Symmetry, existence and regularity results for a class of mixed local-nonlocal semilinear singular elliptic problem via variational characterization

TL;DR

This paper investigates symmetry, existence, and regularity of solutions for a mixed local-nonlocal singular elliptic problem using variational methods, comparison principles, and the moving plane technique.

Contribution

It introduces a variational characterization approach to establish symmetry and regularity results for a class of mixed local-nonlocal singular elliptic problems, including solution decomposition.

Findings

Proved symmetry of weak solutions.

Established existence and nonexistence results.

Analyzed regularity of solutions.

Abstract

In this article, we present the symmetry of weak solutions to a mixed local-nonlocal singular problem. We also establish results related to the existence, nonexistence, and regularity of weak solutions to a mixed local-nonlocal singular jumping problem. A crucial element in proving our main results is the variational characterization of the solutions, which also reveals the decomposition property. This decomposition property, together with comparison principles and the moving plane method, yields the symmetry result. Additionally, we utilize nonsmooth critical point theory alongside the variational characterization to analyze the jumping problem.