Regularizing effects for an elliptic system of singular equations

TL;DR

This paper studies a coupled system of singular elliptic equations inspired by the Schrödinger-Maxwell system, establishing existence of positive solutions with finite energy under certain conditions and demonstrating improved integrability properties.

Contribution

It extends the theory of elliptic equations to a coupled singular system, providing new existence results and integrability improvements through approximation methods.

Findings

Existence of positive finite energy solutions under specific data conditions

Higher integrability of solutions due to data regularity

Improved results compared to classical elliptic equation theory

Abstract

A system of two singular semi-linear elliptic equations, patterned after the Schr\"odinger-Maxwell system, is considered. If the reaction term of the first equation contains a datum $f\in L^m$, existence of positive solutions with finite energy is established for suitable ranges of $m$. In particular, the results from the theory of elliptic single equations are improved. At the same time, thanks to an approach based on approximation schemes and a priori estimates on the approximated sequences of solutions, it is shown that the integrability assumptions on the datum produce higher integrability of the solutions.