Existence and symmetry results for a Schr\"odinger type problem involving the fractional Laplacian

TL;DR

This paper establishes existence and symmetry of solutions for a class of nonlocal fractional Schr"odinger equations, extending classical results to the fractional Laplacian case with s in (0,1).

Contribution

It provides new existence and symmetry results for fractional Schr"odinger equations, bridging the gap between local and nonlocal cases.

Findings

Existence of solutions in fractional Sobolev spaces.

Symmetry properties of solutions.

Consistency with classical s=1 case.

Abstract

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