## 魏军城: Sharp quantitative estimates for Struwe's decomposition

：魏军城 教授

Suppose $u\in \dot{H}^1(\mathbb{R}^n)$. In a seminal work,  Struwe  proved that if $u\geq 0$ and $\Gamma(u):=\|\Delta u+u^{\frac{n+2}{n-2}}\|_{H^{-1}}\to 0$ then $dist(u,\mathcal{T})\to0$,where$dist(u,\mathcal{T})$denotesthe$\dot{H}^1(\mathbb{R}^n)$-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 $dist (u,\mathcal{T}) \leq C \Gamma (u)$. For Struwe's decomposition with two or more bubbles, Figalli and Glaudo  showed a striking  dimensional dependent quantitative estimate, namely  $dist(u,\mathcal{T})\leq C \Gamma(u)$ when $3\leq n\leq 5$ while this is false for $n\geq 6$. In this talk, I will present the following sharp estimate$dist(u,\mathcal{T})\leqC\begin{cases}\Gamma(u)\left|\log\Gamma(u)\right|^{\frac{1}{2}}\quad&\textit{if }n=6, |\Gamma(u)|^{\frac{n+2}{2(n-2)}}\quad&\textit{if }n\geq 7.\end{cases}$. Furthermore, we show that this inequality is sharp. (Joint work with B. Deng and L. Sun)