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July/August 2021 Lyapunov-type inequalities for a Sturm-Liouville problem of the one-dimensional p-Laplacian
Shingo Takeuchi, Kohtaro Watanabe
Differential Integral Equations 34(7/8): 383-399 (July/August 2021). DOI: 10.57262/die034-0708-383

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.

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Shingo Takeuchi. Kohtaro Watanabe. "Lyapunov-type inequalities for a Sturm-Liouville problem of the one-dimensional p-Laplacian." Differential Integral Equations 34 (7/8) 383 - 399, July/August 2021. https://doi.org/10.57262/die034-0708-383

Information

Published: July/August 2021
First available in Project Euclid: 23 June 2021

Digital Object Identifier: 10.57262/die034-0708-383

Subjects:
Primary: 34L10 , 46E35

Rights: Copyright © 2021 Khayyam Publishing, Inc.

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Vol.34 • No. 7/8 • July/August 2021
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