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On the sharp multi-bubble stability for fractional Hardy-Sobolev equations -- A quantitative approach in low dimensions

Souptik Chakraborty, Utsab Sarkar

TL;DR

The paper addresses sharp quantitative stability for the fractional Hardy–Sobolev inequality in the nonlocal, weighted, low-dimensional setting. It develops a robust framework combining a localization scheme, weighted fractional Kato–Ponce commutator estimates, and a bubble-wise spectral gap to prove that near-extremizers are close to a finite sum of weakly interacting fractional bubbles, with the distance controlled linearly by the Euler–Lagrange deficit and optimality confirmed via a counterexample. The results extend one-bubble stability to the multi-bubble regime and introduce analytic tools that may apply to broader nonlocal PDEs with singular potentials. Overall, it yields quantitative rigidity, recovers precise bubble configurations under arbitrary finite weak interactions, and advances the understanding of stability in fractional critical problems.

Abstract

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.

On the sharp multi-bubble stability for fractional Hardy-Sobolev equations -- A quantitative approach in low dimensions

TL;DR

The paper addresses sharp quantitative stability for the fractional Hardy–Sobolev inequality in the nonlocal, weighted, low-dimensional setting. It develops a robust framework combining a localization scheme, weighted fractional Kato–Ponce commutator estimates, and a bubble-wise spectral gap to prove that near-extremizers are close to a finite sum of weakly interacting fractional bubbles, with the distance controlled linearly by the Euler–Lagrange deficit and optimality confirmed via a counterexample. The results extend one-bubble stability to the multi-bubble regime and introduce analytic tools that may apply to broader nonlocal PDEs with singular potentials. Overall, it yields quantitative rigidity, recovers precise bubble configurations under arbitrary finite weak interactions, and advances the understanding of stability in fractional critical problems.

Abstract

We establish sharp quantitative multi-bubble stability for non-sign-changing critical points of the fractional Hardy-Sobolev inequality in the low-dimensional regime . 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.

Paper Structure

This paper contains 14 sections, 14 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Background and Hurdles
  3. Main Results
  4. Notations
  5. Structure of the paper
  6. Stability in low dimensions
  7. Proof of Theorem \ref{['quantitative-stability']}
  8. Localization via bump functions
  9. Spectral gap inequality
  10. Taming interaction integrals and the robustness of bubble coefficients
  11. Proof of Theorem \ref{['energybound-implies-stability']}
  12. Sharpness in low dimensions
  13. Few useful estimates
  14. Integrals involving two bubbles

Key Result

Theorem 1.2

Let $s\in(0,1)$, $t\in(0,2s)$, $N>2s$, and $N,\nu\in{\mathbb N}$. Let $\{u_k\}_{k\ge1}\subset\dot{H}^s(\mathbb R^N)$ be a sequence of nonnegative functions such that Then, up to a subsequence, there exist positive scales $\lambda_k^1,\ldots,\lambda_k^\nu$ such that If $u_k\rightharpoonup u_0\neq0$, then $\lambda_k^1=\lambda_0$ for all $k$; whereas if $u_k\rightharpoonup0$, no scale remains fixed

Theorems & Definitions (19)

  • Remark 1.1
  • Theorem 1.2: Qualitative stability
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: $\delta$-interacting bubbles
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 2.6
  • ...and 9 more