Extremal functions for sharp Moser–Trudinger type inequalities in the whole space [formula omitted

Abstract

In this paper, we prove the existence of maximizers for the sharp Moser–Trudinger type inequalities in whole space RN, 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 RN due to Li and Ruf [30] and Li and Yang [31] without using blow-up analysis argument.