报告题目： The number of positive solutions for n-coupled elliptic systems
时 间：2024年11月7日（星期四）16：00
地 点：理工北楼601
主 办：数学与统计学院、分析数学及应用教育部重点实验室、福建省分析数学及应用重点实验室、福建省应用数学中心（福建师范大学）
参加对象：感兴趣的老师和学生
报告摘要：We study the number of positive solutions to the $n$-coupled elliptic system
$$-\Delta u_i=\mu_iu_i^{2^*-1}+\sum_{j=1,\, j\neq i}^n\beta_{ij} u_i^{p_{ij}-1}u_j^{q_{ij}},\ u_i\in\mathscr{D}^{1,2}(\R^N),\ i=1,2,\cdots,n,$$where $N\geq3$, $n\geq2$, $\mu_i>0$, $\beta_{ij}>0$, $p_{ij}<2^*$, and $p_{ij}+q_{ij}=2^*$ for $i\neq j\in\{1,2,\cdots,n\}$. We prove new multiplicity and uniqueness results for positive solutions of the system, whether the system has a variational structure or not. In some cases we provide a rather complete characterization on the exact number of positive solutions. The results we obtain reveal that the positive solution set of this system has very different structures in the three cases $p_{ij}<2$, $p_{ij}=2$, and $2<p_{ij}<2^*$. Moreover, when $2<p_{ij}<2^*$, very different structures of the positive solution set can also be seen in the case where $p_{ij}$ close to $2$ and the case where $p_{ij}$ close to $2^*$. Similar results are given for elliptic systems with subcritical Sobolev exponents. This is joint work with Yongtao Jing, Haidong Liu, Yanyan Liu and Juncheng Wei.
报告人简介： 刘兆理，首都师范大学数学科学学院教授，博士生导师，北京市特聘教授。从事非线性分析研究，在变分方法和椭圆型偏微分方程方面做出了多项研究成果。曾先后四次获得省部级自然科学奖和科技进步奖，曾获得国家杰出青年科学基金，被评为教育部“长江学者特聘教授”，曾获得首都劳动奖章。