Extremal functions for Adams' inequalities in dimension four

Abstract

Let $\Omega \subset {\mathbb{R}}^4$ be a smooth bounded domain, $W_0^{2,2}(\Omega )$ be the usual Sobolev space. For any positive integer $\ell$, ${\lambda }_{\ell }(\Omega )$ is the $\ell$-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 $E_{{\lambda }_i(\Omega )}$ is eigenfunction space associated with ${\lambda }_i(\Omega )$. $E_{\ell }^{\perp }$ denotes the orthogonal complement of $E_{\ell }$ in $W_0^{2,2}(\Omega )$. For $0\le \alpha <{\lambda }_{\ell +1}(\Omega )$, we define a norm by $\Vert u{\Vert }_{2,\alpha }^2=\Vert \Delta u{\Vert }_2^2-\alpha \Vert u{\Vert }_2^2$ for $u\in E_{\ell }^{\perp }$. In this paper, using the blow-up analysis, we prove the following Adams inequalities $$\underset{u\in E_{\ell }^{\perp },\Vert u{\Vert }_{2,\alpha }\le 1}{\sup }{\int }_{\Omega }{e}^{32{\pi }^2{u}^2}dx<+\infty ;$$ moreover, the above supremum can be attained by a function $u_0\in E_{\ell }^{\perp }\cap C^4(\overline{\Omega })$ with $\Vert u_0{\Vert }_{2,\alpha }=1$. 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).