Adams-Trudinger-Moser inequalities of Adimurthi-Druet type regulated by the vanishing phenomenon and its extremals

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 $0\le \alpha <1$, $0<\gamma <\frac{1}{\alpha }-1$ for $\alpha >0$, ${\mathrm{\nabla }}^{m}u$ is the $m$-th order gradient for $u$, $0\le \beta \le {\beta }_0$, with ${\beta }_0$ being the Adams critical constant, and $\mathrm{\Phi }(t)={\mathrm{e}}^t-{\sum }_{j=0}^{j_{m,n}-2}\frac{t^j}{j!}$ with $j_{m,n}=\min \{j\in \mathbb{N}:j\ge n\mathrm{/}m\}$. In addition, we prove that the constant ${\beta }_0$ is sharp. In the subcritical case $\beta <{\beta }_0$, 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 $\beta ={\beta }_0$. 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 ${\mathbb{R}}^4$.