Calculus of Variations and Geometric Measure Theory

A. Jevnikar

A note on a multiplicity result for the mean field equation on compact surfaces

created by jevnikar on 25 Mar 2015
modified on 09 Jan 2017


Accepted Paper

Inserted: 25 mar 2015
Last Updated: 9 jan 2017

Journal: Advanced Nonlinear Studies
Year: 2015


We are concerned with the following class of equations with exponential nonlinearities on a compact surface:

$ - \Delta u = \rho_1 \left( \frac{h \,e^{u}}{\int_\Sigma h \,e^{u} \,dV_g} - \frac{1}{
} \right) - \rho_2 \left( \frac{h \,e^{-u}}{\int_\Sigma h \,e^{-u} \,dV_g} - \frac{1}{
} \right), $

which describes the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. Here $h$ is a smooth positive function and $\rho_1, \rho_2$ two positive parameters.

We provide the first multiplicity result for this class of equations by using Morse theory.