Multiplicity of positive solutions for mixed local-nonlocal singular critical problems

TL;DR

This paper establishes the existence of multiple positive solutions for complex elliptic problems involving both local and nonlocal operators, addressing singularities and critical growth conditions in a unified framework.

Contribution

It introduces new existence results for positive solutions in mixed local-nonlocal singular and critical problems, extending previous work on singular and critical elliptic equations.

Findings

Proves at least two positive weak solutions exist.

Extends previous results to more general mixed local-nonlocal problems.

Addresses singularities and critical growth simultaneously.

Abstract

We prove the existence of at least two positive weak solutions for mixed local-nonlocal singular and critical semilinear elliptic problems in the spirit of [Haitao, 2003], extending the recent results in [Garain, 2023] concerning singular problems and, at the same time, the results in [Biagi, Dipierro, Valdinoci, Vecchi, 2022] regarding critical problems.