On Coron problems with Choquard term and mixed operator

TL;DR

This paper investigates a nonlinear elliptic problem with a critical Choquard term and mixed differential operators, establishing existence, regularity, and compactness results in annular domains.

Contribution

It introduces new existence and regularity results for solutions to a mixed operator problem with critical nonlinearity involving the Choquard term.

Findings

Existence of positive solutions when the inner hole is small

Global compactness for Palais-Smale sequences

Regularity results for weak solutions

Abstract

In this article, we study a Coron-type problem involving a critical Choquard nonlinearity driven by a mixed operator combining the Laplacian and fractional Laplacian. In annular-type domains, we prove the existence of nontrivial positive solutions when the inner hole is sufficiently small. Using variational methods and concentration compactness arguments, we establish a global compactness result for Palais- Smale sequences and obtain high-energy solutions using topological methods. We also derive regularity results for weak solutions.