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://doi.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 - <p>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&nbsp; $$\|u\|_{\mathscr{H}}^2=\int_{\mathbb{B}_1}\le(|\nabla u|^2-\frac{u^2}{(1-|x|^2)^2}){\rm d}x.$$ 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].</p>