On a sharp inequality of Adimurthi–Druet type and extremal functions

Abstract

Our main purpose is to establish the existence and nonexistence of extremal functions for sharp inequality of Adimurthi–Druet type for fractional dimensions on the entire space. Precisely, we extend the sharp Trudinger–Moser type inequality in de Oliveira and do Ó (Calc Var Part Differ Equ 52:125–163, 2015) for the entire space. In addition, we perform the two-step strategy due to L. Carleson and S-Y.A. Chang together with blow up analysis method to ensure the existence of maximizers for the associated extremal problems for both subcritical and critical regimes. We also present a nonexistence result under a subcritical regime for some special cases.