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An existence result for the mean-field equation on compact surfaces in a doubly supercritical regime

Published online by Cambridge University Press:  25 September 2013

Aleks Jevnikar
Aleks Jevnikar*
Affiliation:
Scuola Internazionale Superiore di Studenti Avanzati, via Bonomea 265, 34136 Trieste, Italy, (ajevnika@sissa.it)
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Abstract

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We consider a class of variational equations with exponential nonlinearities on a compact Riemannian surface, describing the mean-field equation of the equilibrium turbulence with arbitrarily signed vortices. For the first time, we consider the problem with both supercritical parameters and we give an existence result by using variational methods. In doing so, we present a new Moser–Trudinger-type inequality under suitable conditions on the centre of mass and the scale of concentration of both eu and e−u, where u is the unknown function in the equation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 143 , Issue 5 , October 2013 , pp. 1021 - 1045
Copyright
Copyright © Royal Society of Edinburgh 2013