Multiple Axially Asymmetric Solutions to a Mean Field Equation on $\mathbb{S}^{2}$

Abstract

We study the following mean field equation

$$\Delta_{g}u+\rho\left(\frac{e^{u}}{\int_{\mathbb{S}^{2}}e^{u}d\mu}-\frac{1}{4\pi}\right)=0\ \  \mbox{in}\ \ \mathbb{S}^{2},$$

where $\rho$ is a real parameter. We obtain the existence of multiple axially asymmetric solutions bifurcating from $u=0$ at the values $\rho=4n(n+1)\pi$ for any  odd integer $n\geq3$.