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 $$-\mathrm{\Delta }v=\lambda V(x)\mathrm{\mid }x{\mathrm{\mid }}^2{e}^{v}in{B}_1,v=0on\mathrm{\partial }{B}_1,$$ 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.