  The main purpose of our paper is to prove sharp Adams-type inequalities in
unbounded domains of $\mathbb{R}^{n}$ for the Sobolev space
$W^{m,\frac{n}{m}}\left(\mathbb{R} ^{n}\right)$ for any positive integer $m$
less than $n$. Our results complement those of Ruf and Sani \cite{RS} where
such inequalities are only established for even integer $m$. Our inequalities
are also a generalization of the Adams-type inequalities in the special case
$n=2m=4$ proved in \cite{Y} and stronger than those in \cite{RS} when $n=2m$
for all positive integer $m$ by using different Sobolev norms.
