On simplicity of spectra of \(p\)-biharmonic equations
type: Item
instance of: scholarly article
title: On simplicity of spectra of \(p\)-biharmonic equations
zbMATH Open document ID: 1072.34096
author: Jiří Benedikt
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-10-01T00:00:00Z
review text: The author investigates the real eigenvalues \(\lambda\) of the following nonlinear (but still homogeneous) problem \[ (\rho(t)|u''(t)|^{p-2} u''(t))''= \lambda\chi(t)|u(t)|^{p- 2}u(t),\quad t\in [0,1],\quad Md_u(0)= 0,\quad Nd_u(1)= 0, \] where \(p> 1\), \(\rho,\chi\in C[0,1]\), \(\rho,\chi> 0\), \(M\), \(N\) are \(2\times 4\)-matrices with rank 2 and \[ d_u(t)= [|u(t)|^{p- 2}u(t),|u'(t)|^{p- 2} u'(t), \rho(t)|u''(t)|^{p- 2}u''(t), (\rho(t)|u''(t)|^{p- 2} u''(t))']^T. \] Examples with positive eigenvalues which are not simple as well as examples with negative eigenvalues are presented. On the other hand, one derives conditions which are sufficient in order that there do not exist negative eigenvalues and all eigenvalues are simple. These conditions are formulated in terms of the solutions of the equations \(Mx= 0\) and \(Ny= 0\).
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 47J10
Mathematics Subject Classification ID: 34L05
zbMATH DE Number: 2105450
zbMATH Keywords: \(p\)-biharmonic operator
zbMATH Keywords: Jumping nonlinearity
zbMATH Keywords: Nonlinear spectral theory
reviewed by: J. Brasche
MaRDI profile type: Publication
full work available at URL: j.na.2003.12.004
OpenAlex ID: W1987029172
cites work: Q4544379
cites work: Uniqueness theorem for quasilinear 2\(n\)th-order equations
cites work: On the closed solution to some nonhomogeneous eigenvalue problem with \(p\)-Laplacian
cites work: Q5330830
cites work: Q5637235
DOI: 10.1016/J.NA.2003.12.004
description: scientific article; zbMATH DE number 2105450