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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.
