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

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 $Omegasubset mathbb{R}^n$, 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 P{o}lya-Szeg{o} 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 $$ S(alpha)=sup_{|u|_{H^2}=1}int_{mathbb{R}^4}big(exp(32pi^2|u|^2)-1-alpha|u|^2big)dx,$$ where $alpha in (-infty, 32pi^2)$. We establish the existence of the threshold $alpha^{ast}$, where $alpha^{ast}geq frac{(32pi^{2})^2B_{2}}{2}$ and $B_2geq frac{1}{24pi^2}$, such that $Sleft( alpharight) $ is attained if $32pi^{2}-alpha<alpha^{ast}$, and is not attained if $32pi^{2}-alpha>alpha^{ast}$. 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.