On the Anisotropic Moser-Trudinger inequality for unbounded domains in ℝⁿ

Changliang Zhou and Chunqin Zhou

Contact the author: Please use for correspondence this email.
Submission date: 29. Aug. 2019
Pages: 38
published in: Discrete and continuous dynamical systems / A, 40 (2020) 2, p. 847-881 
DOI number (of the published article): 10.3934/dcds.2020064
Bibtex
Keywords and phrases: moser-trudinger inequality, Anisotropic Sobolev norm, Blow up analysis
Download full preprint: PDF (515 kB)

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 ≥ 3 to prove the existence of the extremal functions, which is different from the arguments of isotropic case.