Trivial solution and symmetries of nontrivial solutions to a mean field equation

We consider the mean field equation α 2 ∆gu+ e u − 1 = 0 on S. We show that under some technical conditions, u has to be constantly zero for 1/3 ≤ α < 1. In particular, this is the case if u(x) = −u(−x) and u is odd symmetric about a plane. In the cases u(x) = −u(−x) with 1/3 ≤ α < 1 and u(x) = u(−x) with 1/4 ≤ α < 1, we analyze the additional symmetries of the nontrivial solution in detail.