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)\}.

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