A tool for symmetry breaking and multiplicity in some nonlocal problems

Abstract

We prove some basic inequalities relating the Gagliardo-Nirenberg seminorms of a symmetric function $u$ on ${\mathbb{R}}^n$ and of its perturbation $u{\phi }_{\mu }$, where ${\phi }_{\mu }$ is a suitably chosen eigenfunction of the Laplace-Beltrami operator on the sphere ${\mathbb{S}}^{n-1}$, thus providing a technical but rather powerful tool to detect symmetry breaking and multiplicity phenomena in variational equations driven by the fractional Laplace operator. A concrete application to a problem related to the fractional Caffarelli-Kohn-Nirenberg inequality is given.

18 pages