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On the supercritical mean field equation on pierced domains


Authors: Mohameden Ould Ahmedou and Angela Pistoia
Journal: Proc. Amer. Math. Soc. 143 (2015), 3969-3984
MSC (2010): Primary 35J60, 35B33, 35J25, 35J20, 35B40
DOI: https://doi.org/10.1090/S0002-9939-2015-12596-8
Published electronically: March 18, 2015
MathSciNet review: 3359586
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Abstract: We consider the problem

$\displaystyle (P_\epsilon )\qquad \Delta u +\lambda { e^{u }\over \int \limits ... ... u =0\ \hbox {on}\ \partial \left (\Omega \setminus B(\xi ,\epsilon )\right ), $

where $ \Omega $ is a smooth bounded open domain in $ \mathbb{R}^2$ which contains the point $ \xi .$ We prove that if $ \lambda >8\pi ,$ problem $ (P_\epsilon )$ has a solutions $ u_\epsilon $ such that

$\displaystyle u_\epsilon (x)\to {8\pi + \lambda \over 2} G(x,\xi ) \ \hbox {uniformly on compact sets of $\Omega \setminus \{\xi \}$ }$

as $ \epsilon $ goes to zero. Here $ G$ denotes Green's function of Dirichlet Laplacian in $ \Omega .$ If $ \lambda \not \in 8\pi \mathbb{N}$ we will not make any symmetry assumptions on $ \Omega ,$ while if $ \lambda \in 8\pi \mathbb{N}$ we will assume that $ \Omega $ is invariant under a rotation through an angle $ { 8\pi ^2\over \lambda } $ around the point $ \xi .$

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Additional Information

Mohameden Ould Ahmedou
Affiliation: Mathematisches Institut, Justus-Liebig-University Giessen, Arndtstraße 2, 35392 Giessen, Germany
Email: Mohameden.Ahmedou@math.uni-giessen.de

Angela Pistoia
Affiliation: Dipartimento SBAI, Università di Roma “La Sapienza”, via Antonio Scarpa 16, 00161 Roma, Italy
Email: pistoia@dmmm.uniroma1.it

DOI: https://doi.org/10.1090/S0002-9939-2015-12596-8
Keywords: Mean field equation, blow-up solutions, supercritical problem, pierced domain
Received by editor(s): December 12, 2013
Received by editor(s) in revised form: May 24, 2014
Published electronically: March 18, 2015
Additional Notes: The authors have been supported by Vigoni Project E65E06000080001
Communicated by: Joachim Krieger
Article copyright: © Copyright 2015 American Mathematical Society

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