On the sharp multi-bubble stability for fractional Hardy-Sobolev equations -- A quantitative approach in low dimensions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908725248262144 |
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| author | Chakraborty, Souptik Sarkar, Utsab |
| author_facet | Chakraborty, Souptik Sarkar, Utsab |
| contents | We establish sharp quantitative multi-bubble stability for non-sign-changing critical points of the fractional Hardy-Sobolev inequality in the low-dimensional regime $2s<N<6s-2t$. For functions whose energy is close to that of a finite superposition of bubbles, we prove that the Euler-Lagrange deficit controls linearly the distance, in the homogeneous fractional Sobolev norm, to the multi-bubble manifold, and we recover the precise bubble configuration. This yields quantitative rigidity under arbitrary finite weak interactions. The proof combines a localization scheme adapted to the Hardy weight, weighted fractional Kato-Ponce commutator estimates, a bubble-wise spectral gap inequality, and a sharp interaction analysis. We also show that the linear rate is optimal by constructing a matching counterexample. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_18350 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the sharp multi-bubble stability for fractional Hardy-Sobolev equations -- A quantitative approach in low dimensions Chakraborty, Souptik Sarkar, Utsab Analysis of PDEs 35R11, 35B35, 35B33, 35P30, 35J20, 35A23 We establish sharp quantitative multi-bubble stability for non-sign-changing critical points of the fractional Hardy-Sobolev inequality in the low-dimensional regime $2s<N<6s-2t$. For functions whose energy is close to that of a finite superposition of bubbles, we prove that the Euler-Lagrange deficit controls linearly the distance, in the homogeneous fractional Sobolev norm, to the multi-bubble manifold, and we recover the precise bubble configuration. This yields quantitative rigidity under arbitrary finite weak interactions. The proof combines a localization scheme adapted to the Hardy weight, weighted fractional Kato-Ponce commutator estimates, a bubble-wise spectral gap inequality, and a sharp interaction analysis. We also show that the linear rate is optimal by constructing a matching counterexample. |
| title | On the sharp multi-bubble stability for fractional Hardy-Sobolev equations -- A quantitative approach in low dimensions |
| topic | Analysis of PDEs 35R11, 35B35, 35B33, 35P30, 35J20, 35A23 |
| url | https://arxiv.org/abs/2512.18350 |