  This paper is mainly concerned with the existence of extremals for the Adams
inequality. We first establish an upper bound for the classical Adams
functional along of all concentrated sequences in
$W^{m,\frac{n}{m}}_{\mathcal{N}}(\Omega)$, in particular in
$W^{m,\frac{n}{m}}_{0}(\Omega)$, where $\Omega$ is a smooth bounded domain in
Euclidean $n$-space. Secondly, based on the Concentration-compactness
alternative due to Do \'{O} and Macedo, we prove the existence of extremals for
the Adams inequality under Navier boundary conditions for second order
derivatives at least for higher dimensions when $\Omega$ is an Euclidean ball.
