Energy Estimate Related to a Hardy-Trudinger-Moser Inequality

Abstract

Let B1 be a unit disc of R2, and H be a completion of C∞ 0 (B1) under the norm ∥u∥H = ∫ B1 ( |∇u|2− u 2 (1−|x|2)2 ) dx. Using blow-up analysis, Wang-Ye [1] proved existence of extremals for a Hardy-TrudingerMoser inequality. In particular, the supremum sup u∈H ,∥u∥H ≤1 ∫ B1 e4πu 2 dx can be attained by some function u0 ∈H with ∥u0∥H =1. This was improved by the author and Zhu [2] to a version involving the first eigenvalue of the Hardy-Laplacian operator −∆−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]. AMS Subject Classifications: 35A01, 35B33, 35B44, 34E05 Chinese Library Classifications: O17