Boundary value problems for higher-order nonlinear ordinary differential equations

研究了高阶常微分方程边值问题解的存在性、唯一性、零点个数和零点的分布，并在常微分方程解的基础上得到了高阶椭圆型微分方程解的详细信息。新的结果如下：1.对于四阶或偶数阶拟线性常微分方程解的存在的充要条件。对于二阶常微分方程组和二阶椭圆型方程组，讨论了正解的存在性。2.在外域上考虑了一类具有任意非线性项的退化椭圆型方程，得到了所有解振动的充要条件。在带状区域上研究了退化椭圆型方程，讨论了正解的不存在性。3.对于热方程初值问题，讨论了定常等温面和一致稠密区域，讨论了退化扩散和区域形状之间的相互作用。4.对于无穷区间上的高阶常微分方程奇特征值问题，证明了存在可数个特征函数列，且恰好有n个零点。5.对于一个含多重调和算子的四阶非线性椭圆型方程，得到了存在和不存在的结果。

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.