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

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