On a study and applications of the Concentration-compactness type principle for Systems with critical terms in $\mathbb{R}^{N}$

TL;DR

This paper extends the concentration-compactness principle to fractional Sobolev spaces with variable exponents, enabling the analysis of nonlinear elliptic systems with critical growth and nonlocal operators.

Contribution

It introduces new variants of the concentration-compactness principle for variable exponent fractional Sobolev spaces and applies them to establish existence and asymptotic behavior of solutions.

Findings

Established variants of the concentration-compactness principle for fractional Sobolev spaces with variable exponents.

Proved existence of nontrivial solutions for elliptic systems with nonlocal operators and critical growth.

Analyzed the asymptotic behavior of solutions in the context of variable exponent spaces.

Abstract

In this paper, we obtain some important variants of the Lions and Chabrowski Concentration-compactness principle, in the context of fractional Sobolev spaces with variable exponents, especially for nonlinear systems. As an application of the results, we show the existence and assymptotic behaviour of nontrivial solutions for elliptic systems involving a new class of general nonlocal integrodifferential operators with exponent variables and critical growth conditions in $\mathbb{R}^{N}$.