On the stability of critical points of the Hardy-Littlewood-Sobolev inequality

This paper is concerned with the quantitative stability of critical points of the Hardy-Littlewood-Sobolev inequality. Namely, we give quantitative estimates for the Choquard equation: $$-\Delta u=(I_{\mu}\ast|u|^{2_\mu^*}) u^{2_\mu^*-1}\ \ \text{in}\ \ \R^N,$$ where $u>0,\ N\geq 3,\ \mu\in(0,N)$, $I_{\mu}$ is the Riesz potential and $2_\mu^* \coloneqq \frac{2N-\mu}{N-2}$ is the upper Hardy-Littlewood-Sobolev critical exponent. The Struwe's decomposition (see M. Struwe: Math Z.,1984) showed that the equation $\Delta u + u^{\frac{N+2}{N-2 }}=0$ has phenomenon of ``stable up to bubbling'', that is, if $u\geq0$ and $\|\Delta u+u^{\frac{N+2}{N-2}}\|_{(\mathcal{D}^{1,2})^{-1}}$ approaches zero, then $d(u)$ goes to zero, where $d(u)$ denotes the $\mathcal{D}^{1,2}(\R^N)$-distance between $u$ and the set of all sums of Talenti bubbles. Ciraolo, F{}igalli and Maggi (Int. Math. Res. Not.,2017) obtained the f{}irst quantitative version of Struwe's decomposition with single bubble in all dimensions $N\geq 3$, i.e, $\displaystyle d(u)\leq C\|\Delta u+u^{\frac{N+2}{N-2}}\|_{L^{\frac{2N}{N+2}}}.$ For multiple bubbles, F{}igalli and Glaudo (Arch. Rational Mech. Anal., 2020) obtained quantitative estimates depending on the dimension, namely $$ d(u)\leq C\|\Delta u+u^{\frac{N+2}{N-2}}\|_{(\mathcal{D}^{1,2})^{-1}}, \hbox{ where } 3\leq N\leq 5,$$ which is invalid as $N\geq 6.$ \vskip0.1in \quad In this paper, we prove the quantitative estimate of the Hardy-Littlewood-Sobolev inequality, we get $$d(u)\leq C\|\Delta u +(I_{\mu}\ast|u|^{2_\mu^*})|u|^{2_\mu^*-2}u\|_{(\mathcal{D}^{1,2})^{-1}}, \hbox{ when } N=3 \hbox{ and } 5/2< \mu<3.$$