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 $(p^{+}{)}^{\ast }$, $p_s^{\ast }(x)$ with the assumption that the critical set $\{x\in \Omega :p_s^{\ast }(x)=(p^{+}{)}^{\ast }\}$ is nonempty.