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

Abstract

We study the following mean field equation ∆gu + ρ ( eu ∫ S2 e udμ − 1 4π ) = 0 in S2, 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.