# A study and an application of the concentration compactness type principle

## TL;DR

This paper develops a concentration compactness principle for variable exponent spaces and applies it to fractional p-Laplacian problems with critical nonlinearities, advancing the analysis of variable exponent PDEs.

## Contribution

It introduces a new concentration compactness principle tailored for variable exponent settings and applies it to fractional p-Laplacian problems with critical exponents.

## Key findings

- Established a concentration compactness principle in variable exponent spaces.
- Applied the principle to fractional p-Laplacian problems with critical nonlinearities.
- Provided insights into the existence and behavior of solutions in variable exponent PDEs.

## Abstract

In this article we develop a concentration compactness type principle in a variable exponent setup. As an application of this principle we discuss a problem involving fractional `{\it $(p(x),p^+)$-Laplacian}' and power nonlinearities with exponents $(p^+)^*$, $p_s^*(x)$ with the assumption that the critical set $\{x\in\Omega:p_s^*(x)=(p^+)^*\}$ is nonempty.