广州数学大讲坛第十一期

第一百零七讲——华中师范大学王春花教授学术报告

题目：Quantization analysis of Moser-Trudinger equations in the Poincar\'e disk and applications

时间: 2024年12月28日(星期六) 14:00-16:30

地点: 腾讯会议（会议ID:113-682-991）

报告人: 王春花 教授

摘要：In this talk, we first study the quantitative properties for positive solutions to the Moser-Trudinger equations in the two-dimensional Poincare disk $\mathbb{B}^2$:

\begin{equation*}\label{mt1}

\left\{

\begin{aligned}

&-\Delta_{\mathbb{B}^2}u=\lambda ue^{u^2},\ x\in\mathbb{B}^2,\\

&u\to0,\ \text{when}\ \rho(x)\to\infty,\\

&||\nabla_{\mathbb{B}^2} u||_{L^2(\mathbb{B}^2)}^2\leq M_0,

\end{aligned}

\right.

\end{equation*}

where

$0<\lambda<\frac{1}{4}=\inf\limits_{u\in W^{1,2}(\mathbb{B}^2)\backslash\{0\}}

\frac{\|\nabla_{\mathbb{B}^2}u\|_{L^2(\mathbb{B}^2)}^2}{\|u\|_{L^2(\mathbb{B}^2)}^2}$,

$\rho(x)$ denotes the geodesic distance between $x$ and the origin and $M_0$ is a fixed large positive constant. Then by doing a delicate expansion for Dirichlet energy

$\|\nabla_{\mathbb{B}^2}u\|_{L^2(\mathbb{B}^2)}^2$

when $\lambda$ approaches to $0$, we prove that there exists $\Lambda^\ast>4\pi$ such that the Moser-Trudinger functional

$F(u)=\int_{\mathbb{B}^2}\left(e^{u^2}-1\right) dV_{\mathbb{B}^2}$

under the constraint

$\int_{\mathbb{B}^2}|\nabla_{\mathbb{B}^2}u|^2 dV_{\mathbb{B}^2}=\Lambda$

has at least one positive critical point for $\Lambda\in(4\pi,\Lambda^{\ast})$ up to some M\"{o}bius transformation. Finally, when $\lambda\rightarrow 0$, by doing a more accurate expansion for $u$ near the origin and away from the origin, applying a local Pohozaev identity around the origin and the uniqueness of the Cauchy initial value problem for ODE, we prove that the Moser-Trudinger equation only has one positive solution when $\lambda$ is close to $0$.