Multi-peak solutions for a singularly perturbed semilinear elliptic problem
type: Item
instance of: scholarly article
title: Multi-peak solutions for a singularly perturbed semilinear elliptic problem
zbMATH Open document ID: 0966.35039
author: E. S. Noussair
author: Daomin Cao
published in: Journal of Differential Equations
publication date: 2001-08-16T00:00:00Z
review text: The existence of single (multi)-peak positive solutions of the Dirichlet problem for the equation \(-\varepsilon^2\Delta u+ u= u^{p-1}\), when \(\varepsilon\downarrow 0\), in a bounded smooth domain \(\Omega\subset\mathbb{R}^N\), for \(p\in (2;2N/(N- 2))\) if \(N\geq 3\) and \(p\in (2;\infty)\) if \(N= 2\), depends on the topology of \(\Omega\). Some sufficient conditions on \(\Omega\) in order that such solutions exist where given in \textit{Y. Y. Li} and \textit{L. Nirenberg} [Commun. Pure Appl. Math. 51, No. 11-12, 1445-1490 (1998; Zbl 0933.35083)] and in \textit{E. S. Noussair} and \textit{S. Yan} [Proc. Lond. Math. Soc., III. Ser. 76, No. 2, 427-452 (1998; Zbl 0905.35035)]. In the paper under review, new conditions on \(\Omega\), sufficient for the existence of single (multi)-peak positive solutions of the above-mentioned problem are given. Namely, solutions concentrating when \(\varepsilon\downarrow 0\) near saddle points (defined in the paper) of the distance function \(\text{dist}(\cdot,\partial\Omega)\) are constructed.
reviewed by: Mihai Pascu
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 1530273
zbMATH Keywords: multi-peak solutions
zbMATH Keywords: semilinear
zbMATH Keywords: saddle point
MaRDI profile type: Publication
full work available at URL: jdeq.2000.3795
OpenAlex ID: W2092914310
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DOI: 10.1006/JDEQ.2000.3795
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description: scientific article; zbMATH DE number 1530273