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数学学科现代分析及其应用研究所(2021非线性分析与偏微分方程系列报告会三十五)

发布者:付慧娟   发布时间:2021-01-18  浏览次数:10

报告题目67 Sign-changing solutions for some nonlinear Schrödinger equations

报告人:王征平  武汉理工大学

会议时间:121日(周四),8:30-9:05

腾讯会议ID:  904 227 764,

https://meeting.tencent.com/s/NywtARdxwSmZ

要:In this talk, we will present some recent results on the existence of sign-changing solutions for some nonlinear Schrödinger equations. For the Schrödinger-Poisson equation and fractional Schrödinger equation, by variational methods we prove that there exists at least one least energy nodal solution.

报告人简介武汉理工大学数学科学研究中心教授。2001年本科毕业于华中师范大学数学系,2007年于中国科学院武汉物理与数学研究所获博士学位,2015年入选中科院青年创新促进会会员,2018年获湖北省自然科学奖二等奖,主持国家自然科学基金面上项目2项。在非线性薛定谔方程孤立波解的数学理论研究等方面取得了一些重要结果并发表在国际权威学术刊物上,如Archive for Rational Mechanics and Analysis, Calculus of Variations and Partial Differential EquationsJournal of the European Mathematical Society等。研究结果被SCI他引300余次。


报告题目68: Existence and asymptotic behavior of Normalized Solutions for the Kirchhoff equation with singular potential

报告人:张贻民  武汉理工大学

会议时间:121日(周四),9:05-9:40

腾讯会议ID:  904 227 764,

https://meeting.tencent.com/s/NywtARdxwSmZ

摘要:In this talk, we first consider the existence on normalized solutions for a class Kirchhoff equation with singular potential. Then, we can find some interesting things. If $p<p^*=8/N$, the minimization problem exists minimizers for any $\beta$. But, for $p=p^*$, the minimization problem exists minimizers for $\beta\in (0,\beta_{p^*})$ or does not exist for $\beta>\beta_{p^*}$. Then for $\beta>\beta_{p^*}$, we can obtain a blow-up analysis for $p$ tends to $p^*$.  But in the case $p=p^*$ and $\beta=\beta_{p^*}$, there is minimizers for minimization problem with $N=1$, there is no minimizers for minimization problem with $N=2,3$. Hence, for $N=2,3$, we can obtain asymptotic behavior of minimizers not only for $p$ tends to $p^*$, but also for $\beta$ tends to $\beta_{p^*}$.

报告人简介教授,博士生导师2003年四川大学数学科学学院本科毕业,2009年华南理工大学获得理学博士学位,2010年至2012年在中国科学院武汉物理与数学研究所从事博士后研究工作,2012年至2016年任中国科学院武汉物理与数学研究所副研究员,2012年被中国科学院大学聘为硕士导师,招生方向为应用数学,2013年6月访问南开大学陈省身数学研究所王志强教授,2014年3-8月国家公派访问澳大利亚新英格兰大学严树森教授,2016年起任武汉理工大学教授。主持国家自然科学基金青年基金一项,主持国家自然科学基金面上项目一项,作为主要参与人参与国家自然科学基金面上项目两项。主要研究方向为非线性泛函分析和非线性偏微分方程。学术成果主要发表于J. Differ. Equ.J. Math. Phys. Comm. Conte. Mathe.Proceedings Amer. Math. Socie., Top. Meth. Nonlinear Anal.等国际数学期刊上


报告题目69: Large number of bubble solutions for a fractional elliptic equation with almost critical exponents.

报告人:王春花  华中师范大学

会议时间:121日(周四),9:40-10:15

腾讯会议ID: 904 227 764,

https://meeting.tencent.com/s/NywtARdxwSmZ

摘要We study the following nonlinear equation with a fractional Laplacian operator and almost critical exponents

$$

(-\Delta)^{s} u=K(|y'|,y'')u^{\frac{N+2s}{N-2s}\pm\epsilon},\,\,u>0,\,\,u\in

D^{1,s}(\R^{N}),

$$

where $N\geq 4$, $0<s<1$, $(y',y'')\in \R^2\times\R^{N-2},$ $\epsilon>0$ is a small parameter and $K(y)$ is nonnegative and bounded. Under some suitable assumptions of the potential function $K(r,y'')$, we will use the finite dimensional reduction method and some local Pohozaev identities to prove that the above problem has large number of bubble solutions. The concentration points of the bubble solutions include a saddle point of $K(y)$. Moreover, the functional energy of these solutions are in the order $\epsilon^{-\frac{N-2s-2}{(N-2s)^2}}$. This is based on a joint work with Suting Wei.

报告人简介:华中师范大学副教授,硕士生导师。2012年博士毕业于华中师范大学,留校任教。研究方向为非线性泛函分析和非线性椭圆型偏微分方程,主要兴趣是利用非线性泛函分析、变分方法研究椭圆方程解的存在性以及解的相关性质。主持国家自然科学基金青年基金和国家自然科学基金面上基金等。主要成果发表在JFA、JDE、DCDS-A、JMP、Edinburgh Sec.A等国际数学期刊上。

 

报告题目70 Ground States of Pseudo-Relativistic Boson Stars under the Critical Stellar Mass.

报告人曾小雨  武汉理工大学

会议时间:121日(周四),10:15-10:50

腾讯会议ID:  904 227 764,

https://meeting.tencent.com/s/NywtARdxwSmZ

摘要We consider ground states of pseudo-relativistic boson stars with a self-interacting potential $K(x)$ in $\R^3$, which can be described by minimizers of the pseudo-relativistic Hartree energy functional. Under some assumptions on $K(x)$, we prove that minimizers exist if the stellar mass $N$ satisfies $0<N<N^*$, and there is no minimizer if $N>N^*$, where $N^*$ is called the critical stellar mass. we also present a detailed analysis of the behavior of minimizers as $N$ approaches $N^*$ from below, for which the stellar mass concentrates at a unique point. Moreover, we prove that the ground state is unique for the particular case where $N>0$ is small enough.

报告人简介曾小雨,理学博士,武汉理工大学教授。研究方向为非线性泛函分析及椭圆型偏微分方程。主要从事与薛定谔方程以及玻色-爱因斯坦凝聚相关的变分问题研究。主要成果发表在Trans.AMS、JFA、Ann. Inst. H. Poincar'eAnal. Non Lin'eaire、Nolinearity、J. Differential Equations等国际数学期刊上。

 

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