谱理论和非线性分析方法及对奇异微分方程的应用

In this project, several class of singular differential equations with actual background and the involved spectral theory and nonlinear analysis method will be studied. The main research of the project is as follows..The existence of solutions of differential equations including attractive, repulsive and mixed singularities will be considered. Some new methods to give a priori estimate of solution and construct the lower and upper of solution of differential equations will be obtained and the spectrum of linear integrable operator can be computed. Secondly, every singularity of differential equations can be researched by a unified approach and the problems will be solved by the use of new techniques which are the method of lower and upper of solution, operator spectral theory combined with topological degree. .Applying some new methods, it can research the (Fucik) spectral theory, and establish new generalized Rabinowitz global bifurcation theory and the topological degree of non-compact operator, and give the condition of nonresonance of the singular second (higher)-order nonlinear differential equations and singular generalized p-Laplacian equations, where the nonlinearities may have jumping property. Based on these new establishing theory and novel method, the existence of solution, global bifurcation constructure and geometrical property analysis for these singular differential equations can be obtained. Meanwhile, there have also a new approach to solve the singularity..The existence of solution of the singular generalized p-Laplacian equation and this type of other differential equations will be established. There have three main approaches to solve these problems. (i).The continuum and a priori estimate of solution will be given, and the lower and upper of solution of singular differential equation can be constructed by the use of these above new results; Finally, utilizing the method of lower and upper of solution combined with topological degree, the corresponding problems can be well achieved. (ii). The second method is that the differential equations shall be transormed into the optimization problems by the use of convex analysis and optimization theory; Secondly, combined with the fixed point theorem, the existence of solutions will be obtained. Applying the new method can hope to obtain some innovative results. (iii). The third method is that variational method will be utilized or a new generalized Manasevich-Mawhin continuation theorem can be established for overcome the singularity so that these problems will be solved.

研究几类有实际背景的奇异微分方程和所涉及的谱理论与非线性分析方法。考虑含吸收、斥性或混合奇性的微分方程解的存在性，给出解的先验估计和如何构造方程上下解的新方法，再用上下解方法、算子谱结合拓扑度等新技巧解决问题且统一处理方程各类奇性。研究奇异二（高）阶非线性微分方程和广义p-Laplacian方程的(Fucik)谱理论、新建立广义Rabinowitz全局分歧理论、非紧算子的拓扑度和给出非共振条件，其中非线性项有跳跃性质；将这些新理论和方法相结合证明方程解的存在性、全局分歧结构和几何性质分析，这也给出了处理奇性的新途径。研究奇异广义p-Laplacian方程和这种类型的各类方程解的存在性：给出解的连续统和先验估计，利用其构造方程上下解结合拓扑度进行处理；或把原方程转化为优化问题，利用不动点定理证明方程解的存在性，用这种新方法有望获得一些创新性成果；或新建立广义M-M连续性定理、变分法解决问题。

本项目主要研究了若干类有实际应用背景的奇异非线性微分方程。如，非牛顿流、化学异质催化剂和电子传导材料的热传导理论等所对应的数学模型为奇异非线性微分方程(系统)；Lane-Emden方程(系统)用来描述天体物理的运动模型等。我们获得以下几方面研究成果。.一、利用Leray-Schauder全局连续性原理研究了奇异二阶非线性微分方程正解的连续统，从而得到方程正解的存在性。同时，讨论了正解不存在时参数的取值区间。研究了奇异Lane-Emden-Fowler方程，利用常微分方程理论构造出方程的径向上下解，再结合Lebesgue控制收敛定理得到方程对所有的参数$\mu$都存在正解，即得到方程的一个全局分歧结构。同时我们讨论了当$x$趋近于边界时，方程正解的渐近行为。利用变量变换的方法，我们把含$\phi$-Laplacian算子的非齐次微分方程转化为两个一阶微分方程，再利用Krasnosel'skii不动点定理证明了方程解的存在性。这里新颖之处在于边界条件中的部分系数可以取负数。另外，利用正线性算子特征值和不动点定理研究了高阶微分方程解的(非)存在性。.二、利用Pohozaev恒等式、解的单调公式、爆缩序列以及doubling引理相结合研究了四阶Henon方程，给出了方程稳定解和有限Morse指标解的完全分类。另外，利用类似的研究方法，我们讨论了加权的椭圆系统稳定解的Liouville定理成立。利用积分先验估计、Souplet不等式和自助法研究了加权Lane-Emden系统和方程稳定解的Liouville定理. 我们的研究成果受到同行专家的关注及推广。.三、研究了分数阶线性时间不变性中立型动力系统的可控性问题，首先推导出系统状态解。通过构造适当可控函数建立了系统可控性的两个标准。另外，应用Lyapunov第二方法，我们推导了一类非线性分数阶微分方程新的稳定性标准，其成果部分解决了一个公开问题---即非线性分数阶微分系统的稳定性问题。讨论了含多个时滞的中立性微分方程，给出了方程解的不稳定性、稳定性、渐近稳定性和指数稳定性的标准。另外，我们也给出渐近稳定性的充分必要条件。