  In this paper we study a nonlocal critical growth elliptic problem driven by
the fractional Laplacian in presence of jumping nonlinearities. In the main
results of the paper we prove the existence of a nontrivial solution for the
problem under consideration, using variational and topological methods and
applying a new linking theorems recently got by Perera and Sportelli in [10].
The existence results provided in this paper can be seen as the nonlocal
counterpart of the ones obtained in [10] in the context of the Laplacian
equations. In the nonlocal framework the arguments used in the classical
setting have to be refined. Indeed the presence of the fractional Laplacian
operator gives rise to some additional difficulties, that we are able to
overcome proving new regularity results for weak solutions of nonlocal
problems, which are of independent interest.
