A tool for symmetry breaking and multiplicity in some nonlocal problems

We prove some basic inequalities relating the Gagliardo‐Nirenberg seminorms of a symmetric function u on Rn and of its perturbation uφμ, where φμ is a suitably chosen eigenfunction of the Laplace‐Beltrami operator on the sphere Sn−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.