  Our main purpose in this paper 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 (Calc.Var.Partial Differential Equations,
\textbf{52} (2015) 125-163) for the entire space. In addition, we perform the
two-step strategy of Carleson-Chang together 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
subcritical regime for some special cases.
