几类数学物理方程驻波解的存在性与动力学分析

Schrödinger equations, Dirac equations and various related coupled Schodinger systems constitute the core of quantum mechanics, and have been widely employed in other fields such as atomic physics, molecular physics, split state physics, nuclear physics and chemistry. Many other problems in mathematics and physics are also closely related to these equations, including the Brezis-Nirenger problem, the Yamabe problem in geometry, the existence of minimum of the Yang-Mills functional, isoperimetric inequality，and the theory for the balance of gravity in galaxy. In this proposed research, based on the existing literature, we will use the method of variation together with the critical point theory to explore some key issues for these equations. Among these issues are the existence and multiplicity of standing waves, ground state solutions of Nehari-Pankov type and semi-classic solutions, as well as some dynamical properties of the standing waves such as intensity，decay property 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/methods which will enable us to obtain some essentially new results and significantly contribute to the theory of elliptic differential equations.

Schrödinger 方程、Dirac方程和Schrödinger 耦合系统构成了量子力学的核心基础，在原子物理、分子物理、固体物理、核物理、化学等领域中被广泛应用。许多数学物理问题，如 Brezis-Nirenberg 问题，几何中的 Yamabe 问题，Yang-Mills 泛函的非极小解的存在性，等周不等式，星系的重力平衡理论都与其密切相关。本项目将借助变分方法与临界点理论，在已有文献的基础上，重点研究非线性Schrödinger 方程、Dirac方程和Schrödinger 耦合系统的核心问题：驻波解、“Nehari-Pankov型”基态解和半经典解的存在性与多重性，驻波解的集中性、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，推进非线性椭圆方程定性理论的发展。

Schrödinger方程、Dirac方程和Schrödinger耦合系统构成了量子力学的核心基础，在原子物理、分子物理、固体物理、核物理、化学等领域中被广泛应用。许多数学物理问题，如Brezis-Nirenberg问题，几何中的Yamabe问题，Yang-Mills泛函的非极小解的存在性，等周不等式，星系的重力平衡理论都与其密切相关。本项目借助变分方法与临界点理论，在已有文献的基础上，建立了非线性Schrödinger 方程、Dirac方程和Schrödinger 耦合系统的驻波解、“Nehari-Pankov型”基态解和半经典解的存在性与多重性，系统地研究了驻波解的集中性、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成果，提出了新方法，发展了现有的非线性椭圆微分方程定性理论。

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