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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
82/2019

On the Anisotropic Moser-Trudinger inequality for unbounded domains in $\mathbb{R}^{n}$

Changliang Zhou and Chunqin Zhou

Abstract

In this paper, we investigate a sharp Moser-Trudinger inequality which involves the anisotropic Sobolev norm in unbounded domains. Under this anisotropic Sobolev norm, we establish the Lions type concentration-compactness alternative firstly. Then by using a blow-up procedure, we obtain the existence of extremal functions for this sharp geometric inequality. In particular, we combine the low dimension case of $n=2$ and the high dimension case of $n\geq 3$ to prove the existence of the extremal functions, which is different from the arguments of isotropic case.

Received:
Aug 29, 2019
Published:
Sep 2, 2019
Keywords:
moser-trudinger inequality, Anisotropic Sobolev norm, Blow up analysis

Related publications

inJournal
2020 Repository Open Access
Changliang Zhou and Chunqin Zhou

On the anisotropic Moser-Trudinger inequality for unbounded domains in \(\mathbb {R}^n\)

In: Discrete and continuous dynamical systems / A, 40 (2020) 2, pp. 847-881