Abstract

We consider a sequence of multi-bubble solutions $ u_k$ of the following fourth order equation $$\qquad \qquad \! \Delta ^2 u_k = \rho _k \frac{ h e^{u_k}}{ \int _\Omega h e^{u_k}} \ \ \mbox {in} \ \Omega, \ \ u_k=\Delta u_k=0 \ \ \mbox {on} \ \partial \Omega,\qquad \qquad \qquad $$ 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, $\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 $\rho _k$: $$ \begin{aligned} \rho _k\! -\! 32 \sigma _3 m\! &=\! c_0 \sum _{j=1}^m\! \epsilon _{k, j}^2\! \left\! +\! \Delta R_4 \!+\! \frac{1}{32 \sigma _3} \Delta \log h \!\right)\hspace{-2.0pt}\\ &\quad + o\left \end{aligned} $$ where $c_0\!>\!0$, $\log \frac{64}{\epsilon _{k, j}^4 }\!=\!\!\! \max \limits _{x \in B_\delta }\! u_k \!-\!\log $ and $u_k \!\rightarrow \! 32 \sigma _3 \sum \limits _{j=1}^m G_4 $ in $ C^4_{\rm loc} $. This yields a bound of solutions as $\rho _k$ converges to $ 32 \sigma _3 m$ from below provided that $$ \sum _{j=1}^m \left + \Delta R_4 + \frac{1}{32 \sigma _3} \Delta \log h \right)>0.$$ The analytic work of this paper is the first step toward computing the Leray-Schauder degree of solutions of equation $$