Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian

This paper deals with the following class of nonlocal Schrödinger equations(-\Delta)^s u + u = |u|^{p-1}u in \mathbb{R}^N, for s\in (0,1).We prove existence and symmetry results for the solutions $u$ in the fractional Sobolev space H^s(\mathbb{R}^N). Our results are in clear accordance with those for the classical local counterpart, that is when s=1.