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In this paper, we obtain some important variants of the Lions and ChabrowskiConcentration-compactness principle, in the context of fractional Sobolevspaces with variable exponents, especially for nonlinear systems. As anapplication of the results, we show the existence and assymptotic behaviour ofnontrivial solutions for elliptic systems involving a new class of generalnonlocal integrodifferential operators with exponent variables and criticalgrowth conditions in $\mathbb{R}^{N}$.

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