BEST CONSTANTS AND EXISTENCE OF MAXIMIZERS FOR ANISOTROPIC WEIGHTED MOSER-TRUDINGER INEQUALITIES IN RN

In this paper, we mainly consider the best constants for certain classes of anisotropic weighted Moser-Trudinger inequalities in R-N. By using a subtle method of changing variables and convex symmetrization arguments, we establish some different types of anisotropic weighted Moser-Trudinger inequalities with best constants in entire Euclidean space RN. Our results can be seen as an extension of the classical weighted Moser-Trudinger inequalities which is proved in [15]. In contrast with the results in [15], the class of functions under our consideration are not needed to be radially symmetric with respect to Finsler metric. Furthermore, we show that the extremum problems which are related to the corresponding Moser-Trudinger inequalities can be attained by the extremum functions.