薛定谔耦合系统驻波解的存在性及相关问题

The study on standing wave solutions of Schrödinger equation and its coupled systems has been an important research field in Nonlinear Functional analysis and differential equations. Based on the existing literature, applying critical point theory, this project will focus on the following core issues of the coupled Schrödinger systems: the existence, uniqueness and multipilcity of nontrivial standing wave solutions, the ground states solutions of Nehari-Pankov type and the semiclassic solutions, the global exponential decay of semiclassical solutions at infinity, and the estimation of exponential decay rate and the existing results are very rare. By deepening the mathematical tools, we will extend and develop the nonlinear analysis methods and techniques to obtain a number of new and essential results, which promote the development of the theory to coupled Schrödinger systems.

薛定谔方程及其耦合系统的驻波解问题是近年来非线性泛函分析和微分方程理论领域的一个非常热门的课题，本项目将借助临界点理论，在已有工作的基础上，重点研究薛定谔耦合系统的核心问题：非平凡驻波解、Nehari-Pankov型基态解和半经典解的存在性、唯一性和多重性问题，讨论半经典解在无穷远处的整体指数衰减和指数衰减速度，这一工作在文献中还不多见，进一步发展和深化非线性分析方法、技巧，拓展变分思想和临界点理论的应用范围，获得若干新的、本质性结果，对薛定谔耦合系统的理论发展起到促进作用。

薛定谔方程及其耦合系统的驻波解问题是近年来非线性泛函分析和微分方程理论领域的一个非常热门的课题，本项目借助非线性泛函分析的技巧，以及变分法和临界点理论，在前人的研究基础上，寻找新的有效方法和技巧重点研究薛定谔系统的核心问题：非线性Schrödinger方程非平凡解的存在性、非线性Schrödinger-Poisson系统“Nehari-Pankov 型”基态解的存在性、非线性Hamiltonian椭圆系统解的存在性、非局部Schrödinger方程解的集中性和多重性，进一步推广已有的研究成果，解决实际问题，获得了一系列较为深刻的重要成果。项目组成员已发表SCI论文6篇，核心期刊论文1篇，尚有2篇论文已投稿。