离散固定梁方程特征函数的振荡性质及其应用

Oscillation Property for the Eigenfunctions of Discrete Clamped Beam Equation and Its Applications

中文关键词:  特征值  特征函数  振荡性质  分歧点  结点解

英文关键词:eigenvalue  eigenfunctions  oscillation property  bifurcation point  nodal solutions

中文摘要:

      该文建立了带权函数$m:[2, N+1]_\mathbb{Z}\to (0,\infty)$的离散固定梁方程$\Delta^4 u(k-2)=\lambda m(k)u(k),\ k\in[2, N+1]_\mathbb{Z}$, $u(1)=\Delta u(1)=0=u(N+2)=\Delta u(N+2)$的特征值结构和相应特征函数的振荡性质, 其中$[2,N+1]_\mathbb{Z}=\{2,3,\cdots,N+1\}$. 作为应用,当非线性项在零点和无穷远处分别满足适当的增长性条件时, 获得了相应非线性问题结点解的全局结构.

英文摘要:

      In this article, we established the structure of all eigenvalues and the oscillation property of corresponding eigenfunctions for discrete clamped beam equation $\Delta^4 u(k-2)=\lambda m(k)u(k),\ k\in[2, N+1]_\mathbb{Z}$, $u(0)=\Delta u(0)=0=u(N+2)=\Delta u(N+2)$ with the weight function $m:[2, N+1]_\mathbb{Z}\to (0,\infty)$, $[2, N+1]_\mathbb{Z}=\{2,3,\ldots,N+1\}$. As an application, we obtain the global structure of nodal solutions of the corresponding nonlinear problems based on the nonlinearity satisfying suitable growth conditions at zero and infinity.