On the anisotropic Moser-Trudinger inequality for unbounded domains in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb R^{n} $\end{document}</tex-math></inline-formula>

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 \begin{document}$ n = 2 $\end{document} and the high dimension case of \begin{document}$ n\geq 3 $\end{document} to prove the existence of the extremal functions, which is different from the arguments of isotropic case, see [ 5 , 19 ].