On positive \(L^{\infty}\) solutions of a class of elliptic systems
type: Item
instance of: scholarly article
title: On positive \(L^{\infty}\) solutions of a class of elliptic systems
zbMATH Open document ID: 0578.35023
DOI: 10.1007/BF01162722
published in: Mathematische Zeitschrift
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173690
review text: Consider the weakly coupled elliptic system \(\ell_ 0\vec u=-\sum D_ i(a_{ij}D_ j\vec u)+\sum b_ jD_ j\vec u=\vec F(x,\vec u)\) in \(R^ m\), \(m\geq 3\). Under suitable growth conditions on \(\vec F\), we show the existence of infinitely many positive solutions which are bounded and bounded away from zero. Our main tools are a Harnack inequality some bounds on \(\ell_ 0\) in weighted \(L_ p\) spaces and Schauder's fixed point theorem. Our interest in this problem comes from a recent paper of \textit{N. Kawano} and \textit{T. Kusano} [Math. Z. 186, 287-297 (1984; Zbl 0523.34012)] where the case \(\ell_ 0=-\Delta\) and \(m=2\) was considered by means of suitable radial majorant conditions.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 3925367
zbMATH Keywords: growth conditions
zbMATH Keywords: existence
zbMATH Keywords: infinitely many positive solutions
zbMATH Keywords: Harnack inequality
zbMATH Keywords: weighted \(L_ p\) spaces
zbMATH Keywords: Schauder's fixed point theorem
author: Walter Allegretto
MaRDI profile type: Publication
cites work: Positive solutions of elliptic operators in unbounded domains
cites work: Nonlinear elliptic boundary value problems for unbounded domains
cites work: Q5187597
cites work: Linear and quasilinear elliptic equations
cites work: Finiteness of Negative Spectra of Elliptic Operators
cites work: Isomorphism theorems for elliptic operators in Rn
OpenAlex ID: W2504451742
description: scientific article; zbMATH DE number 3925367