Energy Estimate Related to a Hardy-Trudinger-Moser Inequality

Energy Estimate Related to a Hardy-Trudinger-Moser Inequality

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

Author Details

Yunyan Yang

  1. Energy Estimates on Existence of Extremals for Trudinger–Moser Inequalities

    Wang, Ya Min

    Acta Mathematica Sinica, English Series, Vol. 36 (2020), Iss. 7 P.829

    https://doi.org/10.1007/s10114-020-9528-5 [Citations: 0]
  2. 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]