On a Sobolev critical problem for the superposition of a local and nonlocal operator with the "wrong sign''

TL;DR

This paper investigates a critical PDE involving a mixed local and nonlocal operator with an unusual sign, revealing new existence results for solutions in this challenging setting.

Contribution

It introduces the analysis of a Sobolev critical problem with a superposition of Laplacian and fractional Laplacian operators of opposite signs, a novel configuration in the field.

Findings

Existence of nontrivial solutions under the 'wrong sign' fractional Laplacian.

Development of new analytical techniques for mixed local and nonlocal operators.

Insights into the impact of sign changes on critical PDE problems.

Abstract

We study a critical problem for an operator of mixed order obtained by the superposition of a Laplacian with a fractional Laplacian. The main novelty is that we consider a mixed operator of the form $-\Delta- \gamma(-\Delta)^s$, namely we suppose that the fractional Laplacian has the ``wrong sign'' and can be seen as a nonlocal perturbation of the purely local case, which is needed to produce a nontrivial solution of the critical problem.