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

Abstract: This article considers the eigenvalue problem for the Sturm-Liouville problem including $p$-Laplacian \begin{align*} \begin{cases} \left(\vert u'\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 conditions for the existence of nontrivial solutions of the above problem are presented. Results are obtained through the analysis of variational problem related to a sharp Sobolev embedding and generalized trigonometric and hyperbolic functions.