Volume 31, Issue 4
A Remark on Hardy-Trudinger-Moser Inequality

J. Part. Diff. Eq., 31 (2018), pp. 353-373.

Published online: 2019-01

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• 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].

• Keywords

46E35

h.com.ok@163.com (Qianjin Luo)

Fangyu-3066@ruc.edu.cn (Yu Fang)

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@Article{JPDE-31-353, author = {Luo , Qianjin and Fang , Yu}, title = {A Remark on Hardy-Trudinger-Moser Inequality}, journal = {Journal of Partial Differential Equations}, year = {2019}, volume = {31}, number = {4}, pages = {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].

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v31.n4.6}, url = {http://global-sci.org/intro/article_detail/jpde/12949.html} }
TY - JOUR T1 - A Remark on Hardy-Trudinger-Moser Inequality AU - Luo , Qianjin AU - Fang , Yu JO - Journal of Partial Differential Equations VL - 4 SP - 353 EP - 373 PY - 2019 DA - 2019/01 SN - 31 DO - http://doi.org/10.4208/jpde.v31.n4.6 UR - https://global-sci.org/intro/article_detail/jpde/12949.html KW - Hardy-Trudinger-Moser inequality KW - Trudinger-Moser inequality KW - blow-up analysis. AB -

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].

Qianjin Luo & Yu Fang. (2019). A Remark on Hardy-Trudinger-Moser Inequality. Journal of Partial Differential Equations. 31 (4). 353-373. doi:10.4208/jpde.v31.n4.6
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