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--- 分析与偏微分方程讨论班(2021秋季第2讲)

题目: Positive normalized solution to the Kirchhoff equation with general nonlinearities of mass super-critical

报告人: 钟学秀 副研究员 （华南师范大学）

时间： 2021-10-21 14：30-15：30 (周四下午)

地点: E404

摘要: In present paper, we study the normalized solutions $(\lambda_c, u_c)\in \R\times H^1(\R^N)$ to the following Kirchhoff problem

$$

-\left(a+b\int_{\R^N}|\nabla u|^2dx\right)\Delta u+\lambda u=g(u)~\hbox{in}~\R^N,\;1\leq N\leq 3

$$

satisfying the normalization constraint

$

\displaystyle\int_{\R^N}u^2=c,

$

which appears in free vibrations of elastic strings. The parameters $a,b>0$ are prescribed as is the mass $c>0$. The nonlinearities $g(s)$ considered here are very general and of mass super-critical. Under some suitable assumptions, we can prove the existence of ground state normalized solutions for any given $c>0$. After a detailed analysis via the blow up method, we also make clear the asymptotic behavior of these solutions as $c\rightarrow 0^+$ as well as $c\rightarrow+\infty$. This work is jointed with Qihan He (Guangxi University), Zongyan Lv(Beijing Normal University) and Yimin Zhang (Wuhan University of Technology).

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。

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