arXiv2025
Let $W^{m,\frac{n}{m}}(\mathbb{R}^n)$ with $1\le m < n$ be the standard higher order derivative Sobolev space in the critical exponential growth threshold. We investigate a new Adams-Adimurthi-Druet type inequality on the whole space $\mathbb{R}^n$ which is strongly influenced by the vanishing phenomenon. Specifically, we prove \begin{equation}\nonumber \sup_{\underset{\|\nabla^{m} u\|_{\frac{n}{m}}^{^{\frac{n}{m}}}+\|u\|_{\frac{n}{m}}^{\frac{n}{m}} \leq 1}{u\in W^{m,\frac{n}{m}}(\mathbb{R}^n)}} \int_{\mathbb{R}^n}Φ\left(β\left(\frac{1+α\|u\|_{\frac{n}{m}}^{\frac{n}{m}}}{1-γα\|u\|_{\frac{n}{m}}^{\frac{n}{m}}}\right)^{\frac{m}{n-m}}|u|^{\frac{n}{n-m}}\right) \mathrm{d}x<+\infty. \end{equation} where $0\le α<1$, $0<γ<\frac{1}α-1$ for $α>0$, $\nabla^{m} u$ is the $m$-th order gradient for $u$, $0\leβ\le β_0$, with $β_0$ being the Adams critical constant, and $Φ(t) = \operatorname{e}^{t}-\sum_{j=0}^{j_{m,n}-2}\frac{t^{j}}{j!}$ with $j_{m,n}=\min\{j\in\mathbb{N}\;:\: j\ge n/m\}$. In addition, we prove that the constant $β_0$ is sharp. In the subcritical case $β<β_0$, the existence and non-existence of extremal function are investigated for $n=2m$ and attainability is proven for $n=4$ and $m=2$ in the critical case $β=β_0$. Our method to analyze the extremal problem is based on blow-up analysis, a truncation argument recently introduced by DelaTorre-Mancini \cite{DelaTorre} and some ideas by Chen-Lu-Zhu \cite{luluzhu20}, who studied the critical Adams inequality in $\mathbb{R}^4$.