Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian

Abstract

In this paper, we investigate the Moser-Trudinger inequality when it involves a Finsler-Laplacian operator that is associated with functionals containing \begin{document}$F^2(\nabla u)$\end{document} . Here \begin{document}$F$\end{document} is convex and homogeneous of degree 1, and its polar \begin{document}$F^o$\end{document} represents a Finsler metric on \begin{document}$\mathbb{R}^n$\end{document} . We obtain an existence result on the extremal functions for this sharp geometric inequality.