Regularizing effect for a class of Maxwell–Schrödinger systems

Abstract

In this paper we prove existence and regularity of weak solutions for the following system $\left(\begin{array}{c}-\text{div}(M\left(x\right)\nabla u)+g(x,u,v)=f\text{in}\Omega \\ -\text{div}(M\left(x\right)\nabla v)=h(x,u,v)\text{in}\Omega \\ u=v=0\text{on}\partial \Omega ,\end{array}\right)$where $\Omega$ is an open bounded subset of $R^N$, for $N>2$, $f\in L^m\left(\Omega \right)$, $M$ is a matrix with Lipschitz coefficients, $m>1$ and $g$, $h$ are two Carathéodory 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.