Year: 2018
Author: Qianjin Luo, Yu Fang
Journal of Partial Differential Equations, Vol. 31 (2018), Iss. 4 : pp. 353–373
Abstract
Let $\mathbb{B}$ be the unit disc in $\mathbb{R}^2$, $\mathscr{H}$ be the completion of $C_0^{\infty}(\mathbb{B})$ under the norm
$$||u||_{\mathscr{H}}=\Big(\int_{\mathbb{B}}|\nabla u|^2\mathrm{d}x-\int_{\mathbb{B}}\frac{u^2}{(1-|x|^2)^2}\mathrm{d}x\Big)^{\frac{1}{2}},\quad \forall u\in C_0^{\infty}(\mathbb{B}).$$
Using blow-up analysis, we prove that for any $\gamma\leqslant 4\pi$, the supremum
$$\begin{align*}\sup_{u\in\mathscr{H},||u||_{1,h}\leqslant 1}\int_{\mathbb{B}}\mathrm{e}^{\gamma u^2}\mathrm{d}x\end{align*}$$
can be attained by some function $u_0\in \mathscr{H}$ with $||u_0||_{1,h}=1$, where $h$ is a decreasingly nonnegative, radially symmetric function, and satisfies a coercive condition. Namely there exists a constant $\delta>0$ satisfying
$$||u||_{1,h}^2=\|u\|_{\mathscr{H}}^2-\int_{\mathbb{B}}hu^2{\rm d}x\geq \delta\|u\|_{\mathscr{H}}^2,\quad\forall \, u\in\mathscr{H}.$$
This extends earlier results of Wang-Ye [1] and Yang-Zhu [2].
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v31.n4.6
Journal of Partial Differential Equations, Vol. 31 (2018), Iss. 4 : pp. 353–373
Published online: 2018-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 21
Keywords: Hardy-Trudinger-Moser inequality
Author Details
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Extremals for a Hardy–Trudinger–Moser Inequality with Remainder Terms
Yin, Kexin
Bulletin of the Iranian Mathematical Society, Vol. 49 (2023), Iss. 5
https://doi.org/10.1007/s41980-023-00813-4 [Citations: 1]