Volume 30, Issue 1
Extremal Functions for Trudinger-Moser Type Inequalities in ℝN

J. Part. Diff. Eq., 30 (2017), pp. 64-75.

Published online: 2017-03

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• Abstract

Let $N\geq 2$, $\alpha_N=N\omega_{N-1}^{1/(N-1)}$, where $\omega_{N-1}$ denotes the area of the unit sphere in $\mathbb{R}^N$. In this note, we prove that for any $0<\alpha $$\sup_{u\in W^{1,N}(\mathbb{R}^{N}),\|u\|_{W^{1,N}(\mathbb{R}^{N})}\leq 1}\int_{\mathbb{R}^{N}}|u|^\beta\Big(e^{\alpha |u|^{\frac{N}{N-1}}}-\sum_{j=0}^{N-2}\frac{\alpha^{j}}{j!}|u|^{\frac{Nj}{N-1}}\Big){\rm d}x$$ can be attained by some function$u\in W^{1,N}(\mathbb{R}^N)$with$\|u\|_{W^{1,N}(\mathbb{R}^N)}=1$. Moreover, when$\alpha\geq\alpha_{N}$, the above supremum is infinity. • Keywords Extremal function Trudinger-Moser inequality • AMS Subject Headings 46E35 • Copyright COPYRIGHT: © Global Science Press • Email address xmlimath@ruc.edu.cn (Xiaomeng Li) • BibTex • RIS • TXT @Article{JPDE-30-64, author = {Li , Xiaomeng }, title = {Extremal Functions for Trudinger-Moser Type Inequalities in ℝN}, journal = {Journal of Partial Differential Equations}, year = {2017}, volume = {30}, number = {1}, pages = {64--75}, abstract = { Let$N\geq 2$,$\alpha_N=N\omega_{N-1}^{1/(N-1)}$, where$\omega_{N-1}$denotes the area of the unit sphere in$\mathbb{R}^N$. In this note, we prove that for any$0<\alpha

$$\sup_{u\in W^{1,N}(\mathbb{R}^{N}),\|u\|_{W^{1,N}(\mathbb{R}^{N})}\leq 1}\int_{\mathbb{R}^{N}}|u|^\beta\Big(e^{\alpha |u|^{\frac{N}{N-1}}}-\sum_{j=0}^{N-2}\frac{\alpha^{j}}{j!}|u|^{\frac{Nj}{N-1}}\Big){\rm d}x$$

can be attained by some function $u\in W^{1,N}(\mathbb{R}^N)$ with $\|u\|_{W^{1,N}(\mathbb{R}^N)}=1$. Moreover, when $\alpha\geq\alpha_{N}$, the above supremum is infinity.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v30.n1.5}, url = {http://global-sci.org/intro/article_detail/jpde/5071.html} }
TY - JOUR T1 - Extremal Functions for Trudinger-Moser Type Inequalities in ℝN AU - Li , Xiaomeng JO - Journal of Partial Differential Equations VL - 1 SP - 64 EP - 75 PY - 2017 DA - 2017/03 SN - 30 DO - http://doi.org/10.4208/jpde.v30.n1.5 UR - https://global-sci.org/intro/article_detail/jpde/5071.html KW - Extremal function KW - Trudinger-Moser inequality AB -

Let $N\geq 2$, $\alpha_N=N\omega_{N-1}^{1/(N-1)}$, where $\omega_{N-1}$ denotes the area of the unit sphere in $\mathbb{R}^N$. In this note, we prove that for any $0<\alpha $$\sup_{u\in W^{1,N}(\mathbb{R}^{N}),\|u\|_{W^{1,N}(\mathbb{R}^{N})}\leq 1}\int_{\mathbb{R}^{N}}|u|^\beta\Big(e^{\alpha |u|^{\frac{N}{N-1}}}-\sum_{j=0}^{N-2}\frac{\alpha^{j}}{j!}|u|^{\frac{Nj}{N-1}}\Big){\rm d}x$$ can be attained by some function$u\in W^{1,N}(\mathbb{R}^N)$with$\|u\|_{W^{1,N}(\mathbb{R}^N)}=1$. Moreover, when$\alpha\geq\alpha_{N}\$, the above supremum is infinity.

Xiaomeng Li. (2019). Extremal Functions for Trudinger-Moser Type Inequalities in ℝN. Journal of Partial Differential Equations. 30 (1). 64-75. doi:10.4208/jpde.v30.n1.5
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