EVEN SOLUTIONS OF SOME MEAN FIELD EQUATIONS AT NON-CRITICAL PARAMETERS ON A FLAT TORUS

In this paper, we consider the mean field equation Δu + e^u = Σ_3i=0 4πn_iδω_i/2 in E_τ , where ni ∈ ℤ_≥0, E_τ is the flat torus with periods ω₁ = 1, ω₂ = τ and Im τ > 0. Assuming N = Σ³_i=0 n_i is odd, a non-critical case for the above PDE, we prove: (i) If among {ni|i = 0, 1, 2, 3} there is only one odd integer, then there is always an even solution. Furthermore, if n0 = 0 and n₃ is odd, then up to SL₂(Z) action, except for finitely many E_τ , there are exactly n₃+1/2 even solutions. (ii) If there are exactly three odd integers in {ni|i = 0, 1, 2, 3}, then the equation has no even solutions for any flat torus E_τ . Our second result might suggest the symmetric solution of the above mean field equation does not hold in general.