Moser–Trudinger inequality involving the anisotropic Dirichlet norm [formula omitted] on [formula omitted

Abstract

We investigate a sharp Moser–Trudinger inequality which involves the anisotropic Dirichlet norm (∫ΩFN(∇u)dx)1N on W01,N(Ω) for N≥2. Here F is convex and homogeneous of degree 1, and its polar Fo represents a Finsler metric on RN. Under this anisotropic Dirichlet norm, we establish the Lions type concentration-compactness alternative. Then by using a blow-up procedure, we obtain the existence of extremal functions for this sharp geometric inequality.