We consider a sequence of multi-bubble solutions $u_k$ of the following fourth order equation $${\Delta }^2{u}_k={\rho }_k\frac{h\left(x\right)e^{u_k}}{{\int }_{\Omega }h{e}^{u_k}}\text{in}\Omega ,u_k=\Delta u_k=0\text{on}\partial \Omega ,(*)$$ 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 ${\rho }_k\le C$. We show that (after extracting a subsequence), ${lim}_{k\to +\infty }{\rho }_k=32{\sigma }_{3}m$ for some positive integer $m\ge 1$, where ${\sigma }_3$ is the area of the unit sphere in ${ℝ}^4$. Furthermore, we obtain the following sharp estimates for ${\rho }_k$: $$\begin{array}{cc}\hfill {\rho }_k-32{\sigma }_{3}m& =c_0\sum _{j=1}^m{ϵ}_{k,j}^2\left(\sum _{l\ne j}\Delta G_4(p_j,p_l)+\Delta R_4(p_j,p_j)+\frac{1}{32{\sigma }_3}\Delta logh\left(p_j\right)\right)\hfill \\ & +o\left(\sum _{j=1}^m{ϵ}_{k,j}^2\right)\hfill \end{array}$$ where $c_0\>0$, $log\frac{64}{{ϵ}_{k,j}^4}=\underset{x\in B_{\delta }\left(p_j\right)}{max}{u}_k\left(x\right)-log\left(\underset{\Omega }{\int }h{e}^{u_k}\right)$ and $u_k\to 32{\sigma }_3\sum _{j=1}^m{G}_4(·,p_j)$ in $C_{\mathrm{loc}}^4(\Omega \setminus \{p_1,...,p_m\})$. This yields a bound of solutions as...

Let $𝒟$ denote a true dimension function, i.e., a dimension function such that $𝒟\left({ℝ}^n\right)=n$ for all $n$. For a space $X$, we denote the hyperspace consisting of all compact connected, non-empty subsets by $C\left(X\right)$. If $X$ is a countable infinite product of non-degenerate Peano continua, then the sequence ${\left({𝒟}_{\ge n}\left(C\left(X\right)\right)\right)}_{n=2}^{\infty }$ is $F_{\sigma }$-absorbing in $C\left(X\right)$. As a consequence, there is a homeomorphism $h:C\left(X\right)\to Q^{\infty }$ such that for all $n$, $h\left[\{A\in C\left(X\right):𝒟\left(A\right)\ge n+1\}\right]=B^n\times Q\times Q\times \cdots$, where $B$ denotes the pseudo boundary of the Hilbert cube $Q$. It follows that if $X$ is a countable infinite product of non-degenerate...

The fixed infinitely differentiable function $\rho \left(x\right)$ is such that $\left\{n\rho \right(nx\left)\right\}$ is a regular sequence converging to the Dirac delta function $\delta$. The function ${\delta }_{𝐧}\left(𝐱\right)$, with $𝐱=(x_1,\cdots ,x_m)$ is defined by $${\delta }_{𝐧}\left(𝐱\right)=n_1\rho \left(n_1{x}_1\right)\cdots n_m\rho \left(n_m{x}_m\right).$$ The product $f\circ g$ of two distributions $f$ and $g$ in ${𝒟}_m^{\text{'}}$ is the distribution $h$ defined by $$\underset{n_1\to \infty }{error}\cdots \underset{n_m\to \infty }{error}\langle f_{𝐧}{g}_{𝐧},\phi \rangle =\langle h,\phi \rangle ,$$ provided this neutrix limit exists for all $\phi \left(𝐱\right)={\phi }_1\left(x_1\right)\cdots {\phi }_m\left(x_m\right)$, where $f_{𝐧}=f*{\delta }_{𝐧}$ and $g_{𝐧}=g*{\delta }_{𝐧}$.