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

Changliang Zhou; Chunqin Zhou

doi:10.3934/dcds.2020064

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2020, Volume 40, Issue 2: 847-881. Doi: 10.3934/dcds.2020064

This issue Previous Article A curve of positive solutions for an indefinite sublinear Dirichlet problem Next Article Dimensions of $ C^1- $average conformal hyperbolic sets

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

Changliang Zhou^1, , and
Chunqin Zhou^2, ,

1.
School of Science, East China University of Technology, Nanchang, China
2.
School of Mathematical Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai, China

^* Corresponding author: Chunqin Zhou and Changliang Zhou
^* Corresponding author: Chunqin Zhou and Changliang Zhou

Received: January 31, 2019

Revised: July 31, 2019

Published: November 2019

The authors are supported partially by NSFC of China (No. 11771285). The first author is also supported by the funding for the Doctoral Research of ECUT under grant No. DHBK2018053

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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, see [5,19].

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Mathematics Subject Classification: 35A05, 35J65.

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