Bounds for the positive eigenvalues of the \(p\)-Laplacian with decaying potential
type: Item
instance of: scholarly article
title: Bounds for the positive eigenvalues of the \(p\)-Laplacian with decaying potential
zbMATH Open document ID: 1111.34060
author: B. M. Brown
author: M. S. P. Eastham
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-12-07T00:00:00Z
review text: The authors study the equation \[ (-r^{a}{u'}^{(p-1)})' + r^{a}q(r)u^{(p-1)} = \lambda r^{a}u^{(p-1)} \] for \(p>1\), \(a \geq 0\) and \(q(r) \rightarrow 0\) as \(r\rightarrow \infty\). Here, \(u^{(p-1)} = | u | ^{(p-2)} u\). Using a generalized Prüfer transformation, one obtains an explicit upper bound in terms of \(q\) for the positive values of the spectral parameter \(\lambda\) for which a solution \(u \in L^{p}(0,\infty; r^{a})\) of the equation exists. It is shown that this bound is optimal. These results generalize known spectral properties of the classical linear Laplacian [see e.g. \textit{F. V. Atkinson} and \textit{W. N. Everitt}, Proc. R. Soc. Edinb., Sect. A 80, 57--66 (1978; Zbl 0426.34015)].
reviewed by: Mihai Pascu
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34L30
Mathematics Subject Classification ID: 34B40
zbMATH DE Number: 5078962
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: positive eigenvalues
zbMATH Keywords: Prüfer transform
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.01.058
OpenAlex ID: W1973644087
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DOI: 10.1016/J.JMAA.2006.01.058
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description: scientific article; zbMATH DE number 5078962