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

Abstract

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: $$-\Delta u=(I_{\mu }\ast |u{|}^{2_{\mu }^{\ast }})u^{2_{\mu }^{\ast }-1}\text{in}{\mathbb{R}}^N,$$ where $u>0,N\ge 3,\mu \in (0,N)$, $I_{\mu }$ is the Riesz potential and $2_{\mu }^{\ast }:=\frac{2N-\mu }{N-2}$ 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 $u\ge 0$ and $\Vert \Delta u+u^{\frac{N+2}{N-2}}{\Vert }_{({\mathcal{D}}^{1,2}{)}^{-1}}$ approaches zero, then $d(u)$ goes to zero, where $d(u)$ denotes the ${\mathcal{D}}^{1,2}({\mathbb{R}}^N)$-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 $N\ge 3$, i.e, $d(u)\le C\Vert \Delta u+u^{\frac{N+2}{N-2}}{\Vert }_{L^{\frac{2N}{N+2}}}.$ For multiple bubbles, F{}igalli and Glaudo (Arch. Rational Mech. Anal., 2020) obtained quantitative estimates depending on the dimension, namely $$d(u)\le C\Vert \Delta u+u^{\frac{N+2}{N-2}}{\Vert }_{({\mathcal{D}}^{1,2}{)}^{-1}},where\; 3\le N\le 5,$$ which is invalid as $N\ge 6.$ \vskip0.1in \quad In this paper, we prove the quantitative estimate of the Hardy-Littlewood-Sobolev inequality, we get $$d(u)\le C\Vert \Delta u+(I_{\mu }\ast |u{|}^{2_{\mu }^{\ast }})|u{|}^{2_{\mu }^{\ast }-2}u{\Vert }_{({\mathcal{D}}^{1,2}{)}^{-1}},whenN=3\; and\; 5/2<\mu <3.$$