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 $M$: $$-\Delta u={\rho }_1\left(\frac{h_1{e}^u}{{\int }_M{h}_1{e}^u}-\frac{1}{|M|}\right)-{\rho }_2\left(\frac{h_2{e}^{-u}}{{\int }_M{h}_2{e}^{-u}}-\frac{1}{|M|}\right),$$ 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 start by proving a concentration phenomena for the above equation, which leads to a-priori bound for the solutions of this problem provided ${\rho }_i\notin 8\pi \mathbb{N},i=1,2$. Then 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 ${\rho }_2\notin 8\pi \mathbb{N}$. 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.

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Cited by in corpus (1)

• A mean field equation involving positively supported probability measures: blow-up phenomena and variational aspects