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

Chao-Zhong Chen et al. $[{Proc}.$ ${Amer. Math. Soc},2013],$ proved the upper estimate \frac{\lambda _{n}}{\lambda _{m}}\leq \frac{% n^{p}}{m^{p}} $(n>m\geq 1)$ for Dirichlet Shr\"{o}dinger 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 $q(x)\leq 0$ and single-barrier then \frac{\lambda _{n}}{\lambda _{m}}\geq \frac{n^{p}% }{m^{p}} for $\lambda _{n}>\lambda _{m}\geq -2q^{\ast },$ 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 $(\lambda _{n}(\ell)) _{n\geq 1}$ (of the problem defined on $[0,\ell]$) satisfy $\lambda _{1}( \ell)>0$ and $\frac{\lambda _{n}( \ell)}{\lambda _{m}( \ell) }\geq \frac{n^{p}}{m^{p}}$ $n>m\geq 1$. The value $\ell _{0}$ satisfies the following estimate $0<\ell_{0}\leq \sqrt[p]{\frac{-p}{3q^{*}}}$.