A sharp Adams inequality in dimension four and its extremal functions

Let $\Omega$ be a smooth oriented bounded domain in $\mathbb R^4$, $H_0^2(\Omega)$ be the Sobolev space, and $\lambda_1(\Omega)= \inf \{\|\Delta u\|_2^2 : u\in H_0^2(\Omega), \|u\|_2 =1\}$ be the first eigenvalue of the bi-Laplacian operator $\Delta^2$ on $\Omega$. For $\alpha \in [0,\lambda_1(\Omega))$, we define $\|u\|_{2,\alpha}^2 = \|\Delta u\|_2^2 - \alpha \|u\|_2^2$, for $u \in H_0^2(\Omega)$. In this paper, we will prove the following inequality $$\sup_{u\in H_0^2(\Omega),\, \|u\|_{2,\alpha} \leq 1} \int_{\Omega} e^{32 \pi^2 u(x)^2} dx < \infty.$$ This strengthens a recent result of Lu and Yang \cite{LuYang}. We also show that there exists a function $u^*\in H_0^2(\Omega)\cap C^4(\overline{\Omega})$ such that $\|u^*\|_{2,\alpha} =1$ and the supremum above is attained by $u^*$. Our proofs are based on the blow-up analysis method.