A multiplicity result for critical elliptic problems involving differences of local and nonlocal operators

TL;DR

This paper proves the existence of two distinct solutions for certain critical elliptic problems involving differences of local and nonlocal operators, using an abstract variational approach.

Contribution

It introduces new existence results for elliptic problems with operator differences, extending previous theoretical frameworks.

Findings

Existence of two nontrivial solutions with different energy signs

Solutions exist for all sufficiently small parameter values

Utilizes an abstract variational method for proof

Abstract

We study some critical elliptic problems involving the difference of two nonlocal operators, or the difference of a local operator and a nonlocal operator. The main result is the existence of two nontrivial weak solutions, one with negative energy and the other with positive energy, for all sufficiently small values of a parameter. The proof is based on an abstract result recently obtained in [20].