Computation of the difference of topology at infinity for Yamabe-type problems on annuli-domains. I, II
type: Item
instance of: scholarly article
title: Computation of the difference of topology at infinity for Yamabe-type problems on annuli-domains. I, II
zbMATH Open document ID: 0966.35043
DOI: 10.1215/S0012-7094-98-09411-X
author: Mohameden Ould Ahmedou
author: Khalil O. El Mehdi
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: For \(\varepsilon> 0\) let \(A_\varepsilon= \{x\in\mathbb{R}^n; \varepsilon<|x|< 1/\varepsilon\}\), \(n\geq 3\). The authors consider the nonlinear elliptic problem \[ -\Delta u= u^{(n+2)/n-2},\quad u>0\quad\text{on }A_\varepsilon,\quad u= 0\quad\text{on }\partial A_\varepsilon.\tag{P\(_\varepsilon\)} \] The positive critical points of the functional \(J_\varepsilon\) on \(H^1_0(A_\varepsilon)\), \[ J_\varepsilon(u)= (1/2) \int_{A_\varepsilon}|\nabla u|^2- {n-2\over 2n} \int_{A_\varepsilon}|u|^{2n/n- 2}, \] are solutions of \((\text{P}_\varepsilon)\). This problem is delicate from a variational viewpoint because of the possible existence of critical points at infinity, which are orbits of \(J_\varepsilon\) along which \(J_\varepsilon\) remains bounded, the gradient goes to zero, and the orbits do not converge. To find the solutions of \((\text{P}_\varepsilon)\) by studying the difference of topology between the level sets of \(J_\varepsilon\), it becomes essential to evaluate the topological contribution of the critical points at infinity. In the first part of this work one computes the difference of topology at infinity in the particular case of double blow-up for thin annuli-domains. In the second part one computes it for expanding annuli \((\varepsilon\to 0)\).
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 47J30
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 1425141
zbMATH Keywords: positive critical points
zbMATH Keywords: double blow-up
zbMATH Keywords: thin annuli-domains
zbMATH Keywords: expanding annuli
reviewed by: Viorel Iftimie
MaRDI profile type: Publication
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full work available at URL: s0012-7094-98-09411-x
OpenAlex ID: W1493468989
description: scientific article; zbMATH DE number 1425141