Energy Estimate Related to a Hardy-Trudinger-Moser Inequality (Journal of Partial Differential Equations, 2020)
Let $\mathbb{B}_1$ be a unit disc of $\mathbb{R}^2$, and $\mathscr{H}$ be a completion of $C_0^∞(\mathbb{B}_1)$ under the norm $$\|u\|_{\mathscr{H}}^2=\int_{\mathbb{B}_1}\le(| abla u|^2-\frac{u^2}{(1-|x|^2)^2}){\rm d}x.$$ Using blow-up analysis, Wang-Ye [1] proved existence of extremals for a Hardy-Trudinger-Moser inequality. In particular, the supremum $$\sup_{u\in \mathscr{H},\,\|u\|_{\mathscr{H}}≤ 1}\int_{\mathbb{B}_1}e^{4π u^2}{\rm d}x$$ 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 $-Δ-1/(1-|x|^2)^2$. 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.