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

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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.