Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains

Teresa D'Aprile

doi:10.3934/cpaa.2020262

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2021, Volume 20, Issue 1: 159-191. Doi: 10.3934/cpaa.2020262

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Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains

Teresa D'Aprile

Dipartimento di Matematica, Università di Roma "Tor Vergata", via della Ricerca Scientifica 1, 00133 Roma, ITALY

Received: May 31, 2020

Revised: July 31, 2020

Early access: October 2020

Published: January 2021

The research of T. D'Aprile is partially supported by the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome "Tor Vergata", CUP E83C18000100006, and by the group GNAMPA of INdAM Istituto Nazionale di Alta Matematica

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Abstract

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 {\mathit{\boldsymbol{\delta}}}_0\;\hbox{ in } \Omega, \quad u = 0 \;\hbox{ on }\partial \Omega, $

where $ \Omega $ is a smooth and bounded domain in $ \mathbb{R}^2 $ such that $ 0\in\Omega $, $ V $ is a positive smooth potential, $ N $ is a positive integer and $ \lambda>0 $ is a small parameter. Here $ {\mathit{\boldsymbol{\delta}}}_0 $ defines the Dirac measure with pole at $ 0 $. We assume that $ \Omega $ is $ (N+1) $-symmetric and we find conditions on the potential $ V $ and the domain $ \Omega $ under which there exists a solution blowing up at $ N+1 $ points located at the vertices of a regular polygon with center $ 0 $.

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Mathematics Subject Classification: Primary: 35J20, 35J57; Secondary: 35J61.

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