Oscillation property for the eigenfunctions of discrete clamped beam equation and its applications
Received:July 06, 2020  Revised:February 08, 2021
Key Word: eigenvalue   eigenfunctions   oscillation property   bifurcation point   nodal solution
Fund ProjectL:The National Natural Science Foundation of China (Youth Program No.11901464, No.11801453) Northwest Normal University (NWNU-LKQN2020-20)
 Author Name Affiliation Address Yanqiong Lu College of Mathematics and Statistics College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, P R China Rui Wang College of Mathematics and Statistics College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 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 $\Delta^4 u(k-2)=\lambda m(k)u(k),\ k\in[2, N+1]_\mathbb{Z}$, $u(1)=\Delta u(1)=0=u(N+2)=\Delta u(N+2)$ with the weight function $m:[2, N+1]_\mathbb{Z}\to (0,\infty)$, $[2, N+1]_\mathbb{Z}=\{2,3,\cdots,N+1\}$. As an application, we obtain the global structure of positive solutions of the corresponding nonlinear problems based on the nonlinearity satisfying suitable growth conditions at zero and infinity.