Adams-Trudinger-Moser inequalities of Adimurthi-Druet type regulated by the vanishing phenomenon and its extremals
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909628453879808 |
|---|---|
| author | Macedo, Abiel Costa de Oliveira, José Francisco Rocha, Fábio Sodré |
| author_facet | Macedo, Abiel Costa de Oliveira, José Francisco Rocha, Fábio Sodré |
| contents | 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 $\mathbb{R}^4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_24297 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Adams-Trudinger-Moser inequalities of Adimurthi-Druet type regulated by the vanishing phenomenon and its extremals Macedo, Abiel Costa de Oliveira, José Francisco Rocha, Fábio Sodré Analysis of PDEs Functional Analysis 35J60, 35B33, 35J91, 35J30, 31A30, 26D10 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 $\mathbb{R}^4$. |
| title | Adams-Trudinger-Moser inequalities of Adimurthi-Druet type regulated by the vanishing phenomenon and its extremals |
| topic | Analysis of PDEs Functional Analysis 35J60, 35B33, 35J91, 35J30, 31A30, 26D10 |
| url | https://arxiv.org/abs/2505.24297 |