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 \(\begin{aligned} -\Delta u= \lambda V(x) e^u-4\pi N {\varvec{\delta }}_0 \;\hbox { in } B_1,\quad u=0 \;\hbox { on }\partial B_1, \end{aligned}\) $$\begin{array}{c}\hfill -\mathrm{\Delta }u=\lambda V(x)e^u-4\pi N{\mathit{\delta }}_0\text{in}{B}_1,u=0\text{on}\partial B_1,\end{array}$$ where \(B_1\) $B_1$ is the unit ball in \(\mathbb {R}^2\) ${\mathbb{R}}^2$ centered at the origin, V(x) is a positive smooth potential, N is a positive integer ( \(N\ge 1\) $N\ge 1$ ). Here \({\varvec{\delta }}_0\) ${\mathit{\delta }}_0$ defines the Dirac measure with pole at 0, and \(\lambda >0\) $\lambda >0$ is a small parameter. We assume that \(N=1\) $N=1$ 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 \(\lambda \rightarrow 0^+\) $\lambda \to 0^{+}$ .