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

Abstract

This paper 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 $1 < p < \infty$, $\lambda < p-1$, $\pi_{p}$ is the generalized $\pi$ given by $\pi_{p}=2\pi/ \left ( p\sin(\pi/p) \right ) $ and $r\in C[0,\pi_{p}]$. 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.