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

研究了具有Emden-Fowler型非线性的高阶常微分方程解的零点数和零点分布，并在此基础上讨论了高阶椭圆型微分方程解的振动性质。在这两年中获得的新成果和新知识如下：1。对于无限区间[a， +∞)上的线性和次线性高阶常微分方程的奇异特征值问题，证明了存在一个可数的特征值序列，且第n个特征函数恰好有n个零。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，证明了存在一个可数的特征值序列，且第n个特征函数恰好有n个零。对于二阶半线性常微分方程，证明了特定非振荡解的零个数随参数的变化而逐一变化。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种推广和模拟。对于四维Emden-Fowler微分系统和Emden-Fowler型四阶拟线性微分方程，建立了t→∞时具有特定渐近性质的非振荡解存在的充分必要条件，并得到了所有解振荡的充分条件。对于Laplace型二维半线性椭圆型微分系统，建立了一个类似于Liouville定理的方法。对于含多谐算子的四阶非线性椭圆型微分方程，证明了在内/外/整体问题中存在与不存在的对偶性仍然成立。

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.