线性奇异微分算子的谱及相应非线性问题解的分歧

As an important branch of boundary value problems for nonlinear ordinary differential equations, singular boundary value problems have been widely applied in practice. In the research of non-linear problems, the calculations of several important indexes are closely related to the eigenvalues of their corresponding linear problems, such as topological degree, linking number, Morse index and rotation number. It is well known that the spectral theory of non-singular ordinary differential operator has already led to many important and profound results, and has been applied in the research of numerous nonlinear problems. However, the research of the spectral theory of singular linear ordinary differential operator is slower. The aim of the project is to study the spectra of singular linear second-order ordinary differential operator by using linear operator theory and variation principle, further, to consider the global structure of set of solutions as well as the existence, uniqueness, multiplicity of solutions to the corresponding nonlinear problems by using bifurcation theory. The theoretic conclusions of this work will have important significance to the theoretical analysis and numerical calculation of many models, including nuclear physics, aerodynamics, fluid mechanics, boundary layer theory, etc.

奇异边值问题作为非线性常微分方程边值问题的一个重要分支，有着非常丰富的实际应用背景。在非线性问题的研究中，拓扑度、环绕数、Morse指标、旋转数等重要指标的计算均同“与之相应线性问题的特征值”密切相关。众所周知，非奇异线性常微分算子的谱理论已经取得许多重要而深刻的结果，并被广泛地运用到许多非线性问题的研究中。但是，奇异线性微分算子的谱理论发展相对迟缓。因此，本项目试图运用线性算子理论、分歧理论以及变分原理等工具研究奇异二阶线性微分算子的谱及相应非线性问题解集分支的全局结构，进一步得到解的存在性、唯一性以及多解性。本项目所获理论结果对核物理、气体动力学、流体力学及边界层理论等学科中诸多模型的理论分析和数值计算具有重要指导意义。

非线性常微分方程边值问题解集分支的全局结构研究是微分方程领域中一个热点和难点问题，而此类问题的研究与相应线性边值问题的谱理论是否完善密切相关。众所周知，非奇异二阶线性边值问题的谱结构已有许多重要且深刻的结果，并被广泛应用于相应非线性问题解集分支全局结构的研究中。然而，奇异二阶线性边值问题谱理论发展却相对迟缓，仅有的谱结果是在特殊微分算子的情形下建立的，而带一般微分算子的谱结构还处于探索阶段。本项目在研阶段主要完成和取得的成果有：首先在微分算子较一般的情况下获得了线性特征值问题的谱结构，这一结果推广了已有的特殊微分算子情形下的谱结果；其次研究了对应的非线性边值问题解的全局结构，该结果的获得与对应的线性边值问题的谱结果密切相关；最后研究非线性问题结点解的方法讨论了一类非线性Neumann边值问题的全局结构。