几类强不定变分问题解的存在性与动力学研究

It is well known that the coupled Schrödinger systems, Dirac equations and Dirac-Maxwell systems constitute the core of quantum mechanics, and have been widely employed in other fields such as atomic physics, nuclear physics, molecular physics, gravitational physics and chemistry. Formally, these systems have strongly indefinite variational structure. In this project, based on the existing literature, we will use critical point theory for strongly indefinite functionals to explore some key issues for these systems. Among these issues are the existence and multiplicity of standing waves, as well as some dynamical properties of the semi-classical ground state solutions such as the existence, concentration phenomena, convergence, exponential decay and regularity etc. These are very important yet very challenging mathematical problems, as such, we also expect to develop some novel and more effective techniques which will enable us to obtain some essentially new results and significantly contribute to the theory of several kinds of differential systems. Moreover, these studies are beneficial to investigate the other strongly indefinite variational problems.

众所周知，Schrödinger耦合系统、Dirac方程和Dirac-Maxwell系统构成了量子力学的核心基础，在原子物理、核物理、分子物理、引力物理和化学等领域中被广泛应用。形式上，这些系统具有强不定的变分结构。本项目将使用强不定泛函的临界点理论，在已有文献的基础上，重点研究Schrödinger耦合系统、Dirac方程和Dirac-Maxwell系统的核心问题：驻波解的存在性与多重性，半经典基态解的存在性、集中性、收敛性、衰减性和正则性等动力学性态。发展和开拓非线性分析方法、技巧，深化数学工具，对所研究的问题获得若干全新的、本质性的结果，推进这几类微分系统定性理论的发展。并且，这些研究将会有益于解决其他强不定变分问题。

众所周知，Schrödinger耦合系统、Dirac方程和Dirac-Poisson系统构成了量子力学的核心基础，在原子物理、核物理、分子物理、引力物理和化学等领域中被广泛应用。从变分的观点看，这些系统形式上具有强不定的变分结构，故利用变分法研究强不定变分问题是十分困难且具有重要意义的课题。本项目将使用强不定泛函的临界点理论重点研究Schrödinger耦合系统、Dirac方程和Dirac-Poisson系统的核心问题：驻波解的存在性与多重性，半经典基态解的存在性、集中性、收敛性、衰减性和正则性等动力学性态，获得了一系列较为深刻的重要研究成果，提出了新方法，发展了现有理论和技巧，推进这几类微分系统定性理论的发展。并且，这些研究将会有益于解决其他强不定变分问题。项目组成员已发表SCI论文21篇，其中ESI（前1%）高引论文2篇，ESI（前0.1%）高引论文1篇。