# A tool for symmetry breaking and multiplicity in some nonlocal problems

## TL;DR

This paper develops inequalities involving Gagliardo-Nirenberg seminorms to detect symmetry breaking and multiplicity in variational problems with fractional Laplacians, with applications to fractional inequalities.

## Contribution

It introduces a new method using eigenfunction perturbations to analyze symmetry properties in fractional Laplace problems.

## Key findings

- Inequalities relating seminorms of symmetric functions and their perturbations.
- A technique to identify symmetry breaking in fractional variational equations.
- Application to fractional Caffarelli-Kohn-Nirenberg inequality.

## Abstract

We prove some basic inequalities relating the Gagliardo-Nirenberg seminorms of a symmetric function $u$ on $\mathbb R^n$ and of its perturbation $u\varphi_\mu$, where $\varphi_\mu$ 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.