The concentration-compactness principles for $W^{s,p(\cdot,\cdot)}(\mathbb{R}^N)$ and application

TL;DR

This paper develops concentration-compactness principles for fractional Sobolev spaces with variable exponents and applies them to prove the existence of multiple solutions for critical nonlocal problems, advancing the understanding of variable exponent spaces.

Contribution

It introduces new concentration-compactness principles for fractional Sobolev spaces with variable exponents and demonstrates their application to critical nonlocal problems.

Findings

Established critical embedding for fractional Sobolev spaces with variable exponents

Proved concentration-compactness principles in this setting

Proved existence of multiple solutions for critical nonlocal problems

Abstract

We obtain a critical imbedding and then, concentration-compactness principles for fractional Sobolev spaces with variable exponents. As an application of these results, we obtain the existence of many solutions for a class of critical nonlocal problems with variable exponents, which is even new for constant exponent case.