非线性泛函分析团队学术报告

2021.11.04

报告人：钟学秀，华南师范大学

报告时间：2021年11月4日（星期四）上午11：00-12：00

报告地点： 腾讯会议 ID：124 205 336

报告题目：On the positive normalized solution to the Sch\"odinger equation with general nonlinearities

摘要：In the present paper, we study the asymptotic behaviors of nontrivial nonnegative solutions to the following Schr\"odinger equation with very general nonlinearities $$-\Delta u+\lambda u=g(u)\;\hbox{in}\;\R^N,$$ as $\lambda\rightarrow 0^+$ and $\lambda\rightarrow +\infty$.

In the framework of present paper, the well known Rabinowitz global theorem is not applicable since $\lambda=0$ is an essential spectrum and there is also no bifurcation point $\lambda>0$. However, basing on the fixed point index in cones and the continuation method, we can still find out a global branch of the positive solutions. As an application, we present a new approach to study the normalized solutions problem, i.e., solutions satisfying a prescribed mass $\displaystyle\int_{\R^N}u^2=a>0$. We can obtain some exciting results on the normalized solutions under very \textcolor{red}{mild} assumptions on $g$, including existence, nonexistence and multiplicity, etc.

合作者: Louis Jeanjean (Universit\'e de Bourgogne Franche-Comt\'e) and 张建军（重庆交通大学）

关键字: Schr\"odinger equation; positive normalized solution; global branch；essential spetrum.

报告人简介：钟学秀，华南师范大学副研究员，华南数学应用与交叉研究中心青年拔尖引进人才。主要研究领域为椭圆偏微分方程、泛函分析和变分法。2015年博士毕业于清华大学，获清华大学优秀博士学位论文一等奖和优秀博士毕业生。2015-2017年在台湾大学理论科学研究中心跟随林长寿教授做博士后。主持国家基金一项，广东省基金两项，广州市基金一项。已在JDG、Math. Ann.、Ann. Sc. Norm. Super. Pisa Cl. Sci.、CVPDE、JDE等国际重要刊物上发表多篇学术论文。