Equivalence of Subcritical and Critical Adams Inequalities in W^m,2(ℝ^2m) and Existence and Non-existence of Extremals for Adams Inequalities under Inhomogeneous Constraints

Though there have been extensive works on the existence of maximizers for sharp Trudinger-Moser inequalities under homogeneous and inhomogeneous constraints, and sharp Adams inequalities under homogeneous constraints, much less is known for that of the maximizers for Adams inequalities under inhomogeneous constraints. Furthermore, whether exists equivalence between subcritical and critical Adams inequalities in W^m,2(ℝ^2m) also remains open. In this paper, we shall give partial answers to these problems. We first establish the equivalence of subcritical Adams inequalities and critical Adams inequalities under inhomogeneous constraints through exploiting the scaling invariance of Adams inequalities in W^m,2(ℝ^2m) (see Theorem 1.1). Then we consider the existence and non-existence of extremals for sharp Adams inequalities under inhomogeneous constraints in Theorem 1.2, 1.3 and 1.4. Our methods are based on Fourier rearrangement inequality and careful analysis for vanishing phenomenon of radially maximizing sequence for Adams inequalities in W^m,2(ℝ^2m).