A sharp trace Adams inequality in ℝ4 andexistence of the extremals

Let 4 be a bounded domain with smooth boundary .In this paper, we establish the following sharp form of the trace Adams inequality in W 2,2 ( ) 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 biharmonic equations on the half-space 4+ .With this classification result, we can show that S(12 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.