Calculus of Variations and Geometric Measure Theory

A. Jevnikar - J. Wei - Y. Wen

On the Topological degree of the Mean field equation with two parameters

created by jevnikar on 11 Feb 2016
modified on 08 Sep 2016


Accepted Paper

Inserted: 11 feb 2016
Last Updated: 8 sep 2016

Journal: Indiana Univ. Math. J.
Year: 2016


We consider the following class of equations with exponential nonlinearities on a compact surface $M$: $- \Delta u = \rho_1 \left( \frac{h_1 \,e^{u}}{\int_M h_1 \,e^{u} } - \frac{1}{
} \right) - \rho_2 \left( \frac{h_2 \,e^{-u}}{\int_M h_2 \,e^{-u} } - \frac{1}{
} \right), $ which is associated to the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. Here $h_1, h_2$ are smooth positive functions and $\rho_1, \rho_2$ are two positive parameters.

We study the blow up behavior when $\rho_1$ crosses $8\pi$ and $\rho_2 \notin 8\pi\mathbb{N}$. By performing a suitable decomposition of the above equation and using the shadow system that was introduced for the $SU(3)$ Toda system, we can compute the Leray-Schauder topological degree for $\rho_1 \in (0,8\pi) \cup (8\pi,16\pi)$ and $\rho_2 \notin 8\pi\mathbb{N}$.