  In this paper, we derive a sharp version of the singular Trudinger-Moser
inequality, which was originally established by Adimurthi and Sandeep
(Nonlinear Differ. Equ. Appl. 2007). Moreover, extremal functions for those
singular Trudinger-Moser inequalities are also obtained. Our method is the
blow-up analysis. Compared with our previous work (J. Differential Equations
2015), the essential difficulty caused by the presence of singularity is how to
analyse the asymptotic behaviour of certain maximizing sequence near the
blow-up point. We overcome this difficulty by combining two different
classification theorems of Chen and Li (Duke Math. J. 1991; Duke Math. J. 1995)
to get the desired bubble.
