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

Abstract

This paper investigates sharp stability estimates for the fractional Hardy-Sobolev inequality: $${\mu }_{s,t}\left({\mathbb{R}}^N\right){\left({\int }_{{\mathbb{R}}^N}\frac{|u{|}^{2_s^{\ast }(t)}}{|x{|}^t}\mathrm{d}x\right)}^{\frac{2}{2_s^{\ast }(t)}}\le {\int }_{{\mathbb{R}}^N}{\left|(-\Delta {)}^{\frac{s}{2}}u\right|}^2\mathrm{d}x,\text{for all }u\in {\dot{H}}^s\left({\mathbb{R}}^N\right),$$ where $N>2s$, $s\in (0,1)$, $0<t<2s<N$, and $2_s^{\ast }(t)=\frac{2(N-t)}{N-2s}$. Here, ${\mu }_{s,t}\left({\mathbb{R}}^N\right)$ 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.

Minor improvements to the overall exposition

References in corpus (1)

• Stability with explicit constants of the critical points of the fractional Sobolev inequality and applications to fast diffusion