On principal eigenvalues of \(p\)-Laplacian-like operators
type: Item
instance of: scholarly article
title: On principal eigenvalues of \(p\)-Laplacian-like operators
zbMATH Open document ID: 0865.34066
author: Klaus Schmitt
author: Marta Garcia-Huidobro
author: Raúl Manásevich
published in: Journal of Differential Equations
publication date: 1997-07-07T00:00:00Z
review text: The authors consider both homogeneous and the nonhomogeneous boundary-value problems of the form \[ [r^{N-1}\phi(u')]'+\lambda r^{N-1}\psi(u)=0,\quad r\in(0,R),\quad u'(0)=0,\quad u(R)=0 \] and \[ [r^{N-1}\phi(u')]'+\lambda r^{N-1}\psi(u)=r^{N-1}h(r),\quad r\in(0,R),\quad u'(0)=0,\quad u(R)=0, \] where \(N\geq 1\) (not necessarily an integer), \(\phi\) is an increasing homeomorphism of \(R\) and \(\psi\) is nondecreasing with \(\phi(0)=\psi(0)=0\) while \(h\in L^\infty(0,R)\) is a given function, and discuss the existence of the values of \(\lambda\) for which these problems have solutions. Main results concerning the homogeneous problem claim that there exists a smallest \(\lambda_0>0\) such that for \(\lambda<\lambda_0\) this problem has no nontrivial solutions. As to the nonhomogeneous problem, they show that there exists a \(\lambda_0>0\) such that for every \(\lambda<\lambda_0\) the problem has a solution. The proofs are based on fixed point and continuation techniques.
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34L30
zbMATH DE Number: 936697
zbMATH Keywords: homogeneous and nonhomogeneous boundary-value problems
reviewed by: Mithat Idemen
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0140
OpenAlex ID: W2011968437
DOI: 10.1006/JDEQ.1996.0140
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description: scientific article; zbMATH DE number 936697