非线性分析研讨班：Strongly localized semiclassical states for nonlinear Dirac equations

中科院数学与系统科学研究院

数学研究所

非线性分析研讨班

报告人：徐 甜 副教授（天津大学）

题  目：Strongly localized semiclassical states for nonlinear Dirac equations

时  间：2020.10.09（星期五），15:10-16:10

地  点：腾讯会议 788 342 077 

摘  要：We study semiclassical states of the nonlinear Dirac equation\[ -i\hbar\pa_t\psi = ic\hbar\sum_{k=1}^3\al_k\pa_k\psi - mc^2\be \psi - M(x)\psi + f(|\psi|)\psi,\quad t\in\R,\ x\in\R^3,\] where $V$ is a bounded continuous potential function and the nonlinear term $f(|\psi|)\psi$ is superlinear, possibly of critical growth. Our main result deals with standing wave solutions that concentrate near a critical point of the potential. Standard methods applicable to nonlinear Schr\"odinger equations, like Lyapunov-Schmidt reduction or penalization, do not work, not even for the homogeneous nonlinearity $f(s)=s^p$. We develop a variational method for the strongly indefinite functional associated to the problem.