Energy Estimate Related to a Hardy-Trudinger-Moser Inequality

Let Bi be a unit disc of R^2,and H be a completion of C0^∞(B1)under the norm||u||^2H=∫B1(|■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 u∈,H,^sup||u|,H≤|1∫B1^e4πu2dx 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].