题    目：Optimal existence and limiting profile of normalized ground states  for a quasi-linear Schr\"odinger equations: Mass super-critical case

时    间：2024年12月6日（周五）上午9：00-12：00

地    点：数学与统计楼213报告厅

报 告 人：华南师范大学副研究员  钟学秀

摘  要：Consider the existence of normalized ground states of the following quasi-linear elliptic equation:

$$-\Delta u-\Delta(|u|^2)u+\lambda u=|u|^{p-2}u~\quad \hbox{in}~\R^N, N\geq 1,$$

with prescribed mass $\displaystyle\int_{\R^N}|u|^2 \mathrm{d}x=a$. We are interested in the mass super-critical case $4+\frac{4}{N}<p<2\cdot 2^*$, where $2^*:=\frac{2N}{N-2}$ for $N\geq 3$, while $2^*:=+\infty$ for $N=1,2$.

 Our existence results are optimal. Precisely, we can prove the existence of normalized ground state for all mass $a>0$ when $1\leq N\leq 4$. While for $N\geq 5$, we can find a precise number $a_0$ such that the existence of normalized ground state is true if and only if $a\in (0, a_0]$.

 This is the first result in the super $H^1$-critical case (i.e. $p>2^*$). In previous literature, scholars have utilized the key condition $p\leq 2^*$ to obtain the $L^2$-compactness. And thus this is also the first result for the mass super-critical $p>4+\frac{4}{N}$ when $N\geq 4$.

 We also study the asymptotic behavior of the normalized ground states as the mass $a \downarrow 0$, as well as $a$ goes to the upper bound $a^*$, where $a^*=+\infty$ for $1\leq N\leq 4$, while $a^*=a_0$ for $N\geq 5$. For the completeness of this project, we also study the uniqueness of positive radial solutions to a class of overdetermined problem, which extend the results of Serrin and Tang[Indiana Univ. Math. J., 2000, 897--923] for the semilinear case to dimension one and two.

 This is a joint work with Prof. Louis Jeanjean and Prof. Jianjun Zhang.

报告人简介：

钟学秀，2015年博士毕业于清华大学，师从邹文明教授。2015-2017年于台湾大学博士后；2017-2019年于中山大学专职科研人员；2019年至今为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，最新ESI高被引学者。研究方向为运用非线性分析、变分法等方法来研究几何分析学、数学物理中椭圆型偏微分方程和方程组以及某些不等式问题。主持国家青年基金和面上基金各一项。目前已在J.Differential Geom., J. Math. Pures Appl., Math. Ann., Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Calc. Var. PDE, J. Differential Equations等国际重要刊物上发表多篇学术论文。在非线性泛函分析和椭圆偏微分方程领域的Li-Lin 公开问题，Sirakov公开问题，Bartsch-Jeanjean-Soave公开问题等方面获得了重要进展。