Solvability of Dirichlet problems for semilinear elliptic equations on certain domains
type: Item
instance of: scholarly article
title: Solvability of Dirichlet problems for semilinear elliptic equations on certain domains
zbMATH Open document ID: 0865.35052
author: Zhiren Jin
published in: Pacific Journal of Mathematics
publication date: 1996-12-15T00:00:00Z
review text: Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\), \(n\geq3\), \(|\Omega |\) be the volume of the domain. Consider the Dirichlet problem (P) \(-\Delta u= f(x, u)\) in \(\Omega\); \(u=0\) on \(\partial \Omega\), where \(\Delta\) is the standard Laplace operator, \(f(x, u)\) is a local Hölder continuous function defined on \(\overline {\Omega} \times \mathbb{R}\). The main result of the paper is as follows: Assume that there are positive constants \(M_1\), \(M_2\), \(q\geq 1\), such that \(|f(x, t)|\leq M_1+ M_2|t|^q\) for all \(x\in \overline {\Omega}\), \(t\in \mathbb{R}\); then there is a constant \(c(n, q)\) depending only on \(n\), \(q\), such that if \(|\Omega |\leq c(n, q) (M_2 M_1^{q-1} )^{-{n \over 2q}}\), the problem (P) has at least one solution. There is no upper bound restriction on \(q\) in the result. Thus this result improves an earlier similar result obtained by the author [Indiana Univ. Math. J. 42, 259-282 (1993; Zbl 0797.35049)]. The method used here is a combination of the upper and lower solution method and the variational method. A change of variables is used to construct a pair of upper and lower solutions.
reviewed by: Zhen Jin
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 950199
zbMATH Keywords: semilinear elliptic equations
zbMATH Keywords: superlinear linear growth
zbMATH Keywords: upper and lower solution method
MaRDI profile type: Publication
full work available at URL: pjm.1996.176.117
OpenAlex ID: W1570736734
DOI: 10.2140/PJM.1996.176.117
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description: scientific article; zbMATH DE number 950199