Sharp estimates for bubbling solutions of a fourth order mean field equation

Chang-Shou Lin; Juncheng Wei

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

  • Volume: 6, Issue: 4, page 599-630
  • ISSN: 0391-173X

Abstract

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We consider a sequence of multi-bubble solutions u k of the following fourth order equation Δ 2 u k = ρ k h ( x ) e u k ∫ Ω h e u k in Ω , u k = Δ u k = 0 on ∂ Ω , ( * ) where h is a C 2 , β positive function, Ω is a bounded and smooth domain in ℝ 4 , and ρ k is a constant such that ρ k ≤ C . We show that (after extracting a subsequence), lim k → + ∞ ρ k = 32 σ 3 m for some positive integer m ≥ 1 , where σ 3 is the area of the unit sphere in ℝ 4 . Furthermore, we obtain the following sharp estimates for  ρ k : ρ k - 32 σ 3 m = c 0 ∑ j = 1 m ϵ k , j 2 ∑ l ≠ j Δ G 4 ( p j , p l ) + Δ R 4 ( p j , p j ) + 1 32 σ 3 Δ log h ( p j ) + o ∑ j = 1 m ϵ k , j 2 where c 0 > 0 , log 64 ϵ k , j 4 = max x ∈ B δ ( p j ) u k ( x ) - log ( ∫ Ω h e u k ) and u k → 32 σ 3 ∑ j = 1 m G 4 ( · , p j ) in C loc 4 ( Ω ∖ { p 1 , ... , p m } ) . This yields a bound of solutions as ρ k converges to 32 σ 3 m from below provided that ∑ j = 1 m ∑ l ≠ j Δ G 4 ( p j , p l ) + Δ R 4 ( p j , p j ) + 1 32 σ 3 Δ log h ( p j ) > 0 . The analytic work of this paper is the first step toward computing the Leray-Schauder degree of solutions of equation ( * ) .

How to cite

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Lin, Chang-Shou, and Wei, Juncheng. "Sharp estimates for bubbling solutions of a fourth order mean field equation." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.4 (2007): 599-630. <http://eudml.org/doc/272296>.

@article{Lin2007,
abstract = {We consider a sequence of multi-bubble solutions $ u_k$ of the following fourth order equation\[\qquad \qquad \! \Delta ^2 u\_k = \rho \_k \frac\{ h(x) e^\{u\_k\}\}\{ \int \_\Omega h e^\{u\_k\}\} \ \ \mbox\{in\} \ \Omega , \ \ u\_k=\Delta u\_k=0 \ \ \mbox\{on\} \ \partial \Omega ,\qquad \qquad \qquad (*)\]where $h$ is a $C^\{2, \beta \}$ positive function, $\Omega $ is a bounded and smooth domain in $\mathbb \{R\}^4$, and $\rho _k$ is a constant such that $ \rho _k\! \le \! C$. We show that (after extracting a subsequence), $\lim _\{ k\rightarrow +\infty \} \rho _k \!=\! 32 \sigma _3 m $ for some positive integer $m\! \ge \! 1$, where $\sigma _3$ is the area of the unit sphere in $\mathbb \{R\}^4$. Furthermore, we obtain the following sharp estimates for $\rho _k$:\[ \begin\{aligned\} \rho \_k\! -\! 32 \sigma \_3 m\! &=\! c\_0 \sum \_\{j=1\}^m\! \epsilon \_\{k, j\}^2\! \left( \sum \_\{l \ne j\} \Delta G\_4 (p\_j,\! p\_l)\! +\! \Delta R\_4 (p\_j,\! p\_j)\!+\! \frac\{1\}\{32 \sigma \_3\} \Delta \log h(p\_j) \!\right)\hspace\{-2.0pt\}\\ &\quad + o\left(\sum \_\{j=1\}^m \epsilon \_\{k, j\}^2\right) \end\{aligned\} \]where $c_0\!&gt;\!0$, $\log \frac\{64\}\{\epsilon _\{k, j\}^4 \}\!=\!\!\! \max \limits _\{x \in B_\delta (p_j)\}\! u_k (x) \!-\!\log (\int \limits _\Omega h e^\{u_k\}) $ and $u_k \!\rightarrow \! 32 \sigma _3 \sum \limits _\{j=1\}^m G_4 (\cdot , p_j)$ in $ C^4_\{\rm loc\} ( \Omega \backslash \lbrace p_1, \ldots , p_m\rbrace )$. This yields a bound of solutions as $\rho _k$ converges to $ 32 \sigma _3 m$ from below provided that\[ \sum \_\{j=1\}^m \left( \sum \_\{l \ne j\} \Delta G\_4 (p\_j, p\_l) + \Delta R\_4 (p\_j, p\_j)+ \frac\{1\}\{32 \sigma \_3\} \Delta \log h(p\_j) \right)&gt;0.\]The analytic work of this paper is the first step toward computing the Leray-Schauder degree of solutions of equation $(*)$.},
author = {Lin, Chang-Shou, Wei, Juncheng},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {sharp estimates; fourth order mean field equation; Leray-Schauder degree of solution},
language = {eng},
number = {4},
pages = {599-630},
publisher = {Scuola Normale Superiore, Pisa},
title = {Sharp estimates for bubbling solutions of a fourth order mean field equation},
url = {http://eudml.org/doc/272296},
volume = {6},
year = {2007},
}

TY - JOUR
AU - Lin, Chang-Shou
AU - Wei, Juncheng
TI - Sharp estimates for bubbling solutions of a fourth order mean field equation
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2007
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 4
SP - 599
EP - 630
AB - We consider a sequence of multi-bubble solutions $ u_k$ of the following fourth order equation\[\qquad \qquad \! \Delta ^2 u_k = \rho _k \frac{ h(x) e^{u_k}}{ \int _\Omega h e^{u_k}} \ \ \mbox{in} \ \Omega , \ \ u_k=\Delta u_k=0 \ \ \mbox{on} \ \partial \Omega ,\qquad \qquad \qquad (*)\]where $h$ is a $C^{2, \beta }$ positive function, $\Omega $ is a bounded and smooth domain in $\mathbb {R}^4$, and $\rho _k$ is a constant such that $ \rho _k\! \le \! C$. We show that (after extracting a subsequence), $\lim _{ k\rightarrow +\infty } \rho _k \!=\! 32 \sigma _3 m $ for some positive integer $m\! \ge \! 1$, where $\sigma _3$ is the area of the unit sphere in $\mathbb {R}^4$. Furthermore, we obtain the following sharp estimates for $\rho _k$:\[ \begin{aligned} \rho _k\! -\! 32 \sigma _3 m\! &=\! c_0 \sum _{j=1}^m\! \epsilon _{k, j}^2\! \left( \sum _{l \ne j} \Delta G_4 (p_j,\! p_l)\! +\! \Delta R_4 (p_j,\! p_j)\!+\! \frac{1}{32 \sigma _3} \Delta \log h(p_j) \!\right)\hspace{-2.0pt}\\ &\quad + o\left(\sum _{j=1}^m \epsilon _{k, j}^2\right) \end{aligned} \]where $c_0\!&gt;\!0$, $\log \frac{64}{\epsilon _{k, j}^4 }\!=\!\!\! \max \limits _{x \in B_\delta (p_j)}\! u_k (x) \!-\!\log (\int \limits _\Omega h e^{u_k}) $ and $u_k \!\rightarrow \! 32 \sigma _3 \sum \limits _{j=1}^m G_4 (\cdot , p_j)$ in $ C^4_{\rm loc} ( \Omega \backslash \lbrace p_1, \ldots , p_m\rbrace )$. This yields a bound of solutions as $\rho _k$ converges to $ 32 \sigma _3 m$ from below provided that\[ \sum _{j=1}^m \left( \sum _{l \ne j} \Delta G_4 (p_j, p_l) + \Delta R_4 (p_j, p_j)+ \frac{1}{32 \sigma _3} \Delta \log h(p_j) \right)&gt;0.\]The analytic work of this paper is the first step toward computing the Leray-Schauder degree of solutions of equation $(*)$.
LA - eng
KW - sharp estimates; fourth order mean field equation; Leray-Schauder degree of solution
UR - http://eudml.org/doc/272296
ER -

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