(untitled)
Let B_1 be a unit disc of R^2, and H be a completion of C_0^∞(B_1) under the norm \|u\|_H ^2=_B_1(| abla u|^2-u^2(1-|x|^2)^2)dx. Using blow-up analysis, Wang-Ye [1] proved existence of extremals for a Hardy-Trudinger-Moser inequality. In particular, the supremum _uH, \|u\|_H ≤ 1_B_1e^4π u^2dx can be attained by some function u_0H with \|u_0\|_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]. , number=4, journal=Journal of Partial Differential Equations, year=2020, month=May, pages=342–351