几类非线性薛定谔方程及方程组的变分法研究

This project mainly focuses on some nonlinear Schrödinger equations and systems which are related to quantum mechanics such as condensates theory, nonlinear optics etc. We will study the existence,uniqueness and multiplicity of standing wave solutions and the properties of such solutions via variational methods. We aim to solve the following three types of problems. Firstly, for tackling decaying or unbounded potentials, we will establish some Sobolev type embedding theorems between proper weighted function spaces, and also a few of Hardy-LIttlewood-Sobolev type inequalities. Based on these preliminaries, we study the existence of the ground state solution and bound state solution, as well as the geometrical or analytical properties for the local Schrödinger equations or Schrödinger equations with nonlocal term of Hartree type. Secondly, as a special case for the nonlocal equation, the Schrödinger-Poisson system will be studied on the existence of the ground state, sign-changing solution and the uniqueness of positive solution. Thirdly, we will establish an abstract theorem for finding multiple critical points of partially even functionals. By using the theorem, we study the multiplicity of solutions and synchronization or phase separation of solutions for the Schrödinger systems with quadratic coupled nonlinearities. The research will give more extensive interpretation about Schrödinger equation and will make some contributions to the development of critical point theory.

本项目以来源于凝聚态物理、非线性光学等量子理论领域的几类非线性薛定谔方程及方程组为主要研究对象，应用变分方法研究其驻波解的存在性、唯一性、多解性以及解的性态。研究内容包含三方面：一、对于局部的或带有Hartree型非局部项的薛定谔方程，在衰减或无界位势的情形下,建立带权的Sobolev型嵌入定理和带权的Hardy-Littlewood-Sobolev不等式,进而研究其基态解、束缚态解的存在性及其几何分析性质。二、研究Schrödinger-Poisson方程基态解、变号解的存在性和正解的唯一性。三、对于具有二次耦合项的非线性薛定谔方程组，通过建立部分偶泛函的多重临界点定理，研究其多解性以及解的同步或分离性。我们期望通过对本项目问题的研究，加深人们对非线性薛定谔方程的认识，并推进非线性分析理论与应用的发展。

本项目主要应用非线性泛函分析的工具和方法，研究来源于凝聚态物理、非线性光学等量子理论领域的几类非线性偏微分方程。研究课题与拟线性薛定谔方程、具有二次耦合项的薛定谔方程组、分数次薛定谔-泊松方程以及带有非局部项的椭圆型方程密切相关。本项目解决了如下问题: 通过约束极值理论、极大极小定理、集中紧性原理、（PS）序列的精细分析、分歧理论等方法，获得非线性薛定谔方程和方程组解的存在性和多解性并研究其几何分析性质。 基于Ljusternik-Schnirelmann 等理论发展了部分对称性泛函的多重临界点理论。将Morse理论应用于分数次椭圆方程的研究，基于临界群的计算，深入该类椭圆方程边值问题解的存在性和多解性。项目组围绕科研课题展开了认真深入的研究，取得了重要的成果，顺利完成了既定研究目标。项目组在项目执行期间共发表录用SCI科研论文 14 篇，推动了偏微分方程和非线性泛函分析理论及其应用的发展。在培养青年人才方面，培养研究生6名。