Existence of positive solutions of a semilinear elliptic equation in \(\mathbb{R}_+^n\) with a nonlinear boundary condition
type: Item
instance of: scholarly article
title: Existence of positive solutions of a semilinear elliptic equation in \(\mathbb{R}_+^n\) with a nonlinear boundary condition
zbMATH Open document ID: 0932.35086
author: Michel Chipot
author: Miroslav Chlebík
author: Marek Fila
author: Itai Shafrir
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-11-29T00:00:00Z
review text: The authors are interested in the existence and nonexistence of positive classical solutions of the problem \[ -\Delta u= au^p\quad\text{in }\mathbb{R}^n_+,\quad -{\partial u\over\partial x_n}= u^q\quad\text{on }\partial\mathbb{R}^n_+, \] where \(a\geq 0\), \(p>1\), \(q>1\) and \(n>2\). For \(p= (n+2)/(n- 2)\) and \(q= n/(n-2)\), it has been known that positive solutions exist and they can be expressed explicitly. The main aim of the paper is to prove the existence of positive solutions for \(p\geq (n+2)/(n- 2)\) and \(q\geq n/(n- 2)\), and also the nonexistence for (i) \(p\leq (n+2)/(n- 2)\) and \(q\leq n/(n- 2)\) with at least one strict inequality, (ii) \(p< n/(n- 2)\) or (iii) \(q< n/(n- 1)\) in the case \(a>0\). In the existence proof, a positive solution is obtained as a limit of eigenfunctions \(u^R\) of the auxiliary eigenvalue problem with an appropriate boundary condition: \[ -\Delta u= \lambda u+a|u|^{p- 1}u\quad\text{in }B^+_R= \{x\in \mathbb{R}^n:|x|< R, x_n>0\}, \] where the Rabinowitz bifurcation theorem and a Pohozaev identity play a key role. The nonexistence is proved by a variant of the moving plane method in (i), and from some blow-up results for parabolic problems in (ii) and (iii).
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 1215632
zbMATH Keywords: positive solution
zbMATH Keywords: Pohozaev identity
zbMATH Keywords: Rabinowitz bifurcation theorem
zbMATH Keywords: moving plane method
zbMATH Keywords: blow-up results for parabolic problems
MaRDI profile type: Publication
full work available at URL: jmaa.1998.5958
OpenAlex ID: W2314729742
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DOI: 10.1006/JMAA.1998.5958
description: scientific article; zbMATH DE number 1215632