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

TL;DR

This paper investigates the existence of extremals for the Adams inequality, establishing bounds and proving extremal existence under certain boundary conditions in higher-dimensional Euclidean spaces.

Contribution

It provides new bounds for the Adams functional along concentrated sequences and proves the existence of extremals under Navier boundary conditions in higher dimensions.

Findings

Established an upper bound for the Adams functional on concentrated sequences.

Proved the existence of extremals for the Adams inequality in higher dimensions.

Applied concentration-compactness principles to demonstrate extremal existence.

Abstract

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.