On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem
type: Item
instance of: scholarly article
title: On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem
zbMATH Open document ID: 0865.35011
published in: Journal of Differential Equations
publication date: 1997-06-16T00:00:00Z
review text: The author studies a geometrical property of a solution of a singular perturbation problem for a semilinear elliptic boundary value problem \[ \varepsilon^2\Delta u-u+u^p=0,\quad u>0\quad\text{in}\quad\Omega,\quad u=0\quad\text{on}\quad\partial\Omega, \] where \(\varepsilon>0\) is a small parameter. The property of the least energy solution is characterized by the previous work due to W. Ni and J. Wei. Their result stated that the least energy solution has a sharp peak near the point which attains the maximum of the distance function \(d(p,\partial\Omega)\). The problem in this paper is regarded as an inverse part of the characterization. If there is a ``local maximum'' point \(p_*\) of the distance function \(d(p,\partial\Omega)\), can one construct a solution which has a single peak close to \(p_*\)? The author gives an affirmative answer to the problem.
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 933278
zbMATH Keywords: singular solution
zbMATH Keywords: least energy solution
author: Juncheng Wei
reviewed by: Shuichi Jimbo
MaRDI profile type: Publication
OpenAlex ID: W1996322866
DOI: 10.1006/JDEQ.1996.0120
description: scientific article; zbMATH DE number 933278