1.报告题目：Limiting cases in Choquard type equations

报告人：Daniele Cassani（因苏布里亚大学）

时间：2024年 12月 10日（周二）上午9:00-11:00

地点：数理楼135报告厅

报告摘要：Quantitative and qualitative informations on nonlinear Schrödinger equations strongly coupled with Poisson's equation can be derived from nonlocal Choquard type equations. Limiting cases appear when the underlying function space setting is not well defined for the equation, as a consequence of the limiting Sobolev emedding which provides logarithmic kernels competing with exponential nonlinearities. We present two possible approaches to overcome this difficulty. The first one by establishing a suitable weighted Trudinger-Moser type inequality which eventually yields a proper functional setting. Alternatively, one can exploit a uniform approximation of the log-kernel and then pass to the limit in the approximating equations. Both methods reveal new aspects which throw some light on the problem.

报告人简介：丹尼尔・卡萨尼（Daniele Cassani），意大利因苏布里亚大学正教授，其主要研究领域包括偏微分方程和最优泛函不等式，研究成果主要发表在Journal of Functional Analysis, Annales de l'Institut Henri Poincaré Analyse Non Linéaire, Calculus of Variations and Partial Differential Equations, Journal of Differential Equations等国际期刊上。他于2006年在意大利米兰大学获得博士学位，之后随即在温哥华的太平洋数学科学研究所（PIMS）从事博士后研究，导师是伊瓦尔・埃克兰（Ivar Ekeland）。自2016年起，担任黎曼国际bat365官网登录主任。2018年，被任命为因苏布里亚大学董事会成员。同时，曾受邀访问美国、加拿大、欧洲各国、瑞士、巴西和中国的许多重要学术科研机构。目前，他是因苏布里亚大学基金会的首席执行官，并且与陶哲轩（Terence Tao）和马丁・海雷尔（Martin Hairer）一同担任黎曼奖委员会成员。

2. 报告题目：Prescribed mass solutions and limiting profiles for Schrodinger equations with critical exponents and lack of compactness

报告人：张建军（重庆交通大学）

时间：2024年 12月 10日（周二）上午9:00-11:00

地点：数理楼135报告厅

报告摘要：We are concerned with the existence and multiplicity of prescribed mass positive solutions to Schrodinger equations with general critical nonlinearity, in particular, in the pure mass supercritical case and the mass mixed critical case. Precisely, for the pure mass supercritical case, under related mild assumptions, the existence of mountain pass normalized solutions is obtained for all prescribed mass $c>0$. We also capture its precise asymptotic behavior as $c$ goes to zero as well as $c$ goes to infinity. For the mass mixed case, there exist at least two different positive normalized solutions for small $c>0$. One is a local minimizer and the other one is a mountain pass solution. We also establish a sequence of properties for the local minimizer including the uniqueness and asymptotic behavior, etc. The asymptotic behavior of the mountain pass solution as $c\rightarrow 0^$ is also studied. Our results solve a sequence of open problems proposed by Soave( J. Funct. Anal., 279(6):108610, 2020).

报告人简介：张建军，重庆交通大学bat365官网登录教授，重庆市数学会副理事长。2001年本科毕业于中国矿业大学数学系，2012年于清华大学数学科学系获博士学位，2018年获得意大利副教授国家资格认证，2020年入选重庆市高校中青年骨干教师，先后在南开大学陈省身数学研究所、巴西帕拉伊巴联邦大学、意大利因苏布里亚大学从事博士后研究。研究领域主要包括非线性分析中的变分与拓扑方法，非线性椭圆方程等。主持国家自然科学基金面上项目2项、国际合作与交流项目1项以及意大利伦巴第研究员基金（GLOCAL ERC）等。迄今已有一些重要结果发表在《Comm. PDE》，《J. Diff. Eqs》，《J. London Math. Soc.》，《Nonlinearity》等国际主流学术刊物上。