Sharp quantitative estimates of Struwe's Decomposition

Abstract

Suppose $u\in {\dot{H}}^1({\mathbb{R}}^n)$. In a seminal work, Struwe proved that if $u\ge 0$ and $\Vert \Delta u+u^{\frac{n+2}{n-2}}{\Vert }_{H^{-1}}:=\Gamma (u)\to 0$ then $dist(u,\mathcal{T})\to 0$, where $dist(u,\mathcal{T})$ denotes the ${\dot{H}}^1({\mathbb{R}}^n)$-distance of $u$ from the manifold of sums of Talenti bubbles. Ciraolo, Figalli and Maggi obtained the first quantitative version of Struwe's decomposition with one bubble in all dimensions, namely $\delta (u)\le C\Gamma (u)$. For Struwe's decomposition with two or more bubbles, Figalli and Glaudo showed a striking dimensional dependent quantitative estimate, namely $\delta (u)\le C\Gamma (u)$ when $3\le n\le 5$ while this is false for $n\ge 6$. In this paper, we show that \[dist (u,\mathcal{T})\leq C\begin{cases} Γ(u)\left|\log Γ(u)\right|^{\frac{1}{2}}\quad&\text{if }n=6, |Γ(u)|^{\frac{n+2}{2(n-2)}}\quad&\text{if }n\geq 7.\end{cases}\] Furthermore, we show that this inequality is sharp.

49 pages; comments are welcome