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Non simple blow-up phenomena for the singular Liouville equation.

Banff International Research Station for Mathematical Innovation and Discovery

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).

Mathematics; Partial differential equations; Differential geometry

video/mp4

Author affiliation: Università di Roma Tor Vergata

2019-11-03

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/72165

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/