Multiple Axially Asymmetric Solutions to a Mean Field Equation on $\mathbb{S}^{2}$ (Analysis in Theory and Applications, 2022)
We study the following mean field equation$$Δ_{g}u+ρ\left(\frac{e^{u}}{\int_{\mathbb{S}^{2}}e^{u}dμ}-\frac{1}{4π}\right)=0\ \ in\ \ \mathbb{S}^{2},$$where $ρ$ is a real parameter. We obtain the existence of multiple axially asymmetric solutions bifurcating from $u=0$ at the values $ρ=4n(n+1)π$ for any odd integer $n≥3$.
Keywords: Mean field equation; axially asymmetric solutions; bifurcation.