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This article considers the eigenvalue problem for the Sturm-Liouville problemincluding $p$-Laplacian \begin{align*} \begin{cases} \left(\vertu'\vert^{p-2}u'\right)'+\left(\lambda+r(x)\right)\vert u\vert ^{p-2}u=0,\,\,x\in (0,\pi_{p}),\\ u(0)=u(\pi_{p})=0, \end{cases} \end{align*} where$1<p<\infty$, $\pi_{p}$ is the generalized $\pi$ given by$\pi_{p}=2\pi/\left(p\sin(\pi/p)\right)$, $r\in C[0,\pi_{p}]$ and$\lambda<p-1$. Sharp Lyapunov-type inequalities, which are necessary conditionsfor the existence of nontrivial solutions of the above problem are presented.Results are obtained through the analysis of variational problem related to asharp Sobolev embedding and generalized trigonometric and hyperbolic functions.

Lyapunov-type inequalities for a Sturm-Liouville problem of the one-dimensional $p$-Laplacian