Sharp parameter ranges in the uniform anti-maximum principle for second-order ordinary differential operators
type: Item
instance of: scholarly article
title: Sharp parameter ranges in the uniform anti-maximum principle for second-order ordinary differential operators
zbMATH Open document ID: 1047.34018
author: W. Reichel
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2004-01-05T00:00:00Z
review text: The author studies the differential equation \((pu')'- qu+\lambda wu= f\) in \((0,1)\) with the boundary conditions \((\alpha u+ pu')|_{x=0}= 0\), \((\beta u+ pu')|_{x=1}= 0\), where constants \(\alpha,\beta\in\mathbb{R}\), \(p,q,w\in C([0,1],\mathbb{R})\) and \(p\), \(w> 0\) in \([0,1]\). Let \(\lambda^{\alpha\beta}\) be the first eigenvalue of the corresponding Sturm-Liouville problem and let \(\lambda^{\infty\beta}\), \(\lambda^{\alpha,\infty}\) stand for the first eigenvalue with zero Dirichlet boundary condition at \(x= 0\), \(x= 1\), respectively, and unchanged boundary condition at the opposite endpoint. He shows that if the parameter \(\lambda\) in the above equation belongs to the interval \((\lambda^{\alpha,\beta}_1,\,\min\{\lambda^{\infty\beta}, \lambda^{\alpha\infty}\}]\) and if the function \(f\) is \(\leq 0\) and not identically zero, then the solution \(u\) of the above problem is negative in \([0,1]\).
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 34L99
zbMATH DE Number: 2021116
zbMATH Keywords: Uniform anti-maximum principle
zbMATH Keywords: Sturm-Liouville problem
zbMATH Keywords: Radially symmetric \(p\)-Laplacian
reviewed by: Ludvík Janoš
MaRDI profile type: Publication
full work available at URL: s00033-003-3208-z
OpenAlex ID: W1972883358
DOI: 10.1007/S00033-003-3208-Z
description: scientific article; zbMATH DE number 2021116