Extremal functions for Adams' inequalities in dimension four

Let $\mathrm{\Omega }\subset {\mathbb{R}}^4$ be a smooth bounded domain, $W_0^{2,2}(\mathrm{\Omega })$ be the usual Sobolev space. For any positive integer $\mathrm{\ell }$, ${\lambda }_{\mathrm{\ell }}(\mathrm{\Omega })$ is the $\mathrm{\ell }$-th eigenvalue of the bi-Laplacian operator. Define $E_{\ell}=E_{\lambda_1(\Omega)}\oplus E_{\lambda_2(\Omega)}\oplus\cdots\oplus E_{\lambda_{\ell}(\Omega)}$,where$E_{\lambda_i(\Omega)}$iseigenfunctionspaceassociatedwith$\lambda_i(\Omega)$\mathrm{.}$E^{\bot}{\ell}$denotestheorthogonalcomplementof$E\ell$in$W_0^{2,2}(\Omega)$\mathrm{.}For$0\leq\alpha<\lambda_{\ell+1}(\Omega)$, we define a norm by $\mathrm{\parallel }u{\mathrm{\parallel }}_{2,\alpha }^2=\mathrm{\parallel }\mathrm{\Delta }u{\mathrm{\parallel }}_2^2-\alpha \mathrm{\parallel }u{\mathrm{\parallel }}_2^2$ for $u\in E_{\mathrm{\ell }}^{\mathrm{\perp }}$. In this paper, using the blow-up analysis, we prove the following Adams inequalities $$\underset{u\in E_{\mathrm{\ell }}^{\mathrm{\perp }},\mathrm{\parallel }u{\mathrm{\parallel }}_{2,\alpha }\le 1}{\sup }{\int }_{\mathrm{\Omega }}{e}^{32{\pi }^2{u}^2}dx<+\mathrm{\infty };$$ moreover, the above supremum can be attained by a function $u_0\in E_{\ell}^{\bot}\cap C^4(\overline{\Omega})$with$|u_0|_{2,\alpha}=1$. This result extends that of Yang (J. Differential Equations, 2015), and complements that of Lu and Yang (Adv. Math. 2009) and Nguyen (arXiv: 1701.08249, 2017).