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Quantitative Stability in Fractional Hardy-Sobolev Inequalities: The Role of Euler-Lagrange Equations

Souptik Chakraborty, Utsab Sarkar

TL;DR

This work proves sharp quantitative stability results for the fractional Hardy-Sobolev inequality on R^N and analyzes the corresponding Euler–Lagrange equation near a ground-state bubble. Building on the classical BE stability framework, it establishes a BE-type inequality with a sharp quadratic control by the distance to the bubble manifold 𝓜, and it develops a qualitative Struwe-type decomposition for Palais–Smale sequences, capturing single- and multi-bubble phenomena. A detailed one-bubble stability result is proved, linking deficits to the distance from the ground state via the linearized operator spectrum, while laying groundwork for multi-bubble analysis in a forthcoming paper. The results extend the sharp stability theory from local and fractional Sobolev inequalities to the fractional Hardy–Sobolev setting, exploiting dilation invariances and careful spectral analysis of the linearized problem.

Abstract

This paper investigates sharp stability estimates for the fractional Hardy-Sobolev inequality: $$μ_{s,t}\left(\mathbb{R}^N\right) \left(\int_{\mathbb{R}^N} \frac{|u|^{2^*_s(t)}}{|x|^t} \,{\rm d}x \right)^{\frac{2}{2^*_s(t)}} \leq \int_{\mathbb{R}^N} \left|(-Δ)^{\frac{s}{2}} u \right|^2 \,{\rm d}x, \quad \text{for all } u \in \dot{H}^s\left(\mathbb{R}^N\right),$$ where $N > 2s$, $s \in (0,1)$, $0 < t < 2s < N $, and $2^*_s(t) = \frac{2(N-t)}{N-2s}$. Here, $μ_{s,t}\left(\mathbb{R}^N\right)$ represents the best constant in the inequality. The paper focuses on the quantitative stability results of the above inequality and the corresponding Euler-Lagrange equation near a positive ground-state solution. Additionally, a qualitative stability result is established for the Euler-Lagrange equation, offering a thorough characterization of the Palais-Smale sequences for the associated energy functional. These results generalize the sharp quantitative stability results for the classical Sobolev inequality in $\mathbb{R}^N$, originally obtained by Bianchi and Egnell \cite{BE91} as well as the corresponding critical exponent problem in $\mathbb{R}^N$, explored by Ciraolo, Figalli, and Maggi \cite{CFM18} in the framework of fractional calculus.

Quantitative Stability in Fractional Hardy-Sobolev Inequalities: The Role of Euler-Lagrange Equations

TL;DR

This work proves sharp quantitative stability results for the fractional Hardy-Sobolev inequality on R^N and analyzes the corresponding Euler–Lagrange equation near a ground-state bubble. Building on the classical BE stability framework, it establishes a BE-type inequality with a sharp quadratic control by the distance to the bubble manifold 𝓜, and it develops a qualitative Struwe-type decomposition for Palais–Smale sequences, capturing single- and multi-bubble phenomena. A detailed one-bubble stability result is proved, linking deficits to the distance from the ground state via the linearized operator spectrum, while laying groundwork for multi-bubble analysis in a forthcoming paper. The results extend the sharp stability theory from local and fractional Sobolev inequalities to the fractional Hardy–Sobolev setting, exploiting dilation invariances and careful spectral analysis of the linearized problem.

Abstract

This paper investigates sharp stability estimates for the fractional Hardy-Sobolev inequality: where , , , and . Here, represents the best constant in the inequality. The paper focuses on the quantitative stability results of the above inequality and the corresponding Euler-Lagrange equation near a positive ground-state solution. Additionally, a qualitative stability result is established for the Euler-Lagrange equation, offering a thorough characterization of the Palais-Smale sequences for the associated energy functional. These results generalize the sharp quantitative stability results for the classical Sobolev inequality in , originally obtained by Bianchi and Egnell \cite{BE91} as well as the corresponding critical exponent problem in , explored by Ciraolo, Figalli, and Maggi \cite{CFM18} in the framework of fractional calculus.

Paper Structure

This paper contains 14 sections, 15 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Notations
  4. Structure of the paper
  5. Preliminaries
  6. Fractional Sobolev space and its properties
  7. Symmetries of the Problem
  8. Action of dilations on and asymptotic orthogonality
  9. Proof of Theorem \ref{['BE']}
  10. Palais-Smale decomposition
  11. Proof of Theorem \ref{['SDHS']}
  12. Proof of Theorem \ref{['onebubble']}
  13. Compact embedding of into
  14. Spectrum of the linearized operator

Key Result

Theorem 1.1

If $v\in\dot{H}^s$R^N$$ is a weak solution of then $v= c \partial_{\lambda}{|}_{\lambda=\lambda_0}U_{s,t}^{\lambda}$ for some $c\in{\mathbb R}\setminus\{0\}$. Moreover, the first two eigenvalues of the linearized operator $\mathcal{L}_{s,t}^{\lambda_0}$$($see Lin-Oper-bub for definition$)$ are $\eta_1=1,\,\eta_2=2^*_s(t)-1$ respectively.

Theorems & Definitions (17)

  • Theorem 1.1: Non-degeneracy MUNA21
  • Theorem 1.2: Regularity, decay estimates and uniqueness MUNA21
  • Theorem 1.3: Quantitative stability near a minimizer
  • Theorem 1.4: Qualitative stability
  • Theorem 1.5: Quantitative stability - one bubble
  • Lemma 2.1
  • Lemma 2.2: Action of dilations on weak convergence in $\dot{H}^s$R^N$$
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 7 more