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

Aleks Jevnikar

Type: Article

Publication Date: 2013-09-25

Citations: 23

DOI: https://doi.org/10.1017/s030821051200042x

Abstract

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 e u and e −u , where u is the unknown function in the equation.

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Proceedings of the Royal Society of Edinburgh Section A Mathematics - View
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