*Accepted Paper*

**Inserted:** 25 mar 2015

**Last Updated:** 9 jan 2017

**Journal:** Advanced Nonlinear Studies

**Year:** 2015

**Abstract:**

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}{

\Sigma

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

\Sigma

} \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.

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