Conditions for two-peaked solutions of singularly perturbed elliptic equations
type: Item
instance of: scholarly article
title: Conditions for two-peaked solutions of singularly perturbed elliptic equations
zbMATH Open document ID: 0901.35003
published in: Manuscripta Mathematica
publication date: 1998-11-26T00:00:00Z
review text: Let \(\Omega\) be a bounded smooth domain in \(\mathbb{R}^n\). This paper deals with spike-layer solutions \(u_\varepsilon\) of the perturbed problem \[ \varepsilon^2\Delta u-u+u^p=0, \quad u>0 \quad \text{in } \Omega, \quad u=0 \quad \text{on } \partial \Omega, \tag{1} \] with \(1<p< (n+2)/(n-2)\) if \(n\geq 3\), \(1<p< \infty\) if \(n=2\). In a previous paper [Commun. Pure Appl. Math. 48, 731-768 (1995; Zbl 0838.35009)], the author, in collaboration with \textit{W. M. Ni}, studied single-peaked solutions of (1). Here he investigates two-peaked solutions of (1). He obtains necessary geometric conditions for the existence and the locations of the two peaks. In particular, he shows that, if \(\Omega\) is strictly convex, then (1) does not admit two-peaked solutions. The proofs are based on a sharp analysis of the asymptotic behavior of \(u_\varepsilon\) around the peaks, up to the second order.
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1160803
zbMATH Keywords: spike-layer solutions
zbMATH Keywords: geometric conditions
author: Juncheng Wei
reviewed by: Denise Huet
MaRDI profile type: Publication
full work available at URL: s002290050057
OpenAlex ID: W2030419636
DOI: 10.1007/S002290050057
description: scientific article; zbMATH DE number 1160803