Oscillation Property for the Eigenfunctions of Discrete Clamped Beam Equation and Its Applications
Received:July 06, 2020  Revised:March 11, 2021
Key Words: eigenvalue   eigenfunctions   oscillation property   bifurcation point   nodal solutions
Fund Project:Supported by the National Natural Science Foundation of China (Grant Nos.11901464; 11801453) and the Young Teachers' Scientific Research Capability Upgrading Project of Northwest Normal University (Grant No.NWNU-LKQN2020-20).
 Author Name Affiliation Yanqiong LU College of Mathematics and Statistics, Northwest Normal University, Gansu 730070, P. R. China Rui WANG College of Mathematics and Statistics, Northwest Normal University, Gansu 730070, P. R. China
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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.