2023年椭圆偏微分方程

与变分法青年学者论坛

深圳大学 数学与统计学院

2023 年 4 月 14 日-17日

会议组织：余晓辉（深圳大学）

椭圆偏微分方程与变分法青年学者论坛

 

非线性椭圆方程在现代科学中具有重要的地位，比如，在生物学中，它可以用来刻

画细菌和物种的分布行为，可用于建立肿瘤模型。在经济学中，平均场方程可以用来描述

众多个体的博弈行为。在非线性光学中，薛定谔方程用于刻画光子的动力学行为。除此之

外，非线性椭圆方程在等离子物理、广义相对论和材料科学中也有着重要的应用。针对椭

圆方程领域的一些公开问题，为了加强国内椭圆偏微分方程与变分法相关领域青年学者之

间的沟通，交流各自的最新研究成果，促进彼此间的相互合作，深圳大学数学与统计学院

将于 2023 年 4 月 14 日至 4 月 17 日在深圳大学举办“椭圆偏微分方程与变分法青年

学者论坛”。具体事宜如下：

 

一、主办单位：深圳大学数学与统计学院

二、组织委员会：（姓氏字母顺序）

龚华均、黎泉荣、余晓辉

三、会议安排：

时间：4 月 14 日：注册报到

4 月 15 日和 4 月 16 日：学术报告及研讨

4 月 17 日：离会

地点：深圳大学

四、联系人：

余晓辉，电话：13424362984，E-mail:xiaohui.yu@szu.edu.cn

 

会议日程安排

2023年4月15日 上午
时间	报告	主持人
8:50-9:00	开幕式  院领导（深圳大学）
9:00-9:40	唐仲伟（北京师范大学） Compactness and existence results of the   prescribing fractional Q-curvature problem	杨敏波 （浙江师范大学）
9:40-10:20	陈建清（福建师范大学） 关于广义Kadomtsev-Petviashvili (I)方程的一些结果
10:20-10:40	茶歇
10:40-11:20	王聪（四川大学） On the critical elliptic equations and the resonant   problems	王大斌 （汕头大学）
11:20-12:00	钟学秀（华南师范大学） On   Bartsch-Jeanjean-Soave’s open problem
12:15-13:30	午餐
下午

14:00-14:40	钮维生（安徽大学） Asymptotic expansion of fundamental   solutions for parabolic operators with non-self-similar scales	赵富坤 （云南师范大学）
14:40-15:20	何其涵（广西大学） Existence and blow up   behavior of positive normalized solution to the Kirchhoff equation with   general nonlinearities: Mass super-critical case
15:20-15:40	茶歇
15:40-16:20	王春花（华中师范大学） Existence and local uniqueness   of multi-peak solutions for the Chern-Simons-Schrodinger system	蒋永生 （中南财经政法大学）
16:20-17:00	张贻民（武汉理工大学） Normalized solutons for some quasilinear schrodinger equations
17:00-17:50	孙玉华（南开大学） Sharp criteria for nonlocal elliptic inequalities on manifolds	高琦 （武汉理工大学）
18:30	晚餐

会议地点：深圳大学汇星楼

 

会议日程安排

2023年4月16日 上午
时间	报告	主持人
8:50-9:30	郭玉劲（华中师范大学） Ground States of the Planar Schrodinger-Poisson System   with Logarithmic Potentials	王明新 （河南理工大学）
9:30-10:10	戴蔚（北京航空航天大学） Method of scaling   spheres: Liouville theorems in general bounded or unbounded domains,   blowing-up analysis on non- domains and other applications
10:10-10:30	茶歇
10:30-11:10	邓圣兵（西南大学） Some results about the weighted elliptic   problems	王征平 （武汉理工大学）
11:10-11:50	罗鹏（华中师范大学） Qualitative analysis on the critical points of solutions of nonlinear   elliptic equations
12:30	午餐
2023年4月16日 下午
14:30-16:30	自由讨论

会议地点：深圳大学汇星楼

会议报告摘要

报告1: Compactness and existence results of the prescribing fractional Q-curvature problem.

报告人：唐仲伟 北京师范大学

报告摘要：In this talk, I will present some results of the prescribing fractional Q-curvature problem, we are devoted to establishing the compactness and existence results of the solutions to the prescribing fractional Q-curvature problem of order on n-dimensional standard sphere when , , . The compactness results are novel and optimal. In addition, we prove a degree-counting formula of all solutions to achieve the existence. From our results, we can know where blow up occur. Furthermore, the sequence of solutions that blow up precisely at any finite distinct location can be constructed. It is worth noting that our results include the case of multiple harmonic. This is a joint work with Dr. Yan Li and Ning Zhou

 

 

报告2：关于广义Kadomtsev-Petviashvili (I)方程的一些结果

报告人： 陈建清 福建师范大学

报告摘要：本报告首先回顾一些半线性椭圆方程的结果;接着介绍一些广义Kadomtsev-Petviashvili (I)方程的一些已有结论；最后介绍旋转的广义Kadomtsev-Petviashvili (I)方程和广义Kadomtsev-Petviashvili (I)方程与BO方程耦合的一些结论。

 

报告3： On the critical elliptic equations and the resonant problems

报告人：王聪 四川大学

报告摘要：This talk is about the existence of nontrivial solutions for elliptic equations involving single and double weighted critical exponents in the bounded domain or , as well as multiple nontrivial solutions for the resonant problems in , the methods depend on the mountain pass lemma, Nehari manifold and Morse theory. Based on the joint papers with Professor Jiabao Su(CNU).

 

 

报告4: On Bartsch-Jeanjean-Soave’s open problem

报告人：钟学秀 华南师范大学

报告摘要：Bose-Einstein凝聚(BEC)是椭圆偏微分方程领域的前沿热门课题。对应方程的解可以帮助人们详细地分析拓扑物体的结构和性质。当前冷原子系统提供了一个理想的平台去研究复杂的拓扑结构，作为代表，两组分的玻色爱恩斯坦凝聚(BEC)已经在实验中得到了广泛的应用。众所周知，固定频率BEC中的Sirakov 公开问题是非线性泛函分析和椭圆偏微分方程领域的一个重要公开问题。在当前比较关注的规范解问题研究中，Bartsch-Jeanjean-Soave在[JMPA,2016]也提出了一个与之平行的公开问题。报告人将汇报一下跟合作者在这个公开问题方面的一些进展(成果涉及合作者：Thomas Bartsch, 邹文明教授, Louis Jeanjean, 张建军教授)。

 

 

报告5: Asymptotic expansion of fundamental solutions for parabolic operators with non-self-similar scales

报告人：钮维生 安徽大学

报告摘要：This talk is devoted to quantitative homogenization of fundamental solutions for second-order parabolic systems with rapidly oscillating periodic coefficients involving non-self-similar scales in spatial and temporal variables. We establish the asymptotic expansions of the fundamental solution for the operator with precise error estimates.

 

 

报告6: Some results about the weighted elliptic problems

报告人：邓圣兵 西南大学

报告摘要：In this talk, I will introduce some results about the weighted elliptic problems, which includes the existence of the sharp constants and optimizers for the weighted Caffarelli Kohn Nirenberg type inequalities, and the classification and non-degenerate of the solutions for some weighted equations.

 

 

报告7: Existence and local uniqueness of multi-peak solutions for the Chern-Simons-Schrodinger system

报告人：王春花 华中师范大学

报告摘要：In this talk, we will disucss the Chern-Simons-Schrodinger system. Under some mild assumptions on the potential function, applying the finite dimensional reduction method we construct multi-peak solutions of it. Also, we prove all the concentrated solutions of the problem have the same form. Meanwhile, we prove that the concentrated solutions are local unique by various local Pohozaev identities, blow-up analysis and the maximum principle. This is a joint work with Qiaoqiao Hua and Jing Yang.

 

 

报告8 : Normalized solutions for some quasilinear Schrodinger equations

报告人：张贻民 武汉理工大学

报告摘要：Quasilinear Schrodinger equation is used to describe the wave functions of Bose-Einstein condensations with higher order interactions, as well as other nonlinear topics. In this talk, we will discuss the existence and asymptotic behavior of ground state as the parameter of modified term tends to zero when the parameter of interaction term great than or equal to critical case. Moreover, if the parameter of interaction term equals to critical case, the local uniqueness of ground state was studied too.

 

 

报告9: Sharp criteria for nonlocal elliptic inequalities on manifolds

报告人：孙玉华 南开大学

报告摘要：Let M be a complete non-compact Riemannian manifold and be a Radon measure on M, we study the existence and non-existence of positive solutions to a nonlocal elliptic inequality. When the Green function of the fractional Laplacian exists and satisfies the quasi-metric property, we obtain necessary and sufficient criteria for existence of positive solutions. In particular, explicit conditions in terms of volume growth and the growth of Radon measure are given, when M admits Li-Yau Gaussian type heat kernel estimates. This is based on joint work with Qingsong Gu and Xueping Huang.

 

 

报告10: Ground States of the Planar Schrodinger-Poisson System with Logarithmic Potentials

报告人：郭玉劲 华中师范大学

报告摘要：In this talk, we focus on ground states of a planar Schrodinger-Poisson system with Logarithmic Potentials. We address the classification of ground states. In particular, the local uniqueness of positive ground states is also discussed by overcoming the sign-changing property of the logarithmic convolution potential and the non-invariance under translations of the logarithmic external potential.

 

 

报告11: Method of scaling spheres: Liouville theorems in general bounded or unbounded domains, blowing-up analysis on non- domains and other applications.

报告人：戴蔚 北京航空航天大学

报告摘要：In this talk, we aim to introduce the method of scaling spheres (MSS) as a unified approach to Liouville theorems on general domains and apply it to establish Liouville theorems on arbitrary unbounded or bounded MSS applicable domains for general ≤n-th order PDEs and integral equations without translation invariance or with singularities. The set of MSS applicable domains includes any unbounded or bounded generalized radially convex domains and any complementary sets of their closures, which is invariant under Kelvin transforms and is the maximal collection of domains such that the MSS works. For instance, , , balls, bounded or unbounded cone-like domains, exterior domains, convex domains, star-shaped domains and all the complements of their closures are MSS applicable domains. One should note that, MSS applicable domains is to the MSS what convex domains (at least in one direction) is to the famous method of moving planes. As applications, we derive a priori estimates and hence existence of positive solutions from the boundary Hölder estimates for ≤n-th order elliptic equations by applying the blowing-up argument on domains with blowing-up cone boundary (BCB domains for short). After the blowing-up procedure, the BCB domains allow the limiting shape of the domain to be a cone (half space is a cone). While the classical blowing-up techniques in previous works work on -smooth domains, we are able to apply blowing-up analysis on more general BCB domains on which the boundary Hölder estimates hold (can be guaranteed by uniform exterior cone property etc).

 

 

报告12: Existence and blow up behavior of positive normalized solution to the Kirchhoff equation with general nonlinearities: Mass super-critical case

报告人：何其涵 广西大学

报告摘要：In present paper, we study the normalized solutions to the following Kirchhoff problem

satisfying the normalization constraint ,which appears in free vibrations of elastic strings. The parameters are prescribed as is the mass. The nonlinearities 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 . After a detailed analysis via the blow up method, we also make clear the asymptotic behavior of these solutions as as well as .

 

 

报告13：Qualitative analysis on the critical points of solutions of nonlinear elliptic equations

报告人：罗鹏 华中师范大学

报告摘要：In this talk, we consider the following problem

where , with is a smooth bounded domain, is the ball centered at and radius and is a smooth nonlinearity. By some computations involving the Green function and degree theory, we compute the number and location of critical points of solutions for small . This is based on a jointed work with Massimo Grossi.

 

 

学院简介

深圳大学数学学科设立于1983年。1996年成立师范学院数学教育系，1998年成立理学院数学系，2006年两系合并成立数学与计算科学学院，2015年更名为数学与统计学院。

 

数学与统计学院下设4个系1个部：数学系、应用数学系、信息与计算科学系、统计系和大学数学教学部； 拥有1个数学一级学科博士点，2个一级学科硕士点：数学、统计学，1个专业硕士点：应用统计学，3个本科专业：数学与应用数学、信息与计算科学、统计学，其中信息与计算科学专业入选国家级一流本科专业建设点。学院现有国家天元数学东南中心-深圳大学、深圳国家应用数学中心-深圳大学、粤港澳（国家）应用数学中心-深圳分中心、广东省高校高清视频智能分析与处理工程技术中心、深圳市现代机器学习与应用重点实验室、深圳大学数学研究所等科研平台，已成为高水平的数学科学研究和人才培养基地。

 

数学与统计学院师资队伍优良，现有专任教师100人，其中教授21人，副教授44人，博士生导师17人，国家杰出青年基金获得者3人，教育部重要人才奖励计划入选者2人，国家优秀青年基金获得者2人，享受国务院政府特殊津贴专家2人，钟家庆数学奖获得者1人，珠江人才计划青年拔尖人才1人以及一批深圳市高层次人才；在复分析、偏微分方程理论与计算、运筹与优化、代数与数论、统计学、信息学及交叉应用等领域研究成果突出，近10年，主持国家自然科学基金84项，重大重点类项目3项，科技部重点项目2项及省市级科研项目百余项，科研经费总金额近亿元。在J.Eur. Math. Soc.、 Adv. in Math、《中国科学》等高水平学术期刊发表论文800多篇。

 

数学与统计学院高度重视教学研究，近年来获批省级以上教研教改项目14项，出版（含修订）教材、参考书、专著等共计十余部，其中2部入选教育部普通高等教育“十二五”和“十三五”规划教材，3门课程入选国家一流本科课程，指导学生参加省级以上学科竞赛获奖近500项，获高校数学教学竞赛国家奖7项，省奖10项。学院不断提高人才培养质量，1983年以来，先后培养4800余名毕业生，他们奋斗在国家建设的各行各业，其中不乏史玉柱、黄舜、陈昳茹、陈海平等一批杰出校友。

 

数学与统计学院坐落于深圳大学粤海校区，毗邻深圳市高新技术产业园区，具有卓越的产学研融合条件，校园内绿树成萌，环境宜人，是醉心数学研究，潜心求学的绝佳选择。

 

数学与统计学院始终恪守“自立、自律、自强”的校训，脚踏实地，开拓进取，积极服务粤港澳大湾区和深圳社会主义先行示范区建设，为建设一流高水平数学学科不懈努力。