Blow-up analysis concerning singular Trudinger-Moser inequalities in dimension two

In this paper, we derive a sharp version of the singular Trudinger-Moser inequality, which was originally established by Adimurthi and Sandeep [2]. Moreover, extremal functions for those singular Trudinger-Moser inequalities are also obtained. Our method is the blow-up analysis. Compared with our previous work ([32]), the essential difficulty caused by the presence of singularity is how to analyse the asymptotic behavior of certain maximizing sequence near the blow-up point. We overcome this difficulty by combining two different classification theorems of Chen and Li [6,7] to get the desired bubble.