Extremal functions for sharp Moser-Trudinger type inequalities in the whole space R-N

摘要

In this paper, we prove the existence of maximizers for the sharp Moser-Trudinger type inequalities in whole space R-N, N >= 2 with more general nonlinearity. The main key in our proof is a precise estimate of the concentrating level of the Moser-Trudinger functional associated with our inequalities on the normalized concentrating sequences. This estimate solves a heavily non-trivial and open problem related to the sharp Moser-Trudinger inequality. Our method gives an alternative proof of the existence of maximizers for the Moser-Trudinger inequality and singular Moser-Trudinger inequality in whole space R-N due to Li and Ruf [30] and Li and Yang [31] without using blow-up analysis argument. (C) 2020 Elsevier Inc. All rights reserved.