Nonlocal critical growth elliptic problems with jumping nonlinearities

In this paper we study a nonlocal critical growth elliptic problem driven by the fractional Laplacian in the presence of a jumping nonlinearity. By using variational and topological methods and applying some new linking theorems recently proved by Perera and Sportelli in [19], we prove the existence of a nontrivial solution for the problem under consideration. The results we obtain here are the nonlocal counterparts of the ones obtained in [19] in the context of a local equation. Due to the nonlocal nature of our problem, some additional difficulties arise, and the arguments employed in the local setting need to be improved or reconceived. In fact, the proofs of our main theorems require some refined techniques and new regularity results for weak solutions of nonlocal problems that are of independent interest. We would like to point out that our results are specifically for a nonlocal problem with the fractional operator in integral form. However, we do not exclude the possibility that our results may have a counterpart for the spectral operator studied in [27]. Since nonlocal operators in integral form are being widely investigated in the current literature, especially in connection with geometric problems, we have restricted ourselves to elliptic equations driven by a fractional operator in integral form here.