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

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 $\mathrm{\mid }x\mathrm{\mid }$. 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.