Calculus of Variations and Geometric Measure Theory

S. Dipierro - G. Palatucci - E. Valdinoci

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

created by palatucci on 12 Mar 2012
modified on 28 Jun 2013


Published Paper

Inserted: 12 mar 2012
Last Updated: 28 jun 2013

Journal: Le Matematiche (Catania)
Volume: 68
Number: 1
Pages: 201–216
Year: 2013


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

Keywords: fractional Laplacian, critical Sobolev exponent, fractional Sobolev spaces, Nonlinear problems, spherical solutions, ground states