| Abstract: |
| In this talk, we first refine a fractional Sobolev space with variable exponent, as investigated in some recent works, and obtain fundamental imbeddings in our space that is new or is an improvement of known results. With these imbeddings, we then provide a sufficient condition guaranteeing global a-priori bounds for weak solutions of non-local problems involving the fractional $p(\cdot)$-Laplacian. It is worth mentioning that such sufficient condition is new and our result is the first regularity result of the fractional $p(\cdot)$-Laplace problems to the best of authors` knowledge. The existence of infinitely many solutions of a class of problems involving the fractional $p(\cdot)$-Laplacian is also established as an application of our regularity result. |
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