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

TL;DR

This paper investigates the existence and nonexistence of extremal functions for critical Adams inequalities in four-dimensional space and Trudinger-Moser inequalities in two dimensions, introducing new methods to address open problems.

Contribution

It develops novel approaches using Fourier rearrangement and sharp inequalities to determine extremal existence thresholds in unbounded domains.

Findings

Existence of a threshold  for extremal attainment in  inequalities.

Proof of existence and nonexistence of extremals depending on .

Symmetry properties of extremal functions deduced.

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 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 $$ S(\alpha)=\sup_{\|u\|_{H^2}=1}\int_{\mathbb{R}^4}\big(\exp(32\pi^2|u|^2)-1-\alpha|u|^2\big)dx,$$ where $\alpha \in (-\infty, 32\pi^2)$. We establish the existence of the threshold $\alpha^{\ast}$, where $\alpha^{\ast}\geq \frac{(32\pi^{2})^2B_{2}}{2}$ and $B_2\geq \frac{1}{24\pi^2}$, 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.