Existence of positive radial solutions for semilinear elliptic equations in the annulus
type: Item
instance of: scholarly article
title: Existence of positive radial solutions for semilinear elliptic equations in the annulus
zbMATH Open document ID: 0651.35033
DOI: 10.1016/0022-0396(87)90169-0
author: Xabier Garaizar
published in: Journal of Differential Equations
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://hdl.handle.net/2027.42/26548
review text: This paper studies the existence of positive radial solutions of the Dirichlet problem for semilinear elliptic equations in an annular region in R n, i.e., in \(\Omega =\{x\in R\) n: \(0<\lambda <| x| <R\}\). The equation has the form \(\Delta u(x)+f(u(x))=0,\) \(x\in \Omega\), and the boundary condition is \(u(x)=0\), \(x\in \partial D\). The author shows that for such regions, certain growth conditions at infinity on f that are needed for the existence of positive radial solutions of \(\Omega\) is a ball are not needed here. Since these radial solutions satisfy the second order ordinary differential equation \[ u''(r)+((n+1)/r)u'(r)+f(u(r))=0 \] where \(r=| x|\), the paper contains a very detailed study of positive solutions u(r), \(x<r<R\), satisfying the boundary conditions \(u(\lambda)=u(R)=0\). A basic condition on f is that there exists an \(A\geq 0\) such that \(f(u)>0\) for \(u>A\) and F(u)\(\leq 0\) for \(0\leq u<A\), where \(F(u)=\int^{x}_{0}f(s)ds\) and that there exist constants \(b>0\), \(d_ 1>0\), \(d_ 2>0\), and \(k>-1\) such that \(d_ 1u\) \(k\leq f(u)\leq \delta_ 2u\) k for \(u\geq b\). The case \(R=+\infty\), the ``exterior'' problem, is also dealt with.
reviewed by: G.Seifert
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 4061765
zbMATH Keywords: positive radial solutions
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: semilinear elliptic equations
zbMATH Keywords: annular region
zbMATH Keywords: growth conditions at infinity
MaRDI profile type: Publication
OpenAlex ID: W2037911926
cites work: Q3736147
cites work: Q5516356
cites work: Existence, uniqueness, and nondegeneracy of positive solutions of semilinear elliptic equations
cites work: Existence of positive solutions for semilinear elliptic equations in general domains
description: scientific article; zbMATH DE number 4061765