Solutions to a nonlinear Neumann problem in three-dimensional exterior domains
type: Item
instance of: scholarly article
title: Solutions to a nonlinear Neumann problem in three-dimensional exterior domains
zbMATH Open document ID: 1429.35102
author: Adélaïde Olivier
author: Olivier Rey
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2018-09-06T00:00:00Z
review text: The authors prove the existence of multipeak solutions to a nonlinear elliptic Neumann problem involving nearly critical Sobolev exponent, in three-dimensional exterior domains. More precisely, consider the problem: \[\begin{cases} -\Delta u=u^{2^\star-1-\varepsilon},\; u>0 \quad&\text{ in } \mathbb{R}^N\setminus\Omega, \\ \frac{\partial u}{\partial \nu}=0 &\text{ on }\partial\Omega,\end{cases} \] where \(\Omega\) is a smooth and bounded domain in \( \mathbb{R}^N\), \(N\geq 3\), such that \( \mathbb{R}^N\setminus\Omega\) is connected, \(2^\ast=2N/(N-2)\) is the critical Sobolev exponent and \(\varepsilon\) is a strictly positive number, assumed to be small. This problem could be considered asymptotically critical. In fact the techniques used in its study as \(\varepsilon\) goes to zero are the same as those to be used in the case of critical nonlinearities. The existence of multipeak solution was studied by \textit{S. Yan} [Adv. Differ. Equ. 7, No. 8, 919--950 (2002; Zbl 1208.35064)] in dimensions \(N\geq 4\). In this paper the authors complete these results considering the case of dimension \(N=3\). Specifically, by considering the case of positive local maximum of the mean curvature of \(\partial \Omega\) (resp. local minimum) they obtain the existence of a multipeak solution of the previous problem for small \(\varepsilon>0\). An asymptotic behaviour of this solutions as \(\varepsilon\) goes to zero is also obtained.
reviewed by: Salvador Villegas
Mathematics Subject Classification ID: 35J91
Mathematics Subject Classification ID: 35B33
zbMATH DE Number: 6933370
zbMATH Keywords: multipeak solutions
zbMATH Keywords: nonlinear elliptic problems
zbMATH Keywords: critical Sobolev exponent
MaRDI profile type: Publication
full work available at URL: j.crma.2018.07.005
OpenAlex ID: W2887831357
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DOI: 10.1016/J.CRMA.2018.07.005
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description: scientific article; zbMATH DE number 6933370