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

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 $\mathbb{R}^{N}$.