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

This paper deals with the following class of nonlocal Schr\"odinger equations $$(-\Delta)s u + u = u^{p-1}u \ \ \text{in} \ \mathbb{R}N, \quad \text{for} \ s\in (0,1).$$ 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 $s=1$.