  In this paper, first we study existence results for a linearly perturbed
elliptic problem driven by the fractional Laplacian. Then, we show a
multiplicity result when the perturbation parameter is close to the
eigenvalues. This latter result is obtained by exploiting the topological
structure of the sublevels of the associated functional, which permits to apply
a critical point theorem of mixed nature due to Marino and Saccon.
