# Ngô Quốc Anh

## January 11, 2011

### An example of sequence of blow-up solutions with finite limiting mass

Filed under: Nghiên Cứu Khoa Học, PDEs — Ngô Quốc Anh @ 14:35

In this note, we recall an example adapted from an elegant paper due to Y.Y. Li and I. Shafrir published in the Indiana Univ. Math. J. in 1994 [here].

Let us consider the asymptitic behavior of sequences of solutions of

$-\Delta u_n=V_n(x)e^{u_n}$

on a bounded domain $\Omega \subset \mathbb R^2$ with $V_n$ a non-negative continuous function. For each solution $u_n$, we call

$\displaystyle \alpha_n := \int_{B_R}V_ne^{u_n}dx$

the mass of $u_n$ (over a ball $B_R$). The terminology limiting mass will be referred to the limit $\lim_{n \to \infty} \alpha_n$.

For simplicity, we assume $V_n \equiv 1$. Given $m$, we are going to construct a sequence of solutions $\{u_n\}$ which blows up exactly at $m$ points, say at $a_1,...,a_m \in D$ where $D$ the unit disc of $\mathbb C$. Our equation reads as

$-\Delta u=e^{u}$

in $D$. Using the Liouville formula for solutions of the above equation, we get

$\displaystyle u(z) = \log \frac{{8|f'(z){|^2}}}{{{{(1 + |f(z){|^2})}^2}}}$

with $f$ an holomorphic function such that $f'(z) \ne 0$.