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Stability of Hardy-Sobolev Inequality

Souptik Chakraborty

Abstract

Given $N\geq 3,$ we consider the critical Hardy-Sobolev equation $-Δu-\fracγ{|x|^2}u=\frac{|u|^{2^*(s)-2}u}{|x|^s}$ in $\mathbb{R}^N\setminus \{0\},$ where $0<γ<γ_{H}:=\left(\frac{N-2}{2}\right)^2,\,s\in (0,2)$ and $2^*(s)=\frac{2(N-s)}{(N-2)}.$ We prove a stability estimate for the corresponding Hardy-Sobolev inequality in the spirit of Bianchi-Egnell (1991). Also, we obtain a Struwe-type decomposition (1984) for the corresponding Euler-Lagrange equation. Finally, we prove a quantitative bound for one bubble, namely $\operatorname{dist}(u,\mathcal{M})\lesssim Γ(u)$ in the spirit of Ciraolo-Figalli-Maggi (2017).

Stability of Hardy-Sobolev Inequality

Abstract

Given we consider the critical Hardy-Sobolev equation in where and We prove a stability estimate for the corresponding Hardy-Sobolev inequality in the spirit of Bianchi-Egnell (1991). Also, we obtain a Struwe-type decomposition (1984) for the corresponding Euler-Lagrange equation. Finally, we prove a quantitative bound for one bubble, namely in the spirit of Ciraolo-Figalli-Maggi (2017).
Paper Structure (14 sections, 15 theorems, 145 equations)

This paper contains 14 sections, 15 theorems, 145 equations.

Table of Contents

  1. introduction
  2. The Hardy-Sobolev inequality on $\mathbb R^N$
  3. The Euclidean Sobolev inequality
  4. The Quantitative Stability problem and known results
  5. Quantitative Stability of Hardy-Sobolev
  6. Main results
  7. Structure of the paper
  8. Bianchi-Egnell's result for the H-S inequality
  9. Profile decomposition for Hardy-Sobolev equation
  10. Stability for the equation
  11. Appendix
  12. Nondegeneracy and spectral properties of the linearized operator
  13. Interaction estimates
  14. Symmetries of the problem

Key Result

Theorem 1.1

(BE) There exists a positive constant $\alpha>0$ ($\alpha$ depends only on $N$), The exponent $\frac{1}{2}$ is sharp in the sense that the inequality fails if we replace $\frac{1}{2}$ by any other exponent strictly bigger than $\frac{1}{2}$ as we consider $\delta_{Eu}(u)\to 0$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['BEHSE']}
  • Theorem 3.1
  • ...and 15 more