几类非线性椭圆型方程组的多解性问题研究

Thanks to many important applications to Bose-Einstein condensates and nonlinear optics, Schrodinger equations and systems have received a lot of attention from many mathematicians in recent years. This project will be focusing on the multiplicity of standing wave solutions to nonlinear Schrodinger systems. Precisely, we shall consider two types of elliptic systems: (A) nonlinearly coupled Schrodinger systems; (B) double coupled Schrodinger system. Many established results have demonstrated the importance of "symmetry", from the perspective of both system structure and domain geometry, in the study of multiplicity of solutions. Our main goal is to get multiple solutions of (A) and (B) under weaker, or more complicate, symmetric assumptions. For problem (A), we consider the multiplicity of solutions of multicomponent symmetric systems on asymmetric domains, and also the multiplicity of solutions on multicomponent, partially symmetric system on radial domains. Comparing with (A), the linearly coupling terms introduced in problem (B) will change the spectrum of related linear operator. Thus to tackle similar multiplicity problems, we need more careful investigations on the function spaces. Moreover, for both problems, the increasing of unknown components will also bring up new difficulties. We hope to discover new connections between various types of symmetric conditions and the multiplicity of solutions, and also find concrete applications to nonlinear Schrodinger systems.

本项目以在玻色-爱因斯坦凝聚和非线性光学等物理问题中有着重要应用的薛定谔方程组为研究对象，主要考虑其驻波解的多重性问题。具体的，我们考虑两类方程组：(A)非线性耦合的薛定谔方程组；(B)双耦合的薛定谔方程组。在已知结果中，方程组结构的对称性和空间区域的对称性，在解决多解性问题时发挥了重要的作用。本项目尝试在较弱的或更复杂的对称性条件下，研究问题(A)和(B)的驻波解多重性。对于(A)，我们考虑非对称区域上的、多分量的全对称方程组解的多重性，以及对称区域上的、多分量的部分对称方程组解的多重性。对于(B)，由于引入线性耦合项，其系数变化将影响线性算子的谱。因而在考虑类似的驻波解多重性问题时，需要对函数空间作更为细致的刻画。此外，方程组中未知分量个数的增加，也给对称性条件的应用增加了难度。我们期望在本项目的研究过程中，推进对于对称性与方程组解的多重性联系的理解，促进关于薛定谔方程组的理论研究。

本项目研究了：(1)以物理学中的Bose-Einstein凝聚和非线性光学等为背景的耦合薛定谔方程组；(2)以相变问题，分层材料问题，晶体错位以及量子力学中超相对论极限等问题为背景的分数阶方程；(3)描述考虑了横向振动对长度影响的弹性弦运动的Kirchhoff方程。主要研究内容包括：(1)薛定谔方程组基态解、向量解在一定参数范围内的存在性、多重性、渐近行为以及向量解的分歧现象。(2)在零点处和无穷远处满足不同增长条件，特别是同时共振，或者正、负无穷远处渐近极限不同等多种非线性项条件组合时，分数阶方程的非平凡解多重性以及Morse指标的计算。(3)Kirchhoff方程非齐次项在零点满足非共振条件，或只在第一特征值处共振条件时的零点临界群计算，以及相应的非平凡解多重性问题。上述问题所得结论，都是相关研究领域内的新结果，完善了人们对于此类问题的认识，所得到的技术型定理及证明可以推广到类似问题的研究中。