Abstract weak Harnack inequality, multiple fixed points and \(p\)-Laplace equations
type: Item
instance of: scholarly article
title: Abstract weak Harnack inequality, multiple fixed points and \(p\)-Laplace equations
zbMATH Open document ID: 1277.47070
published in: Journal of Fixed Point Theory and Applications
publication date: 2013-05-13T00:00:00Z
review text: In this paper, the author establishes an existence and localization result for fixed points of a completely continuous operator \(N\) defined on a cone \(K_0\) of a Banach space \(X\). In this result, the key assumption is the abstract weak Harnack inequality \[ Nu\geq \|Nu\|\chi\quad\text{for every}\quad u\in K_0, \] where \(\chi \in K_0\setminus\{0\}\), \(\leq\) is the partial order relation associated to \(K_0\) and \(\|\cdot\|\) is a continuous seminorm which is monotone with respect to \(\leq\). With some additional technical assumptions on \(N\), the authors proves the existence of at least a fixed point for \(N\) belonging to a conical annular region of \(K_0\). Moreover, exploiting the localization property of the fixed points, the author obtains, in a natural way, the existence of multiple fixed points for the operator \(N\). The obtained abstract results are applied to the Dirichlet problem for nonlinear elliptic equations involving the \(p\)-Laplace operator. In particular, the author proves the existence of sequences of positive weak solutions assuming that the nonlinearity has a suitable oscillatory behavior.
reviewed by: Giovanni Anello
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 6162216
zbMATH Keywords: weak Harnack inequality
zbMATH Keywords: fixed point
zbMATH Keywords: fixed point index
zbMATH Keywords: p-Laplace operator
zbMATH Keywords: quasilinear elliptic equation
author: Radu Precup
MaRDI profile type: Publication
full work available at URL: s11784-012-0091-2
OpenAlex ID: W2063873084
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DOI: 10.1007/S11784-012-0091-2
description: scientific article; zbMATH DE number 6162216