Existence and Non-existence of Extremals for Critical Adams Inequality in any Even Dimension

Abstract

Let Ω ⊆ R 4 be a bounded domain with smooth boundary ∂Ω. In this paper, we establish the following sharp form of the trace Adams' inequality in W 2,2 (Ω) with zero mean value and zero Neumann boundary condition:Moreover, we prove a classification theorem for the solutions of a class of nonlinear boundary value problem of bi-harmonic equations on the half space R 4 + . With this classification result, we can show that S(12π 2 ) is attained by using the blow-up analysis and capacitary estimate. As an application, we prove a sharp trace Adams-Onofri type inequality in general four dimensional bounded domains with smooth boundary.