Failure of an a priori estimate for a Dirichlet problem
type: Item
instance of: scholarly article
title: Failure of an a priori estimate for a Dirichlet problem
zbMATH Open document ID: 1097.35058
DOI: 10.1007/BF02921427
author: Ould Ahmed Izid Bih Isselkou
publication date: 2006-04-26T00:00:00Z
review text: Let \(\Omega\) be a bounded open set in \(\mathbb R^N\) (\(N\geq 3\)) with smooth boundary. The author is concerned with the study of positive solutions of the nonlinear elliptic equation \(-\Delta u=u^\alpha\) in \(\Omega\), under the Dirichlet condition \(u=\varepsilon\) on \(\partial\Omega\), where \(1<\alpha <(N+2)/(N-2)\) and \(\varepsilon >0\). The main result of the paper establishes that there exists \(\varepsilon^*>0\) depending on \(\Omega\) and \(\alpha\) such that the above problem admits a solution \(u_\varepsilon\in C^2(\Omega)\cap C(\overline\Omega )\) for all \(0\leq\varepsilon\leq\varepsilon^*\) and no solution exists if \(\varepsilon >\varepsilon^*\). In the particular case where \(\Omega\) is the unit ball in \(\mathbb R^N\) (\(N>3\)) it is obtained the lower bound \(\varepsilon^*\geq \left[N(N-2)/4\right]^{(N-2)/4}\). The proofs combine elliptic estimates with the maximum principle for elliptic equations.
reviewed by: Vicenţiu D. Rădulescu
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 5019589
MaRDI profile type: Publication
cites work: Dual variational methods in critical point theory and applications
cites work: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth
cites work: A priori bounds for positive solutions of nonlinear elliptic equations
cites work: Q4052566
cites work: Q5829983
full work available at URL: https://doi.org/10.1007/bf02921427
OpenAlex ID: W2391147055
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article