Adams-Trudinger-Moser inequalities of Adimurthi-Druet type regulated by the vanishing phenomenon and its extremals (2505.24297v1)
Abstract: 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}\Phi\left(\beta \left(\frac{1+\alpha|u|{\frac{n}{m}}{\frac{n}{m}}}{1-\gamma\alpha|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 $\alpha>0$, $\nabla{m} u$ is the $m$-th order gradient for $u$, $0\le\beta\le \beta_0$, with $\beta_0$ being the Adams critical constant, and $\Phi(t) = \operatorname{e}{t}-\sum_{j=0}{j_{m,n}-2}\frac{t{j}}{j!}$ with $j_{m,n}=\min{j\in\mathbb{N}\;:: j\ge n/m}$. In addition, we prove that the constant $\beta_0$ 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 $\mathbb{R}4$.