Existence and nonexistence of extremals for critical Adams inequalities in \(\mathbb{R}^4\) and Trudinger-Moser inequalities in \(\mathbb{R}^2\)
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 attained; \item[(ii)] if $32\pi^2-\alpha > \alpha^*$, then $S(\alpha) = 32\pi^2-\alpha$ and $S(\alpha)$ is not attained. \end{itemize} Explicit upper and lower bounds for $\alpha^*$ are given. Furthermore, some qualitative properties of the extremal functions in the case (i) are proved. \textbf{The Trudinger-Moser inequality.} When $\Omega = \mathbb{R}^2$, the Trudinger-Moser inequality \textit{with exact growth condition} was first established by \textit{S. Ibrahim} et al. [J. Eur. Math. Soc. (JEMS) 17, No. 4, 819--835 (2015; Zbl 1317.35032)] and reads as follows: \[ \sup_{\begin{subarray}{c} u\in H^1(\mathbb{R}^2) \\ \|\nabla u\|_{L^2(\mathbb{R}^2)}\leq 1 \end{subarray}} \int_{\mathbb{R}^2}\frac{\exp(4\pi|u|^2)-1}{(1+|u|^2)^p}\,\operatorname{d} x\leq C_p\,\int_{\mathbb{R}^2}|u|^2\,\operatorname{d} x \quad\text{iff $p\geq 2$}. \] Setting, for $\alpha\in (-\infty,4\pi)$, \[ \widetilde{S}(\alpha) := \sup_{\begin{subarray}{c} u\in H^1(\mathbb{R}^2) \\ \|u\|_{H^1(\mathbb{R}^2)} = 1 \end{subarray}} \int_{\mathbb{R}^4} \Big(\exp\big(4\pi|u|^2\big)-1-\alpha|u|^2\Big)\operatorname{d} x, \] the authors prove that there exists a threshold $\beta^*>4\pi$ such that \begin{itemize} \item[(i)] if $4\pi-\alpha < \beta^*$, then $\widetilde{S}(\alpha)<\beta^*$ and $\widetilde{S}(\alpha)$ is attained; \item[(ii)] if $4\pi-\alpha > \beta^*$, then $\widetilde{S}(\alpha) = 4\pi-\alpha$ and $\widetilde{S}(\alpha)$ is not attained. \end{itemize} Furthermore, some qualitative properties of the extremal functions in the case (i) are also established. Quoting the abstract of the paper, the previously described results are obtained by developing ``some new ideas and approaches based on a sharp Fourier rearrangement principle, sharp constants of the higher-order Gagliardo-Nirenberg inequalities and optimal poly-harmonic truncations''.
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