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arXiv · 2507.04470

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

A. I. Nazarov·N. V. Rastegaev
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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.

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A. I. Nazarov, N. V. Rastegaev. 2025-07-06. Non-radial minimizers for Hardy--Sobolev inequalities in non-convex cones. https://arxiv.org/abs/2507.04470

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