A pseudo--index approach to fractional equations

The aim of this paper is investigating the existence of weak solutions to non--local equations involving a general integro--differential operator of fractional type, when the nonlinearity is subcritical and asymptotically linear at infinity. These equations admit a variational structure and, in presence of an odd symmetric nonlinearity, we prove multiplicity results by using a pseudo--index theory related to the genus. As a particular case we derive existence and multiplicity results for non--local equations involving the fractional Laplacian operator.