双耦合薛定谔方程组正规化解的存在性研究

The study on the existence of normalized solutions to Schrödinger equations (systems) is not only a hot topic in recent years, but also a very important research field in partial differential equations. In this project, we focus on the normalized solutions to doubly coupled Schrödinger system in both bounded domain and the whole space. When the region is bounded, we propose to obtain the normalized solution to doubly coupled Schrödinger system by using the Gagliardo-Nirenberg inequality, the minimization theory and concentration compactness principle. For the doubly coupled Schrödinger system in the whole space, we would use the minimax theory, construct linking of energy function on the Pohozaev manifolds to obtain the critical value. Furthermore, we would use the symmetry of the system, combined with topological method, genus theory, Lusternik-Schnirelman theory and the tools such as symmetric mountain pass theorem to study the existence and multiplicity of normalized solutions to the Schrödinger system. This project would make a contribution to the development of the theory of nonlinear functional analysis and the theory of Schrödinger systems in quantum mechanics.

薛定谔方程（组）正规化解的存在性问题是近年来的研究热点,也是目前偏微分方程中十分重要的研究领域，本项目拟分有界区域和全空间两种情形，对双耦合薛定谔方程组的正规化解进行研究。对于有界区域情形，我们拟利用Gagliardo-Nirenberg不等式、极小化理论以及集中紧性原理，证明双耦合薛定谔方程组正规化解的存在性。对于全空间上的双耦合薛定谔方程组，拟利用极小极大理论，通过在Pohozaev流形上构造能量泛函的环绕结构，证明临界点的存在性。我们还将进一步利用方程组的对称性，结合拓扑方法、亏格理论、Lusternik-Schnirelman理论以及对称山路引理等工具，得到多重正规化解的存在性。本项研究对于非线性泛函分析理论以及量子力学的薛定谔方程组理论的发展，将做出贡献。

本项目属天元数学访问学者联合项目，访问学者的派出单位是曲阜师范大学（项目编号：12126347），接受单位是首都师范大学（项目编号：12126353）。本项目设定的任务是关于薛定谔方程组正规化解的存在性。在项目执行期间，我们达到了预期目标，超额完成了任务，完成学术论文4篇，其中正式发表2篇，接受发表1篇（已经在线发表），正式投稿1篇。另外，还有研究工作有待进一步完善。