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

Abstract:

      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.