A sharp Adams inequality in dimension four and its extremal functions

Let $\mathrm{\Omega }$ be a smooth oriented bounded domain in ${\mathbb{R}}^4$, $H_0^2(\mathrm{\Omega })$ 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 ${\mathrm{\Delta }}^2$ on $\mathrm{\Omega }$. For $\alpha \in [0,\lambda_1(\Omega))$,wedefine$|u|{2,\alpha}^2 = |\Delta u|_2^2 - \alpha |u|2^2$,for$u \in H_0^2(\Omega)$. In this paper, we will prove the following inequality $$\underset{u\in H_0^2(\mathrm{\Omega }),\mathrm{\parallel }u{\mathrm{\parallel }}_{2,\alpha }\le 1}{\sup }{\int }_{\mathrm{\Omega }}{e}^{32{\pi }^{2}u(x{)}^2}dx<\mathrm{\infty }\mathrm{.}$$ This strengthens a recent result of Lu and Yang \cite{LuYang}. We also show that there exists a function $u^{\ast }\in H_0^2(\mathrm{\Omega })\cap C^4(\bar{\mathrm{\Omega }})$ such that $|u^*|{2,\alpha} =1$andthesupremumaboveisattainedby$u^*$. Our proofs are based on the blow-up analysis method.