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

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