A sharp trace Adams' inequality in $\mathbb{R}^{4}$ and Existence of the extremals

Abstract

Let $Ω\subseteq \mathbb{R}^{4}$ be a bounded domain with smooth boundary $\partialΩ$. 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: \begin{equation*} S(α)=\underset{\int_Ωudx=0,\frac{\partial u}{\partialν}|_{\partialΩ}=0,\VertΔu\Vert_{2}\leq{1}}{\underset {u\in{W^{2,2}(Ω)\setminus\{0\}}}{\sup}}\int_{\partial Ω} e^{αu^{2}}dσ<\infty \end{equation*} holds if and only if $ α\leq12π^2$. 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 $\mathbb{R}^4_{+}$. 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.