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

TL;DR

This paper establishes the existence and regularity of weak solutions for a nonlocal Maxwell-Schrödinger system involving p-Laplacian operators, demonstrating a regularizing effect under certain conditions.

Contribution

It proves the existence and regularity of solutions for a coupled nonlocal PDE system with non-monotone nonlinearities, extending previous results to more general conditions.

Findings

Solutions exhibit Sobolev regularity gain

Solutions exhibit Lebesgue regularity gain

Existence of weak solutions under broad conditions

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.