The Ratio of Eigenvalues of the Dirichlet Eigenvalue Problem for Equations with One-Dimensional p-Laplacian

Abstract

Chao-Zhong Chen et al. $[Proc.$ $Amer.Math.Soc,2013],$ proved the upper estimate $\frac{λ_{n}}{λ_{m}}\leq \frac{% n^{p}}{m^{p}}$ $(n>m\ge 1)$ for Dirichlet Shrödinger operators with nonnegative and single-well potentials. In this paper we discuss the case of nonpositive potentials $q(x)$ continuous on the interval $[0,1]$. We prove that if $q(x)\le 0$ and single-barrier then $\frac{λ_{n}}{λ_{m}}\geq \frac{n^{p}% }{m^{p}}$ for ${\lambda }_n>{\lambda }_m\ge -2q^{\ast },$ where $q^{\ast }=\inf \{q(0),q(1)\}$. Furthermore, we show that there exists ${\ell }_0\in (0,1]$ such that for all $\ell \in (0,{\ell }_0],$ the associated eigenvalues $({\lambda }_n(\ell ){)}_{n\ge 1}$ (of the problem defined on $[0,\ell ]$) satisfy ${\lambda }_1(\ell )>0$ and $\frac{{\lambda }_n(\ell )}{{\lambda }_m(\ell )}\ge \frac{n^p}{m^p}$ $n>m\ge 1$. The value ${\ell }_0$ satisfies the following estimate $0<{\ell }_0\le \sqrt[p]{\frac{-p}{3q^{\ast }}}$.

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