Second order classification for singular Liouville equations with a coefficient function

In this article we are concerned with the existence of blow-up solutions to the following boundary value problem $$-\Delta v= \lambda V(x) |x|^2e^v\;\mbox{in}\quad B_1,\quad v=0 \;\mbox{ on }\quad \partial B_1,$$ where $B_1$ is the unit ball in $\mathbb R^2$ centered at the origin, $V(x)$ is a positive smooth potential, and $\lambda>0$ is a small parameter. We find necessary and sufficient conditions on the potential $V$ for the existence of a blow-up sequence of solutions tending to infinity near the origin as $\lambda\to 0^+$. In particular, we obtain a second-order classification of the coefficient function $V$ for which (simple) blow-up occurs at the origin.