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(λ+r(x)\right)\vert u\vert ^{p-2}u=0,\,\, x\in (0,π_{p}),\\ u(0)=u(π_{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.

15 pages, 1 figure

References in corpus (1)

• Applications of generalized trigonometric functions with two parameters