Sharp estimates for bubbling solutions of a fourth order mean field equation
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 6 (2007) no. 4, p. 599-630
We consider a sequence of multi-bubble solutions ${u}_{k}$ of the following fourth order equation $\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\Delta }^{2}{u}_{k}={\rho }_{k}\frac{h\left(x\right){e}^{{u}_{k}}}{{\int }_{\Omega }h{e}^{{u}_{k}}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\Omega ,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{u}_{k}=\Delta {u}_{k}=0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{on}\phantom{\rule{4pt}{0ex}}\partial \Omega ,\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\left(*\right)$ where $h$ is a ${C}^{2,\beta }$ positive function, $\Omega$ is a bounded and smooth domain in ${ℝ}^{4}$, and ${\rho }_{k}$ is a constant such that ${\rho }_{k}\phantom{\rule{-0.166667em}{0ex}}\le \phantom{\rule{-0.166667em}{0ex}}C$. We show that (after extracting a subsequence), ${lim}_{k\to +\infty }{\rho }_{k}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}32{\sigma }_{3}m$ for some positive integer $m\phantom{\rule{-0.166667em}{0ex}}\ge \phantom{\rule{-0.166667em}{0ex}}1$, where ${\sigma }_{3}$ is the area of the unit sphere in ${ℝ}^{4}$. Furthermore, we obtain the following sharp estimates for ${\rho }_{k}$: $\begin{array}{cc}\hfill {\rho }_{k}\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}32{\sigma }_{3}m\phantom{\rule{-0.166667em}{0ex}}& =\phantom{\rule{-0.166667em}{0ex}}{c}_{0}\sum _{j=1}^{m}\phantom{\rule{-0.166667em}{0ex}}{ϵ}_{k,j}^{2}\phantom{\rule{-0.166667em}{0ex}}\left(\sum _{l\ne j}\Delta {G}_{4}\left({p}_{j},\phantom{\rule{-0.166667em}{0ex}}{p}_{l}\right)\phantom{\rule{-0.166667em}{0ex}}+\phantom{\rule{-0.166667em}{0ex}}\Delta {R}_{4}\left({p}_{j},\phantom{\rule{-0.166667em}{0ex}}{p}_{j}\right)\phantom{\rule{-0.166667em}{0ex}}+\phantom{\rule{-0.166667em}{0ex}}\frac{1}{32{\sigma }_{3}}\Delta logh\left({p}_{j}\right)\phantom{\rule{-0.166667em}{0ex}}\right)\phantom{\rule{-2.0pt}{0ex}}\hfill \\ & \phantom{\rule{1em}{0ex}}+o\left(\sum _{j=1}^{m}{ϵ}_{k,j}^{2}\right)\hfill \end{array}$ where ${c}_{0}\phantom{\rule{-0.166667em}{0ex}}>\phantom{\rule{-0.166667em}{0ex}}0$, $log\frac{64}{{ϵ}_{k,j}^{4}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\underset{x\in {B}_{\delta }\left({p}_{j}\right)}{max}\phantom{\rule{-0.166667em}{0ex}}{u}_{k}\left(x\right)\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}log\left(\underset{\Omega }{\int }h{e}^{{u}_{k}}\right)$ and ${u}_{k}\phantom{\rule{-0.166667em}{0ex}}\to \phantom{\rule{-0.166667em}{0ex}}32{\sigma }_{3}\sum _{j=1}^{m}{G}_{4}\left(·,{p}_{j}\right)$ in ${C}_{\mathrm{loc}}^{4}\left(\Omega \setminus \left\{{p}_{1},...,{p}_{m}\right\}\right)$. This yields a bound of solutions as ${\rho }_{k}$ converges to $32{\sigma }_{3}m$ from below provided that $\sum _{j=1}^{m}\left(\sum _{l\ne j}\Delta {G}_{4}\left({p}_{j},{p}_{l}\right)+\Delta {R}_{4}\left({p}_{j},{p}_{j}\right)+\frac{1}{32{\sigma }_{3}}\Delta logh\left({p}_{j}\right)\right)>0.$ The analytic work of this paper is the first step toward computing the Leray-Schauder degree of solutions of equation $\left(*\right)$.
Classification:  35B40,  35B45,  35J40
@article{ASNSP_2007_5_6_4_599_0,
author = {Lin, Chang-Shou and Wei, Juncheng},
title = {Sharp estimates for bubbling solutions of a fourth order mean field equation},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 6},
number = {4},
year = {2007},
pages = {599-630},
zbl = {1185.35067},
mrnumber = {2394412},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2007_5_6_4_599_0}
}

Lin, Chang-Shou; Wei, Juncheng. Sharp estimates for bubbling solutions of a fourth order mean field equation. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 6 (2007) no. 4, pp. 599-630. http://www.numdam.org/item/ASNSP_2007_5_6_4_599_0/

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