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Description

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^2$ containing the origin. We are concerned with the following Liouville equation with Dirac mass measure $$ \left\{ \begin{aligned}&- \Delta u = \lambda e^u-4\pi N_\lambda {\mathcal \delta}_0& \hbox{ in }& \Omega,\\ & \ u=0 & \hbox{ on }& \partial \Omega. \end{aligned} \right. $$ Here $\lambda$ is a positive small parameter, ${ \mathbb \delta}_0$ denotes Dirac mass supported at $0$ and $N_\lambda $ is a positive number close to an integer $N$ ($N\geq 2$) from the right side. We assume that $\Omega$ is $(N+1)$-symmetric and the regular part of the Green's function satisfies a nondegeneracy condition (both assumptions are verified if $\Omega$ is the unit ball) and we provide an example of non-simple blow-up as $\lambda \to 0^+$ exhibiting a non-symmetric scenario. More precisely we construct a family of solutions split in a combination of $N+1$ bubbles concentrating at 0 arranged on a tiny polygonal configuration centered at $0$. This is a joint work with Juncheng Wei (University of British Columbia).