Nonexistence of positive entire solutions for elliptic inequalities of the mean curvature type
type: Item
instance of: scholarly article
title: Nonexistence of positive entire solutions for elliptic inequalities of the mean curvature type
zbMATH Open document ID: 0808.35009
author: Hiroyuki Usami
published in: Journal of Differential Equations
publication date: 1995-03-05T00:00:00Z
review text: The author gives conditions on \(p(x)\) and on \(f(u)\) that ensure the nonexistence of positive entire solutions of the equation \[ \text{div} \biggl[ {{Du} \over {(1+| Du|^ 2)^ \alpha}} \biggr]\geq p(x) f(u), \qquad x\in \mathbb{R}^ N \tag{\(*\)} \] with \(N\geq 2\), \(\alpha\in [0,1,2]\). The results extend those of the author's earlier paper [J. Math. Anal. Appl. 164, No. 1, 59-82 (1992; Zbl 0761.35025)]. The proofs are obtained by comparing with radial supersolutions. The reviewer remarks that although the author's statement that the capillarity equation is included in the general class \((*)\) is formally correct, that equation is excluded from the theorem by the conditions imposed on \(f(u)\). The result remains nevertheless correct for that case, as can be shown by an independent reasoning.
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35R45
zbMATH DE Number: 635894
zbMATH Keywords: nonexistence of positive entire solutions
reviewed by: Robert Finn
MaRDI profile type: Publication
full work available at URL: jdeq.1994.1091
OpenAlex ID: W2031452706
DOI: 10.1006/JDEQ.1994.1091
description: scientific article; zbMATH DE number 635894