Locating the peaks of the least energy solutions to an elliptic system with Neumann boundary conditions
type: Item
instance of: scholarly article
title: Locating the peaks of the least energy solutions to an elliptic system with Neumann boundary conditions
zbMATH Open document ID: 1246.35089
author: Angela Pistoia
author: Miguel Ramos
published in: Journal of Differential Equations
publication date: 2004-10-01T00:00:00Z
review text: The authors consider positive solutions of the Neumann boundary value problem for the elliptic system \(-\varepsilon^2\Delta u+u=g(v)\), \(-\varepsilon^2\Delta v+v=f(u)\) in \(\Omega\), where \(\Omega\) is a bounded domain in \(\mathbb R^N\) with smooth boundary \(\partial\Omega\), \(N\geq3\), and \(f,g\) are power-type nonlinearities with superlinear and subcritical growth at infinity. They prove that the least energy solutions concentrate, as \(\varepsilon\) goes to zero, at a point of \(\partial\Omega\) which maximizes the mean curvature of \(\partial\Omega\). This statement generalizes and improves the result by \textit{A. I.~Ávila} and \textit{J.~Yang} [J. Differ. Equations 191, No. 2, 348--376 (2003; Zbl 1109.35325)] who considered the power nonlinearities \(f(u)=| u| ^{p-1}u\) and \(g(v)=| v| ^{q-1}v\) with \(p,q>1\) satisfying the subcriticality condition \(1/(p+1)+1/(q+1)>(N-2)/N\). In fact, Ávila and Yang had to assume \(p,q<(N+2)/(N-2)\) and they proved a weaker concentration result.
reviewed by: Pavol Quittner
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 2104530
zbMATH Keywords: superlinear elliptic system
zbMATH Keywords: spike-layered solution
MaRDI profile type: Publication
full work available at URL: j.jde.2004.02.003
OpenAlex ID: W2047391061
cites work: Q4525512
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DOI: 10.1016/J.JDE.2004.02.003
description: scientific article; zbMATH DE number 2104530