Solutions with simultaneous synchronized and segregated components for nonlinear Schrodinger systems

主讲人：叶东 华东师范大学教授

时间：2024年6月18日14：00

地点：三号楼332室

举办单位：数理学院

主讲人介绍：叶东，现任华东师范大学数学科学学院教授。1990年毕业于武汉大学中法数学班，1994年在法国卡尚高等师范学院获得博士学位，后长期在法国大学任职，回国前是法国洛林大学的一级教授。主要研究领域是非线性偏微分方程和几何分析。2018年入选国家级高层次人才计划，于当年9月全职回到华东师范大学工作。

内容介绍：We consider a nonlinear Schr\odinger system in ${\mathbb R}^3$: \begin{align*} -\Delta u_j +P_j(x) u=\mu_j u_j^3+\sum\limits_{i=1,i\neq j}^N\beta_{ij}u_i^2u_j, \end{align*} where $N\geq3$, $P_j$ are nonnegative radial potentials; $\mu_j>0$, $\beta_{ij}=\beta_{ji}$ are coupling constants. This type of systems has been widely studied in the last decade, many purely synchronized or segregated solutions are constructed, but few considerations for simultaneous synchronized and segregated solutions exist. On the other hand, there are new challenges in dealing with the existence of multiple sign-changing solutions or semi-nodal solutions. Using Lyapunov-Schmidt reduction method, we construct new type of positive and sign-changing solutions with simultaneous synchronization and segregation. We prove the existence of infinitely many non-radial positive or also sign-changing vector solutions, where some components are synchronized but segregated with other components; the energy level can be arbitrarily large; and our approach works for general any number of components $N \geq 3$. This is a joint work with Qingfang Wang.