Growth conditions for oscillation of nonlinear differential equations with \(p\)-Laplacian
type: Item
instance of: scholarly article
title: Growth conditions for oscillation of nonlinear differential equations with \(p\)-Laplacian
zbMATH Open document ID: 1085.34029
author: Jitsuro Sugie
author: Naoto Yamaoka
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-06-01T00:00:00Z
review text: The authors give criteria for the oscillation of all solutions of the nonlinear differential equation \[ \left( \phi_p(x')\right)'+\frac{1}{t^p}g(x)=0, \;t>0, \tag{1} \] where \(\phi_p(y)\) is a real-valued function defined by \(\phi_p(y)=| y| ^{p-2}y\) with \(p>1\) a fixed real number, and \(g(x)\) is a continuous function on \(\mathbb{R}\) satisfying \(xg(x)>0\), if \(x\neq0\), and a suitable smoothness condition for the uniqueness of solutions of the initial value problem. The equation includes the famous Euler differential equation and half-linear differential equations. The main result of the paper is the following theorem: Assume \(xg(x)>0\) for \(x\neq0\) and suppose that \[ \frac{g(x)}{\phi_p(x)}\geq \left( \frac{p-1}{p} \right)^p+\frac{\lambda}{(\log| x| )^2} \] for \(| x| \) sufficiently large, where \[ \lambda>\frac12 \left(\frac{p-1}{p} \right)^{p+1}. \] Then, all nontrivial solutions of (1) are oscillatory. The authors conjecture a sufficient condition for all nontrivial solutions of (1) to be nonoscillatory and apply their results to elliptic equations.
reviewed by: Zuzana Došlá
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 2173083
zbMATH Keywords: nonlinear oscillation
zbMATH Keywords: p-Laplacian operator
zbMATH Keywords: Sturm's comparison method
zbMATH Keywords: Riccati technique
zbMATH Keywords: Elliptic equations
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.10.009
OpenAlex ID: W2066925764
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DOI: 10.1016/J.JMAA.2004.10.009
description: scientific article; zbMATH DE number 2173083