Scalar field equation with non-local diffusion

Abstract

In this paper we are interested on the existence of ground state solutions for fractional field equations of the form $$\left\{\begin{array}{ll} (I - \Delta)^{\alpha}u = f(x, u) & \quad {\rm in} \, u > 0 & \quad {\rm in} \, I\!R^N, \quad \displaystyle\lim_{|x| \to \infty}u(x) = 0,\end{array}\right.$$ where \({\alpha \in (0,1)}\) 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.