Concentration of low energy extremals
type: Item
instance of: scholarly article
title: Concentration of low energy extremals
zbMATH Open document ID: 0938.35042
DOI: 10.1016/S0294-1449(99)80015-8
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2000-06-13T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78466
review text: The authors investigate the asymptotic behaviour of solutions and approximate solutions \(u_\varepsilon\) of the variational problem \[ \sup\Big\{\int_\Omega F(u) : \int_\Omega |\nabla u|^2 \leq \varepsilon^2, \;\;u=0 \;\;on \;\;\partial\Omega\Big\} \] in the limit \(\varepsilon \rightarrow 0\). The integrand \(F\) (that can be nonconvex and discontinuous) is supposed to satisfy \(0\leq F(t)\leq c|t|^{2n/(n-2)}\) on a domain of dimension \(n\geq 3\). For smooth \(F\) the extremals \(u_\varepsilon\) satisfy the corresponding Euler-Lagrange equation \(-\Delta u_\varepsilon=\lambda_\varepsilon f(u_\varepsilon)\) in \(\Omega, u_\varepsilon=0\) on \(\partial\Omega\) with \(f=F'\). The main result says 1) The extremals \((u_\varepsilon)\) concentrate at a single point \(x_0 \in \overline{\Omega}\). 2) On a microscopic scale near the concentration point they tend to an entire extremal, i.e. to a solution of the variational problem on \({\mathbb R}^n\) with \(\varepsilon = 1\). These results extend some work by P. L. Lions, using a new proof based on a local generalized Sobolev inequality. The identification of the concentration point \(x_0\) will be discussed in a forthcoming paper.
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1287843
zbMATH Keywords: variational problem
zbMATH Keywords: concentration
zbMATH Keywords: critical Sobolev exponent
zbMATH Keywords: generalized Sobolev inequality
author: Martin Flucher
author: Stefan Müller
reviewed by: Luis Alberto Fernández
MaRDI profile type: Publication
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OpenAlex ID: W2002057735
description: scientific article; zbMATH DE number 1287843