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 $f'(z) \ne 0$.

For some fixed distinct points $b_1,...,b_m \in \mathbb C$, we consider the functions $\{f_n\}_{n=1}^\infty$ as follows $f_n(z)=np(z)=n(z-b_1)....(z-b_m)$,

and the corresponding $\{v_n\}$ given by the Liouville formula.

It is obvious to see that $\{v_n\}$ does blow up at $b_1,...,b_m$ but it is not defined at the points $c_1,...,c_{m-1}$ the zeros of the polynomial $p'(z)$. So we choose a smooth bounded and simply connected domain $\Omega$ containing $b_1,..,b_m$ but avoiding $c_1,...,c_{m-1}$.

Let $\varphi$ be the Riemann mapping of $D$ onto $\Omega$. We define on $D$ the sequence $u_n(z)=v_n(\varphi(z))+2\log (|\varphi'(z)|)$.

It is easy to see that sequence $\{u_n\}$ is a sequence of solutions on $D$ which blows up at $\{a_1,...,a_m\}=\{\varphi^{-1}(b_1),...,\varphi^{-1}(b_m)\}$.