Estimate for concentration level of the Adams functional and extremals for Adams-type inequality

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.