutions for the following system \begin{align*} \begin{cases} -\mbox{div}(M(x)\nabla u) + g(x,u,v) = f \ \ \mbox{in} \ \ \Omega\\ -\mbox{div}(M(x)\nabla v) = 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 $\mathbb{R}^N$, for $N>2$, $f\in L^m(\Omega)$, where $m>1$ and $h,\ g$ are two Carath\'eodory functions. We prove that under appropriate conditions on $g$ and $h$ there exist solutions which escape the predicted regularity by the classical Stampacchia's theory causing the so-called regularizing effect.