Boundary value problems for higher-order nonlinear ordinary differential equations

研究高阶常微分方程边值问题解的存在性、唯一性、零的个数和零的分布，并在常微分方程结果的基础上得到高阶椭圆型微分方程解集的详细信息。新的研究结果如下：对于一类四阶或偶阶拟线性常微分方程，给出了正解存在的充分必要条件。讨论了一类二阶常微分方程的二阶系和一类二阶椭圆方程的二阶系正解的存在性。研究了一类具有任意非线性的退化椭圆型方程的外域问题，给出了所有解振动的充分必要条件。在类条形区域上考虑了一类退化椭圆方程，讨论了其正解的不存在性。对于热方程的初值问题，讨论了固定等温面和均匀致密区域，并讨论了简并扩散与区域形状的相互作用。对于无限区间上的高阶常微分方程的奇异特征值问题，证明了存在一个恰好有n个零的特征函数的可数序列。对于含多谐算子的四阶非线性椭圆型微分方程，得到了存在性和不存在性的结果。

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.