Volume 32, Issue 4
Energy Estimate Related to a Hardy-Trudinger-Moser Inequality

J. Part. Diff. Eq., 32 (2019), pp. 342-351.

Published online: 2020-01

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• Abstract

Let $\mathbb{B}_1$ be a unit disc of $\mathbb{R}^2$, and $\mathscr{H}$ be a completion of $C_0^\infty(\mathbb{B}_1)$ under the norm  $$\|u\|_{\mathscr{H}}^2=\int_{\mathbb{B}_1}\le(|\nabla u|^2-\frac{u^2}{(1-|x|^2)^2}){\rm d}x.$$ Using blow-up analysis, Wang-Ye [1] proved existence of extremals for a Hardy-Trudinger-Moser inequality. In particular, the supremum $$\sup_{u\in \mathscr{H},\,\|u\|_{\mathscr{H}}\leq 1}\int_{\mathbb{B}_1}e^{4\pi u^2}{\rm d}x$$ can be attained by some function $u_0\in\mathscr{H}$ with $\|u_0\|_{\mathscr{H}}= 1$. This was improved by the author and Zhu [2] to a version involving the first eigenvalue of the Hardy-Laplacian operator $-\Delta-1/(1-|x|^2)^2$. In this note, the results of [1, 2] will be reproved by the method of energy estimate, which was recently developed by Malchiodi-Martinazzi [3] and Mancini-Martinazzi [4].

• Keywords

Hardy-Trudinger-Moser inequality energy estimate blow-up analysis.

35A01, 35B33, 35B44, 34E05

yunyanyang@ruc.edu.cn (Yunyan Yang)

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@Article{JPDE-32-342, author = {Yang , Yunyan }, title = {Energy Estimate Related to a Hardy-Trudinger-Moser Inequality}, journal = {Journal of Partial Differential Equations}, year = {2020}, volume = {32}, number = {4}, pages = {342--351}, abstract = {

Let $\mathbb{B}_1$ be a unit disc of $\mathbb{R}^2$, and $\mathscr{H}$ be a completion of $C_0^\infty(\mathbb{B}_1)$ under the norm  $$\|u\|_{\mathscr{H}}^2=\int_{\mathbb{B}_1}\le(|\nabla u|^2-\frac{u^2}{(1-|x|^2)^2}){\rm d}x.$$ Using blow-up analysis, Wang-Ye [1] proved existence of extremals for a Hardy-Trudinger-Moser inequality. In particular, the supremum $$\sup_{u\in \mathscr{H},\,\|u\|_{\mathscr{H}}\leq 1}\int_{\mathbb{B}_1}e^{4\pi u^2}{\rm d}x$$ can be attained by some function $u_0\in\mathscr{H}$ with $\|u_0\|_{\mathscr{H}}= 1$. This was improved by the author and Zhu [2] to a version involving the first eigenvalue of the Hardy-Laplacian operator $-\Delta-1/(1-|x|^2)^2$. In this note, the results of [1, 2] will be reproved by the method of energy estimate, which was recently developed by Malchiodi-Martinazzi [3] and Mancini-Martinazzi [4].

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v32.n4.4}, url = {http://global-sci.org/intro/article_detail/jpde/13613.html} }
TY - JOUR T1 - Energy Estimate Related to a Hardy-Trudinger-Moser Inequality AU - Yang , Yunyan JO - Journal of Partial Differential Equations VL - 4 SP - 342 EP - 351 PY - 2020 DA - 2020/01 SN - 32 DO - http://dor.org/10.4208/jpde.v32.n4.4 UR - https://global-sci.org/intro/article_detail/jpde/13613.html KW - Hardy-Trudinger-Moser inequality KW - energy estimate KW - blow-up analysis. AB -

Let $\mathbb{B}_1$ be a unit disc of $\mathbb{R}^2$, and $\mathscr{H}$ be a completion of $C_0^\infty(\mathbb{B}_1)$ under the norm  $$\|u\|_{\mathscr{H}}^2=\int_{\mathbb{B}_1}\le(|\nabla u|^2-\frac{u^2}{(1-|x|^2)^2}){\rm d}x.$$ Using blow-up analysis, Wang-Ye [1] proved existence of extremals for a Hardy-Trudinger-Moser inequality. In particular, the supremum $$\sup_{u\in \mathscr{H},\,\|u\|_{\mathscr{H}}\leq 1}\int_{\mathbb{B}_1}e^{4\pi u^2}{\rm d}x$$ can be attained by some function $u_0\in\mathscr{H}$ with $\|u_0\|_{\mathscr{H}}= 1$. This was improved by the author and Zhu [2] to a version involving the first eigenvalue of the Hardy-Laplacian operator $-\Delta-1/(1-|x|^2)^2$. In this note, the results of [1, 2] will be reproved by the method of energy estimate, which was recently developed by Malchiodi-Martinazzi [3] and Mancini-Martinazzi [4].

Yunyan Yang . (2020). Energy Estimate Related to a Hardy-Trudinger-Moser Inequality. Journal of Partial Differential Equations. 32 (4). 342-351. doi:10.4208/jpde.v32.n4.4
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