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.