On the Anisotropic Moser-Trudinger inequality for unbounded domains in ℝn
Changliang Zhou and Chunqin Zhou
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Submission date: 29. Aug. 2019
published in: Discrete and continuous dynamical systems / A, 40 (2020) 2, p. 847-881
DOI number (of the published article): 10.3934/dcds.2020064
Keywords and phrases: moser-trudinger inequality, Anisotropic Sobolev norm, Blow up analysis
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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.