On the discreteness of the spectra of the Dirichlet and Neumann \(p\)-biharmonic problems
type: Item
instance of: scholarly article
title: On the discreteness of the spectra of the Dirichlet and Neumann \(p\)-biharmonic problems
zbMATH Open document ID: 1081.34018
DOI: 10.1155/S1085337504311115
author: Jiří Benedikt
published in: Abstract and Applied Analysis
publication date: 2005-04-29T00:00:00Z
full work available at URL: https://eudml.org/doc/52083
review text: The author studies the structure of eigenvalues and eigenfunctions of the nonlinear boundary value problems for \[ (| u^{\prime \prime }(t)| ^{p-2}u^{\prime \prime }(t))^{\prime \prime }=\lambda | u(t)| ^{p-2}u(t),\quad t\in [ 0,1],\;p>1,\text{ and }\lambda >1, \] with Dirichlet and Neumann boundary conditions. This work is a continuation of the one done by \textit{P. Drábek} and \textit{M. Ôtani} [Electron. J. Differ. Equ. 2001, Paper No. 48, 19 p., electronic only (2001; Zbl 0983.35099)]\ which deals with the Navier boundary value problem. For the Dirichlet case, he proves that eigenvalues are positive, simple, isolated and form an increasing unbounded sequence. An eigenfunction, corresponding to the \(n\)th eigenvalue, has precisely \(n-1\) zeros in \( (0,1).\) For the Neumann case, the author proves that positive eigenvalues are isolated and simple. An eigenfunction, corresponding to the \(n\)th eigenvalue, has precisely \(n+1\) zeros in \((0,1).\) \(0\) is also an isolated eigenvalue but not simple. Finally, he establishes a relation between the \(n\)th positive eigenvalues of Dirichlet problem and Neumann problem and he cited some open problems.
reviewed by: Mohammed Bouchekif
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 47J10
Mathematics Subject Classification ID: 34L10
Mathematics Subject Classification ID: 34L15
zbMATH DE Number: 2163687
zbMATH Keywords: nonlinear boundary value problem
zbMATH Keywords: eigenvalue
zbMATH Keywords: eigenfunction
zbMATH Keywords: Dirichlet boundary conditions
zbMATH Keywords: Neumann boundary conditions
zbMATH Keywords: p-biharmonic operator
MaRDI profile type: Publication
OpenAlex ID: W2125192641
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description: scientific article; zbMATH DE number 2163687