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Study on the number of zeros of solutions to higher-order nonlinear differential equations

高阶非线性微分方程解的零点个数研究

基本信息

批准号：
13640178
负责人：
NAITO Manabu
金额：
$ 1.92万
依托单位：
EHIME UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

项目摘要

The aim of this research is to study the number of zeros and the distribution of zeros of solutions of higher-order ordinary differential equations with Emden-Fowler type nonlinearity, and to discuss the oscillatory properties of solutions of higher-order elliptic differential equations on the base of the results for ordinary differential equations.The new results and knowledge obtained in the two years are as follows :1. For a singular eigenvalue problem to the linear and sublinear higher-order ordinary differential equations on an infinite interval [a, +∞), it is shown that there is a countable sequence of eigenvalues and that the n-th eigenfunction has exactly n zeros.2. For a regular eigenvalue problem to the sublinear higher-order ordinary differential equations on a finite interval, it is shown that there is a countable sequence of eigenvalues and that the n-th eigenfunction has exactly n zeros.3. For the second-order half-linear ordinary differential equations, it is shown that the number of zeros of specific nonoscillatory solutions changes one by one as a parameter varies.4. For the second-order half-linear ordinary differential equations, a generalization and an analogue of the Sturm-Liouville linear regular eigenvalue problem are obtained.5. For the four-dimensional Emden-Fowler differential systems and the fourth-order quasilinear differential equations of Emden-Fowler type, a necessary and sufficinet condition for the existence of nonoscillatory solutions with specific asymptotic properties as t →∞ is established, and a sufficient condition for oscillation of all solutions is also obtained.6. For the two-dimensional semilinear elliptic differential systems of the Laplace type, an analogue of Liouville's theorem is established.7. For the fourth-order nonlinear elliptic differential equations including the poly-harmonic operator, it is shown that a duality between the existence and nonexistence in an interior/exterior/entire the problems still holds.

本研究的目的是研究具有Emden-Fowler型非线性项的高阶常微分方程解的零点个数和零点分布，并在常微分方程结果的基础上讨论高阶椭圆型微分方程解的振动性，两年来取得的新成果和认识如下：1.对于无穷区间[a，+∞）上的线性和次线性高阶常微分方程的奇异特征值问题，证明了存在可数特征值序列，且第n个特征函数恰好有n个零点.对于有限区间上的次线性高阶常微分方程的正则特征值问题，证明了存在可数的特征值序列，且第n个特征函数恰好有n个零点.对于二阶半线性常微分方程，证明了特定非振动解的零点个数随着参数的变化而一一变化.对于二阶半线性常微分方程，得到了Sturm-Liouville线性正则特征值问题的一个推广和一个类比.对于四维Emden-Fowler微分系统和四阶拟线性Emden-Fowler型微分方程，建立了当t →∞时存在具有特定渐近性质的非振动解的充分必要条件，以及所有解振动的充分条件.对于二维拉普拉斯型半线性椭圆微分系统，建立了Liouville定理的一个类似定理.对于含多调和算子的四阶非线性椭圆型微分方程，证明了该问题在内部/外部/整体上的存在性与不存在性之间的对偶性仍然成立.