Scalar field equation with non-local diffusion

In this paper we are interested on the existence of ground state solutions for fractional field equations of the form integral (I - Delta)(alpha) u = f(x, u) in IRN, u > 0 in IRN, lim(vertical bar x vertical bar ->infinity) u(x) = 0, where and f is an appropriate super-linear sub-critical nonlinearity. We prove regularity, exponential decay and symmetry properties for these solutions. We also prove the existence of infinitely many bound states and, through a non-local Pohozaev identity, we prove nonexistence results in the supercritical case.