On the construction of non-simple blow-up solutions for the singular Liouville equation with a potential

We are concerned with the existence of blowing-up solutions to the following boundary value problem $$-\mathrm{\Delta }u=\lambda V(x)e^u-4\pi N{\delta }_{0}in{B}_1,u=0on\mathrm{\partial }{B}_1,$$ where $B_1$ is the unit ball in ${\mathbb{R}}^2$ centered at the origin, $V(x)$ is a positive smooth potential, $N$ is a positive integer ($N\ge 1$). 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 $\lambda \to 0^{+}$.