Year: 2017
Author: Xiaomeng Li
Journal of Partial Differential Equations, Vol. 30 (2017), Iss. 1 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v30.n1.5
Journal of Partial Differential Equations, Vol. 30 (2017), Iss. 1 : pp. 64–75
Published online: 2017-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 12
Keywords: Extremal function