  We are concerned with the existence of blowing-up solutions to the following
boundary value problem $$-\Delta u= \lambda V(x) e^u-4\pi N \delta_0\;\mbox{ in
} B_1,\quad u=0 \;\mbox{ on }\partial B_1,$$ where $B_1$ is the unit ball in
$\mathbb R^2$ centered at the origin, $V(x)$ is a positive smooth potential,
$N$ is a positive integer ($N\geq 1$). Here $\delta_0$ defines the Dirac
measure with pole at $0$, and $\lambda>0$ is a small parameter. We assume that
$N=1$ and, under some suitable assumptions on the derivatives of the potential
$V$ at $0$, we find a solution which exhibits a non-simple blow-up profile as
$\lambda\to 0^+$.
