The spectrum in \(\mathbb R\) and \(\mathbb R^2\) of nonlinear elliptic equations with positive parameters
type: Item
instance of: scholarly article
title: The spectrum in \(\mathbb R\) and \(\mathbb R^2\) of nonlinear elliptic equations with positive parameters
zbMATH Open document ID: 1336.35263
DOI: 10.1216/RMJ-2015-45-4-1311
author: Imelda Trejo
author: Raúl Felipe
published in: Rocky Mountain Journal of Mathematics
publication date: 2015-11-04T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1446472436
review text: Let \(\Omega\) be an open, convex and bounded subset of \({\mathbb R}^m\), \(m\geq 2\), with smooth boundary \(\partial\Omega\). The authors study the nonlinear parametric problem \[ \begin{cases} Lu(x)+\lambda(x,u(x))+\mu g(x,u(x)=0, & x\in\Omega, \\ u(x)=0 , & x\in\partial\Omega, \end{cases} \] where \(L\) is a formally self-adjoint elliptic partial differential operator of second order and \(\lambda, \mu>0\). The spectrum of this problem is the set of \((\lambda, \mu)\) such that there exists at least one solution of this equation for this pair of positive numbers. From the authors's summary: ``In this paper we study the spectrum of nonlinear elliptic equations with positive parameters in their nonlinear part. In order to investigate the spectrum in these specific cases, we introduce the monotone method which is an extension of the upper and lower solution methods. Using the Picard iterative process we prove some existence theorems for nonlinear elliptic boundary value problems. We work with both positive and negative solutions.''
reviewed by: Michael Perelmuter
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 6504444
zbMATH Keywords: monotone solutions
zbMATH Keywords: Picard iterative process
zbMATH Keywords: spectrum
MaRDI profile type: Publication
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OpenAlex ID: W1924726445
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description: scientific article; zbMATH DE number 6504444