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We obtain a critical imbedding and then, concentration-compactness principlesfor fractional Sobolev spaces with variable exponents. As an application ofthese results, we obtain the existence of many solutions for a class ofcritical nonlocal problems with variable exponents, which is even new forconstant exponent case.

The concentration-compactness principles for $W^{s,p(\cdot,\cdot)}(\mathbb{R}^N)$ and application