Boundary value problems for higher-order nonlinear ordinary differential equations

本文的目的是研究高阶常微分方程边值问题解的存在性、唯一性、零点个数和零点分布，并在常微分方程结果的基础上得到高阶椭圆型方程解集的详细信息，新的结果如下：1.对一类四阶或偶数阶拟线性常微分方程，给出了正解存在的充分必要条件。讨论了2-二阶常微分方程组和2-二阶椭圆型方程组正解的存在性。2.在外部区域上考虑了一类具有任意非线性项的退化椭圆型方程，建立了其所有解振动的充要条件。讨论了带状区域上退化椭圆型方程的正解的不存在性. 3.对于热方程的初值问题，讨论了定常等温面和一致稠密区域，并讨论了退化扩散与区域形状的相互作用. 4.对于无穷区间上高阶常微分方程的奇异特征值问题，证明了存在可数的具有n个零点的特征函数序列. 5.对于一类含多调和算子的四阶非线性椭圆型微分方程，得到了解的存在性和不存在性结果.

The aim of this research is to study the existence, the uniqueness, the number of zeros and the distribution of zeros of solutions of boundary value problems for higher-order ordinary differential equations, and to obtain the detailed information for the set of solutions of higher-order elliptic differential equations on the base of the results for ordinary differential equations.The new results are as follows :1.For a fourth-order or even-order quasilinear ordinary differential equation, necessary and sufficient conditions for the existence of a positive solution are obtained. For a 2-system of the second-order ordinary differential equations and a 2-system of the second-order elliptic equations, the existence of a positive solution is discussed.2.A degenerate elliptic equation with arbitrary nonlinearity is considered on exterior domain, and necessary and sufficient conditions for all solutions to be oscillatory are established. A degenerate elliptic equation is considered on strip-like domain, and the nonexistence of a positive solution is discussed.3.For an initial value problem for the heat equation, stationary isothermic surfaces and uniformly dense domains are discussed, and an interaction between degenerate diffusion and shape of domain is discussed.4.For a singular eigenvalue problem to a higher-order ordinary differential equation on an infinite interval, it is shown that there is a countable sequence of eigenfunctions having exactly n zeros.5.For a fourth-order nonlinear elliptic differential equation including the poly-harmonic operator, the existence results and nonexistence results are obtained.