Non-radial minimizers for Hardy--Sobolev inequalities in non-convex cones

TL;DR

This paper demonstrates that under certain conditions, minimizers for Hardy--Sobolev inequalities in non-convex cones are non-radial, revealing symmetry breaking and multiple solutions for related Neumann problems involving the p-Laplacian.

Contribution

It establishes conditions under which radial minimizers are not extremal, proving the existence of non-radial minimizers and multiple solutions in non-convex cone settings.

Findings

Radial solutions cannot be extremal under certain eigenvalue restrictions.

Symmetry breaking occurs in non-convex cones for Hardy--Sobolev inequalities.

Multiple solutions exist for the Neumann problem in these geometric settings.

Abstract

The symmetry breaking is obtained for Neumann problems driven by $p$-Laplacian in certain non-convex cones. These problems are generated by the Hardy--Sobolev inequalities. In the case of the Sobolev inequality for the ordinary Laplacian this problem was investigated in (Ciraolo, Pacella, Polvara, 2024). Such problems have obvious radial solutions -- Talenti--Bliss type functions of $|x|$. However, under a certain restriction on the first Neumann eigenvalue $\lambda_1(D)$ of the Beltrami--Laplace operator on the spherical cross-section $D$ of the cone we prove this radial solution cannot be an extremal function, therefore minimizer must be non-radial. This leads to multiple solutions for the corresponding Neumann problem.