On the topological degree of the mean field equation with two parameters

Abstract

We consider the following class of equations with exponential nonlinearities on a compact surface M : −∆u = ρ1 ( h1 eu M h1 e u − 1 |M | ) − ρ2 ( h2 e−u M h2 e −u − 1 |M | ) , which is associated to the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. Here h1, h2 are smooth positive functions and ρ1, ρ2 are two positive parameters. We start by proving a concentration phenomena for the above equation, which leads to a-priori bound for the solutions of this problem provided ρi / ∈ 8πN, i = 1, 2. Then we study the blow up behavior when ρ1 crosses 8π and ρ2 / ∈ 8πN. 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 ρ1 ∈ (0, 8π) ∪ (8π, 16π) and ρ2 / ∈ 8πN. As a byproduct our argument, we give new existence results when the underlying manifold is a sphere and a new proof for some known existence result.