Existence and nonexistence of extremals for critical Adams inequalities in \(\mathbb{R}^4\) and Trudinger-Moser inequalities in \(\mathbb{R}^2\)
type: Item
instance of: scholarly article
title: Existence and nonexistence of extremals for critical Adams inequalities in \(\mathbb{R}^4\) and Trudinger-Moser inequalities in \(\mathbb{R}^2\)
zbMATH Open document ID: 1441.35121
author: Lu Chen
author: Guozhen Lu
author: Maochun Zhu
published in: Advances in Mathematics
publication date: 2020-05-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1812.00413
review text: Let $\Omega\subseteq\mathbb{R}^n$ be a fixed open set, and let $W_0^{m,p}(\Omega)$ be the usual Sobolev space obtained as the completion of $C_0^\infty(\Omega)$ with respect to the norm \[ \|u\|_{W_0^{m,p}(\Omega)} := \bigg(\int_\Omega \Big(|u|^p + |\Delta^{m/2}u|^p\Big)\operatorname{d} x\bigg)^{1/p}. \] As is very well-known, when $1<p<n/m$ the classical Sobolev embedding theorem ensures that $W_0^{m,p}(\Omega)$ is continuously embedded in $L^{p^*}(\Omega)$, where \[ p^* = \frac{np}{n-mp}. \] However, in the `limit case' $p = n/m$, \textit{it is not true that} $W_0^{m,p}(\Omega)\hookrightarrow L^\infty(\Omega)$, and the right substitute for the Sobolev embedding theorem is the Trudinger-Moser inequality (for $m = 1$) and the Adams inequality (for $m > 1$). In this paper, the authors investigate existence and non-existence of extremal functions \textit{both for the Adams inequality in $\Omega = \mathbb{R}^4$ with $m = 2$ and for the Trudinger-Moser inequality in $\Omega = \mathbb{R}^2$}. \textbf{The Adams inequality.} When $\Omega = \mathbb{R}^4$ and $m = 2$, the Adams inequality \textit{with exact growth condition} was established by \textit{N. Masmoudi} and \textit{F. Sani} [Commun. Pure Appl. Math. 67, No. 8, 1307--1335 (2014; Zbl 1309.46017)] and reads as follows: \[ \sup_{\begin{subarray}{c} u\in H^2(\mathbb{R}^4) \\ \|\Delta u\|_{L^2(\mathbb{R}^4)}\leq 1 \end{subarray}} \int_{\mathbb{R}^4}\frac{\exp(32\pi^2|u|^2)-1}{(1+|u|^2)^p}\,\operatorname{d} x\leq C_p\,\int_{\mathbb{R}^4}|u|^2\,\operatorname{d} x \quad\text{iff $p\geq 2$}. \] Setting, for $\alpha\in (-\infty,32\pi^2)$, \[ S(\alpha) := \sup_{\begin{subarray}{c} u\in H^2(\mathbb{R}^4) \\ \|u\|_{H^2(\mathbb{R}^4)} = 1 \end{subarray}} \int_{\mathbb{R}^4} \Big(\exp\big(32\pi^2|u|^2\big)-1-\alpha|u|^2\Big)\operatorname{d} x, \] the authors prove that there exists a threshold $\alpha^*<\infty$ such that \begin{itemize} \item[(i)] if $32\pi^2-\alpha < \alpha^*$, then $S(\alpha)<\alpha^*$ and $S(\alpha)$ is attain…
reviewed by: Stefano Biagi
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 7201202
zbMATH Keywords: Trudinger-Moser inequality
zbMATH Keywords: Adams inequality
zbMATH Keywords: blow up analysis
zbMATH Keywords: extremal function
zbMATH Keywords: sharp Fourier rearrangement principle
zbMATH Keywords: sharp constants
zbMATH Keywords: threshold
MaRDI profile type: Publication
OpenAlex ID: W3016458983
arXiv ID: 1812.00413
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Existence and nonexistence of extremals for critical Adams inequalities in \(\mathbb{R}^4\) and Trudinger-Moser inequalities in \(\mathbb{R}^2\) (DOI)
DOI: 10.1016/J.AIM.2020.107143