An elliptic problem involving critical Choquard and singular discontinuous nonlinearity

TL;DR

This paper studies the existence, multiplicity, and regularity of solutions to a complex elliptic problem involving critical Hartree nonlinearity combined with singular and discontinuous nonlinearities, using variational methods.

Contribution

It introduces a novel approach applying generalized gradients to establish solution existence and multiplicity for a challenging elliptic problem with critical and singular features.

Findings

Existence of weak solutions for certain parameter ranges

Multiple solutions under specified conditions

Solutions exhibit Hölder and Sobolev regularity

Abstract

The present article investigates the existence, multiplicity and regularity of weak solutions of problems involving a combination of critical Hartree type nonlinearity along with singular and discontinuous nonlinearity. By applying variational methods and using the notion of generalized gradients for Lipschitz continuous functional, we obtain the existence and the multiplicity of weak solutions for some suitable range of $\lambda$ and $\gamma$. Finally by studying the $L^\infty$-estimates and boundary behavior of weak solutions, we prove their H\"{o}lder and Sobolev regularity.