Super-critical boundary bubbling in a semilinear Neumann problem
type: Item
instance of: scholarly article
title: Super-critical boundary bubbling in a semilinear Neumann problem
zbMATH Open document ID: 1130.35064
author: Monica Musso
author: Angela Pistoia
author: Manuel del Pino
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2005-03-08T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78647
full work available at URL: http://hdl.handle.net/10533/176266
review text: In the present study the authors considers the following problem \[ -\Delta u+ u= u^{{N+2\over N-2}+\varepsilon}\quad\text{in }\Omega,\qquad u> 0\quad\text{in }\Omega,\qquad {\partial u\over\partial\nu}= 0\quad\text{on }\partial\Omega,\tag{1} \] where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\) and \(N\geq 3\). The authors prove the existence of a one-spike solutions to (1) which concentrates around a topologically non-trivial critical point of the mean curvature of the boundary with positive value. They are able to construct solutions with just one maximum point for which multiple bubbling is present. They show, for example if \(\Omega\) is a ball, there exists a solution whose shape is that of a tower, constituted by supersolution of an arbitrary number of single-bubbles of different blow-up orders. This phenomenon actually takes place just provided that \(\Omega\) is symmetric with respect to the first \((N-1)\) variables, and \(0\in\partial\Omega\) is a point with positive mean curvature.
reviewed by: Messoud Efendiev
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 2141611
zbMATH Keywords: critical Sobolev exponent
zbMATH Keywords: Neumann boundary conditions
zbMATH Keywords: mean curvature
zbMATH Keywords: bubble solutions
zbMATH Keywords: Lyapunov-Schmidt reduction
MaRDI profile type: Publication
OpenAlex ID: W2058711126
cites work: Q4863664
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DOI: 10.1016/J.ANIHPC.2004.05.001
description: scientific article; zbMATH DE number 2141611