Non-radial minimizers for Hardy--Sobolev inequalities in non-convex cones
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| Format: | Preprint |
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2025
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| _version_ | 1866908437132083200 |
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| author | Nazarov, A. I. Rastegaev, N. V. |
| author_facet | Nazarov, A. I. Rastegaev, N. V. |
| contents | 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 $λ_1(D)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_04470 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-radial minimizers for Hardy--Sobolev inequalities in non-convex cones Nazarov, A. I. Rastegaev, N. V. Analysis of PDEs 35A15 (Primary) 35J92, 35B06, 46E35 (Secondary) 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 $λ_1(D)$ 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. |
| title | Non-radial minimizers for Hardy--Sobolev inequalities in non-convex cones |
| topic | Analysis of PDEs 35A15 (Primary) 35J92, 35B06, 46E35 (Secondary) |
| url | https://arxiv.org/abs/2507.04470 |