Bubbling on boundary submanifolds for the Lin-Ni-Takagi problem at higher critical exponents
type: Item
instance of: scholarly article
title: Bubbling on boundary submanifolds for the Lin-Ni-Takagi problem at higher critical exponents
zbMATH Open document ID: 1310.35136
DOI: 10.4171/JEMS/473
author: Fethi Mahmoudi
author: Monica Musso
author: Manuel del Pino
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2014-10-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1107.5566
reviewed by: Satoshi Tanaka
Mathematics Subject Classification ID: 35J91
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 35B33
zbMATH DE Number: 6363151
zbMATH Keywords: critical Sobolev exponent
zbMATH Keywords: blowing-up solutions
zbMATH Keywords: nondegenerate minimal submanifolds
MaRDI profile type: Publication
arXiv ID: 1107.5566
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OpenAlex ID: W2964075649
review text: In this paper, the problem NEWLINE\[NEWLINE \begin{cases} d^2 \Delta u -u + u^{\frac{n-k+2}{n-k-2}} = 0 & \text{in} \;\Omega, \\ \frac{\partial u}{\partial \nu} = 0 & \text{on} \;\partial\Omega \end{cases}NEWLINE\]NEWLINE is considered, where \(\Omega \subset {\mathbb R}^n\) is a bounded domain with a smooth boundary \(\partial \Omega\) and \(d\) is a small positive parameter. It is assumed that there exists a \(k\)-dimensional closed, embedded minimal submanifold \(K\) of \(\partial \Omega\) which is nondegenerate, and a certain weighted average of sectional curvatures of \(\partial \Omega\) is positive along \(K\). Then it is shown the existence of a sequence \(d=d_j \to 0\) and a positive solution \(u_d\) such that NEWLINE\[NEWLINE d^2 |\Delta u_d|^2 \rightharpoonup S\delta_k \quad \text{as} \;d \to 0 NEWLINE\]NEWLINE in the sense of measures, where \(\delta_K\) stands for the Dirac measure supported \(K\) and \(S\) is a positive constant.
description: scientific article; zbMATH DE number 6363151