Year: 2019
Author: Yunyan Yang
Journal of Partial Differential Equations, Vol. 32 (2019), Iss. 4 : pp. 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].
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v32.n4.4
Journal of Partial Differential Equations, Vol. 32 (2019), Iss. 4 : pp. 342–351
Published online: 2019-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 10
Keywords: Hardy-Trudinger-Moser inequality