不定权微分算子的谱问题及其结点解研究

Many interesting bifurcation phenomena in natural sciences can be modeled by nonlinear differential equations with parameters. However, the computation of the topological degree, Morse index, rotation number which are usually used to study the nonlinear differential equation, is closely related with the eigenvalue of corresponding linear problem. It is well known that there are many important spectral results for linear ordinary differential equation with definite weight, and these results are already used to study the nonlinear problems. However, the development is slowly for the spectral theory of problems with indefinite weight, and it is only at the beginning for the non-real spectrum of high-order linear differential equations. So, this project is going to study the spectral structrue of high-order linear differential equation with indefinite weight, also study the existence of nodal solutions and the structure of the solution set for the corresponding nonlinear problems, the main mothods are linear operator theory of Krein space, bifurcation theory and variational method. The results of the project will have important theoretical meaning to the modeling of online social networks.

非线性科学中有许多有趣的分歧现象可以通过研究带参数的非线性微分方程解的全局结构来刻画。然而，非线性微分方程研究中经常遇到的拓扑度、Morse指标、旋转数等重要指标的计算均同“与之相对应的线性问题的特征值”密切相关。众所周知,权函数定号的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。然而,不定权特征值问题的谱理论发展相对迟缓,而对高阶线性不定权特征值问题非实特征值的研究还处于探索阶段。因此，本项目试图运用Krein空间的线性算子理论、分歧理论以及变分原理等工具,较为系统地研究由多种边界条件所确定的高阶线性不定权微分算子的谱结构以及相应非线性问题结点解的存在性、唯一性、多解性以及解集分支的全局结构。本项目的理论结果对研究在线社交网络的信息传播模型的理论分析和数值计算具有重要意义。

权函数定号的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。然而,不定权特征值问题的谱理论发展相对迟缓,特别地，对不定权问题非实特征值的研究还处于探索阶段。因此本项目运用Krein空间的线性算子理论等工具,较为系统地研究了由多种边界条件所确定的线性以及半线性不定权微分算子的非实特征值的存在性、不存在性、以及上下界估计，并获得了一些非线性问题的可解性结果。