报告题目: Positive least energy solutions for k-coupled Schrodinger system with critical exponent

报告人：清华大学邹文明教授

报告时间：2020年10月26日星期一 15：00-17：30

报告地点：数学楼1楼145报告厅

报告摘要: In this talk, I will show the following $k$-coupled nonlinear Schr\"odinger system with Sobolev critical exponent:

\begin{equation*}

\left\{

\begin{aligned}

-\Delta u_i & +\lambda_iu_i =\mu_i u_i^{2^*-1}+\sum_{j=1,j\not= i}^{k} \beta_{ij} u_{i}^{\frac{2^*}{2}-1}u_{j}^{\frac{2^*}{2}} \quad \hbox{in}\;\Omega,\\

u_i&>0 \quad \hbox{in}\; \Omega \quad \hbox{and}\quad u_i=0 \quad \hbox{on}\;\partial\Omega, \quad i=1,2,\cdots, k.

\end{aligned}

\right.

\end{equation*}

Here $\Omega $ is a smooth bounded domain, $2^{*}=\frac{2N}{N-2}$ is the Sobolev critical exponent, $-\lambda_1(\Omega)<\lambda_i<0, \mu_i>0$ ~and~$ \beta_{ij}=\beta_{ji}\ne 0$, where $\lambda_1(\Omega)$ is the first eigenvalue of $-\Delta$ with the Dirichlet boundary condition.

We characterize the positive least energy solution of the $k$-coupled system for the purely cooperative case $\beta_{ij}>0$, in higher dimension $N\ge 5$.Since the $k$-coupled case is much more delicated, we shall introduce {\it the idea of induction}. % than $2$-couple case,

We point out that {\it the key idea} is to give a more accurate upper bound of the least energy. It's interesting to see that the least energy of the $k$-coupled system decreases as $k$ grows. Moreover, we establish the existence of positive least energy solution of the limit system in $ \mathbb{R}^N$, as well as classification results. This is a joint work with X. Yin.

报告人简介：邹文明教授现为清华大学数学科学系系主任、中国数学会常务理事；国家杰出青年基金获得者、教育部数学专业教指委委员；荣获政府特殊津贴，曾任清华大学基础数学研究所所长。邹教授目前担任国际SCI刊物 《中国科学-数学》、《Minimax Theory and its Application》和《Advances in Nonlinear Analysis》 编委。邹文明教授在变分与拓扑方法、偏微分方程、 Hamiltonian系统等方面做出了一系列重要的成果：学术上首次建立Multi-Bump解和Morse理论的关系、并解决4维及以上的周期位势和临界指数增长薛方程Multi-Bump解、系统建立了没有PS紧性的无穷维弱环绕理论。在Bahri-Lions-Rabinowitz 著名的扰动问题、Brezis-Nirenberg 临界指数型问题、Li-Lin 的公开问题、四维Bose-Einstein凝聚椭圆方程组基态解、三维Bose-Einstein 凝聚椭圆系统规范解、Lane-Emden方程分类的研究上许多成果处于领先的位置。在美国Springer-New York出版英文专著二部，系统地建立了变号临界点理论框架和一系列新的临界点抽象定理。在欧美的国际刊物上发布SCI论文120余篇， MathSciNet显示文章被引用2500次。引发他人许多后续研究。