Existence and Nonexistence of Extremals for critical Adams inequalities in R4 and Trudinger-Moser inequalities in R2

Abstract

Though much work has been done with respect to the existence of extremals of the critical first order Trudinger-Moser inequalities in $W^{1,n}({\mathbb{R}}^n)$ and higher order Adams inequalities on finite domain $\Omega \subset {\mathbb{R}}^n$, whether there exists an extremal function for the critical higher order Adams inequalities on the entire space ${\mathbb{R}}^n$ still remains open. The current paper represents the first attempt in this direction. The classical blow-up procedure cannot apply to solving the existence of critical Adams type inequality because of the absence of the Pólya-Szegö\ type inequality. In this paper, we develop some new ideas and approaches based on a sharp Fourier rearrangement principle (see \cite{Lenzmann}), sharp constants of the higher-order Gagliardo-Nirenberg inequalities and optimal poly-harmonic truncations to study the existence and nonexistence of the maximizers for the Adams inequalities in ${\mathbb{R}}^4$ of the form $$S(\alpha )=\underset{\Vert u{\Vert }_{H^2}=1}{\sup }{\int }_{{\mathbb{R}}^4}(\exp (32{\pi }^2|u{|}^2)-1-\alpha |u{|}^2)dx,$$ where $\alpha \in (-\infty ,32{\pi }^2)$. We establish the existence of the threshold ${\alpha }^{\ast }$, where ${\alpha }^{\ast }\ge \frac{(32{\pi }^2{)}^2{B}_2}{2}$ and $B_2\ge \frac{1}{24{\pi }^2}$, such that $S\left(\alpha \right)$ is attained if $32{\pi }^2-\alpha <{\alpha }^{\ast }$, and is not attained if $32{\pi }^2-\alpha >{\alpha }^{\ast }$. This phenomena has not been observed before even in the case of first order Trudinger-Moser inequality. Therefore, we also establish the existence and non-existence of an extremal function for the Trudinger-Moser inequality on ${\mathbb{R}}^2$. Furthermore, the symmetry of the extremal functions can also be deduced through the Fourier rearrangement principle.

A substantial revision is made, and a mistake is corrected

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