Existence of Bianchi-Egnell stability extremizer for the Hardy-Sobolev inequality

2025/05/11 by Souptik Chakraborty, Chakraborty, Souptik, Monideep Ghosh +3
Engineering · Mathematics · #26D10 #46E35 #47J20 #49J20 #49J40 #49K40 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in engineering #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2505.07039

openalex publication_date 2025/05/11 · openalex created_date 2025/10/09 · openalex updated_date 2026/07/28

Abstract

In this article, we prove the best Bianchi-Egnell constant for the Hardy-Sobolev (HS) inequality C_\tinyBE(γ) := inf_{u \small not an optimizer} \frac∫_ℝⁿ (|∇ u|² - \fracγ|x|²u²) \rm dx - S_γ‖u‖_L^2^⋆2dist (u, set of optimizers)², is attained, extending the result of König [arXiv:2211.14185] for the classical Sobolev inequality (that corresponds to γ= 0). One of the main difficulties is that the third eigenspace of the linearized operator may contain only spherical harmonics of degree 1, and hence, an essential non-vanishing criterion fails [arXiv:2210.08482]. This non-vanishing criterion is indispensable for proving the best Bianchi-Egnell constant C_\tinyBE(γ) < C_\tinyBE^\tinyloc(γ) that prevents a minimizing sequence converging to one of the optimizers. In addition, not being translation invariant, extracting a non-zero weak limit from a minimizing sequence presents difficulties. We found another hidden critical level C_\tinyBE(γ) <1 - (S_γ)/(S), where S is the best Sobolev constant that plays a significant role in proving the existence of an extremizer. In particular, we show that there exists a γ₀>0 such that for γ≥ γ₀, C_\tinyBE(γ) is attained. Moreover, we remark that there is a region γ₀ ≤ γ< γ_c^⋆, where the third eigenspace of the linearized operator contains only spherical harmonics of degree 1. Our result improves some of the results in Wei-Wu [arXiv:2308.04667] corresponding to the HS inequality.

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