  Though much work has been done with respect to the existence of extremals of
the critical first order Trudinger-Moser inequalities in
$W^{1,n}(\mathbb{R}^n)$ and higher order Adams inequalities on finite domain
$\Omega\subset \mathbb{R}^n$, whether there exists an extremal function for the
critical higher order Adams inequalities on the entire space $\mathbb{R}^n$
still remains open. The current paper represents the first attempt in this
direction. The classical blow-up procedure cannot apply to solving the
existence of critical Adams type inequality because of the absence of the
P\'{o}lya-Szeg\"{o}\ type inequality. In this paper, we develop some new ideas
and approaches based on a sharp Fourier rearrangement principle (see
\cite{Lenzmann}), sharp constants of the higher-order Gagliardo-Nirenberg
inequalities and optimal poly-harmonic truncations to study the existence and
nonexistence of the maximizers for the Adams inequalities in $\mathbb{R}^4$ of
the form $$
S(\alpha)=\sup_{\|u\|_{H^2}=1}\int_{\mathbb{R}^4}\big(\exp(32\pi^2|u|^2)-1-\alpha|u|^2\big)dx,$$
where $\alpha \in (-\infty, 32\pi^2)$. We establish the existence of the
threshold $\alpha^{\ast}$, where $\alpha^{\ast}\geq
\frac{(32\pi^{2})^2B_{2}}{2}$ and $B_2\geq \frac{1}{24\pi^2}$, such that
$S\left( \alpha\right) $ is attained if $32\pi^{2}-\alpha<\alpha^{\ast}$, and
is not attained if $32\pi^{2}-\alpha>\alpha^{\ast}$. This phenomena has not
been observed before even in the case of first order Trudinger-Moser
inequality. Therefore, we also establish the existence and non-existence of an
extremal function for the Trudinger-Moser inequality on $\mathbb{R}^2$.
Furthermore, the symmetry of the extremal functions can also be deduced through
the Fourier rearrangement principle.
