Properties Around the First Eigenvalue of a Partial Discrete Dirichlet Boundary Value Problem

Abstract

In this article, we study some results around the first eigenvalue $\lambda_{1,m}$ for the partial discrete Dirichlet boundary value problem, like the constant sign of the eigenfunction associated with $\lambda_{1,m}$, the simplicity of $\lambda_{1,m}$ and the sign change of any eigenfunction associated with $\lambda > \lambda_{1,m}$.