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A Note on a Multiplicity Result for the Mean Field Equation on Compact Surfaces

Aleks Jevnikar

Abstract

We are concerned with the class of equations with exponential nonlinearities

-Δu=ρ1(heuΣheu𝑑Vg-1|Σ|)-ρ2(he-uΣhe-u𝑑Vg-1|Σ|)

on a compact surface Σ, which describes the mean field equation of equilibrium turbulence with arbitrarily signed vortices. Here, h is a smooth positive function and ρ1,ρ2 are two positive parameters. We provide the first multiplicity result for this class of equations by using Morse theory.

MSC 2010: 35J20; 35J61; 35R01

Funding statement: The author is supported by the PRIN project “Variational and perturbative aspects of nonlinear differential problems”.

The author would like to express his gratitude to Professor Andrea Malchiodi for his support and for his kind help in preparing this paper.

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Received: 2014-12-11
Accepted: 2015-8-10
Published Online: 2016-3-23
Published in Print: 2016-5-1

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