Fractional Q$Q$‐curvature on the sphere and optimal partitions

Abstract We study an optimal partition problem on the sphere, where the cost functional is associated with the fractional ‐curvature in terms of the conformal fractional Laplacian on the sphere. By leveraging symmetries, we prove the existence of a symmetric minimal partition through a variational approach. A key ingredient in our analysis is a new Hölder regularity result for symmetric functions in a fractional Sobolev space on the sphere. As a byproduct, we establish the existence of infinitely many solutions to a nonlocal weakly coupled competitive system on the sphere that remain invariant under a group of conformal diffeomorphisms and we investigate the asymptotic behavior of least‐energy solutions as the coupling parameters approach negative infinity.