Attainability of Second Order Adams Inequalities with Navier Boundary Conditions

We prove the existence of extremal functions for second order Adams inequalities with Navier boundary conditions on balls in ℝn in any dimension n ≥ 4. The proof is based on a symmetrization argument and the ideas introduced by L. Carleson and S.-Y. A. Chang in (Bulletin des Sciences Mathématiques 110, 113–127 1986) to prove the existence of extremal functions in the first order case, i.e. extremal functions for the Trudinger-Moser inequality on balls.