Existence of multiple positive solutions of semilinear elliptic equations in Esteban-Lions domains with holes
type: Item
instance of: scholarly article
title: Existence of multiple positive solutions of semilinear elliptic equations in Esteban-Lions domains with holes
zbMATH Open document ID: 1194.35168
author: Tsing-San Hsu
published in: Taiwanese Journal of Mathematics
publication date: 2009-06-30T00:00:00Z
review text: Let \(\Omega\) be the union of an infinite cylinder in \({\mathbb{R}}^{N}\) with the axis parallel to \((0,0, \dots,0,1)\) and circular basis of radius \(r\) sitting on the plane orthogonal to \((0,0, \dots,0,1)\), and of a hemisphere of radius \(r\) attached to the basis, minus a hole. Let \(h\), \(K\) be real valued functions in \(\Omega\). Let \(\lambda\geq 0\) be a parameter. Let \(N\geq 2\), \(1<p<2^{*}-1\). Here \(2^{*}\) denotes the (Sobolev) conjugate exponent of \(2\). Under sutiable assumptions on \(h\), \(K\), the authors show that there exists \(\lambda^{*}\) such that the problem \[ -\Delta u(x)+u(x)=\lambda K(x) u^{p}+h(x)\qquad\forall x\in\Omega,\qquad u>0,\qquad u\in H^{1}_{0}(\Omega) \] admits a unique solution for \(\lambda=0\) and \(\lambda=\lambda^{*}\), two solutions for \(0<\lambda<\lambda^{*}\), no solution for \(\lambda>\lambda^{*}\). The authors also provide some estimate on \(\lambda^{*}\) and study the behaviour of the solutions corresponding to the parameter \(\lambda\) as \(\lambda\) tends of \(0\).
reviewed by: Massimo Lanza de Cristoforis
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B30
zbMATH DE Number: 5572688
zbMATH Keywords: semilinear elliptic equations
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: Esteban-Lions domains
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500405356
OpenAlex ID: W1528515730
DOI: 10.11650/TWJM/1500405356
description: scientific article; zbMATH DE number 5572688