On a resonant mean field type equation: A 'critical point at Infinity' approach

Mohameden Ahmedou; Mohamed Ben Ayed; Marcello Lucia

doi:10.3934/dcds.2017075

Article Contents

2017, Volume 37, Issue 4: 1789-1818. Doi: 10.3934/dcds.2017075

This issue Previous Article Next Article The stochastic value function in metric measure spaces

On a resonant mean field type equation: A "critical point at Infinity" approach

1.
Mathematisches Institut der Justus-Liebig-Universität Giessen, Arndtsrasse 2, D-35392 Giessen, Germany
2.
Université de Sfax, Faculté des Sciences, Département de Mathématiques, Route de Soukra, Sfax, Tunisia
3.
The City University of New York, CSI, Mathematics Department, 2800 Victory Boulevard, Staten Island New York 10314, USA

Author Bio: E-mail address: Mohameden.Ahmedou@math.uni-giessen.de; E-mail address: Mohamed.Benayed@fss.rnu.tn; E-mail address: marcello.lucia@math.csi.cuny.edu
* Corresponding author: Mohameden.Ahmedou@math.uni-giessen.de.
* Corresponding author: Mohameden.Ahmedou@math.uni-giessen.de.

Received: January 31, 2016

Revised: October 31, 2016

Published: May 2017

This work was partially supported by a grant from the Simons Foundation (Nr. 210368 to Marcello Lucia) and the grant MTM2014-52402-C3-1-P (Spain).

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Abstract

We consider the following mean field type equations on domains of $\mathbb R^2$ under Dirichlet boundary conditions:

$\left\{ \begin{array}{l} - \Delta u = \varrho \frac{{K {e^u}}}{{\int_\Omega {K {e^u}} }}\;\;\;\;\;{\rm{in}}\;\Omega ,\\\;\;\;\;u = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{on}}\;\partial \Omega ,\end{array} \right.$

where $K$ is a smooth positive function and $\varrho$ is a positive real parameter.

A "critical point theory at Infinity" approach of A. Bahri to the above problem is developed for the resonant case, i.e. when the parameter $\varrho$ is a multiple of $8 π$ . Namely, we identify the so-called "critical points at infinity" of the associated variational problem and compute their Morse indices. We then prove some Bahri-Coron type results which can be seen as a generalization of a degree formula in the non-resonant case due to C.C.Chen and C.S.[18].

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Mathematics Subject Classification: Primary:35C60, 35J20;Secondary:35Q70.

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