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yabo登录JIANGZHUNZI068HAO: Normalized solutions to the fractional Schrodinger equations with combined nonlinearities

发布时间: 2021-09-23|浏览次数:

讲座报告主题: Normalized solutions to the fractional Schrodinger equations with combined nonlinearities
专家姓名: 张志涛
日期: 2021-09-26 时间: 09:00
地点: 数学科学学院206室
主办单位:yabo登录 应用系统分析研究院

主讲简介: 张志涛,二级研究员、博士生导师,华罗庚数学首席研究员,中国科学院特聘研究员(核心骨干),中国科学院大学岗位教授。国家杰出青年基金获得者、洪堡学者。长期从事非线性泛函分析理论和应用的前沿研究,出版Springer专著一部,在Journal of Functional Analysis, Annales de l'Institut Henri Poincare Analyse Non Lineaire, J. Differential Equations, Calculus of Variations and PDE, Transactions of the American Mathematical Society等著名学术刊物上发表论文80多篇,其中SCI检索70多篇。在困难的自由边界问题,Bose-Einstein condensates Schrodinger方程组,生物竞争方程组等众多方面取得重要成果,解决了困难的Terracini猜想和3维Henon-Lane-Emden猜想。这些成果产生了广泛的影响,被890多名数学家引用1700多次,有的已成为基本参考文献,在研究领域起着引领作用。多次应邀在重要国际会议上作大会报告。担任Springer Nature-Partial Differential Equations and Applications 主编,以及Boundary Value Problems等4个国际刊物编委。研究专长: 长期从事非线性泛函分析理论和应用的前沿研究。

主讲内容简介: We study the normalized solutions of the fractional nonlinear Schodinger equations with combined nonlinearities. Under different assumptions, we prove some existence and nonexistence results about the normalized solutions. More specifically, in the purely $L^2$-subcritical case, we overcome the lack of compactness by virtue of the monotonicity of the least energy value and obtain the existence of ground state solution for $\mu>0$. While for the defocusing situation $\mu<0$, we prove the nonexistence result by constructing an auxiliary function. We emphasis that the nonexistence result is new even for Laplacian operator. In the purely $L^2$-supercritical case, we introduce a fiber energy functional to obtain the boundedness of the Palais-Smale sequence and get a mountain-pass type solution. In the combined-type cases, we construct different linking structures to obtain the saddle type solutions. Finally, we remark that we prove a uniqueness result for the homogeneous nonlinearity($\mu=0$), which is based on the Morse index of ground state solutions.


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