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Chao-Zhong Chen et al. $[{Proc}.$ ${Amer. Math. Soc},2013],$ proved the upperestimate  $\frac{\lambda _{n}}{\lambda _{m}}\leq \frac{% n^{p}}{m^{p}}$ $ (n>m\geq 1) $for Dirichlet Shr\"{o}dinger operators with nonnegative and single-wellpotentials. In this paper we discuss the case of nonpositive potentials $q(x)$continuous on the interval $[ 0,1] $. We prove that if $q(x)\leq 0$ andsingle-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 associatedeigenvalues $(\lambda _{n}(\ell)) _{n\geq 1}$ (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 $0<\ell_{0}\leq\sqrt[p]{\frac{-p}{3q^{*}}}$.

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