A study and an application of the concentration compactness type principle

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.