Even solutions of some mean field equations at non-critical parameters on a flat torus

Abstract

In this paper, we consider the mean field equation Δ u + e u = ∑ i = 0 3 4 π n i δ ω i 2 in E τ , \begin{equation*} \Delta u+e^{u}=\sum _{i=0}^{3}4\pi n_{i}\delta _{\frac {\omega _{i}}{2}}\text { in }E_{\tau }, \end{equation*} where n i ∈ Z ≥ 0 , n_{i}\in \mathbb {Z}_{\geq 0}, E τ E_{\tau } is the flat torus with periods ω 1 = 1 \omega _{1}=1 , ω 2 = τ \omega _{2}=\tau and Im ⁡ τ > 0 \operatorname {Im}\tau >0 . Assuming N = ∑ i = 0 3 n i N=\sum _{i=0}^{3}n_{i} is odd, a non-critical case for the above PDE, we prove: (i) If among { n i | i = 0 , 1 , 2 , 3 } \{n_{i}|i=0,1,2,3\} there is only one odd integer, then there is always an even solution. Furthermore, if n 0 n_{0} = = 0 0 and n 3 n_{3} is odd, then up to S L 2 ( Z ) SL_{2}(\mathbb {Z}) action, except for finitely many E τ E_{\tau } , there are exactly n 3 + 1 2 \frac {n_{3}+1}{2} even solutions. (ii) If there are exactly three odd integers in { n i | i = 0 , 1 , 2 , 3 } \{n_{i}|i=0,1,2,3\} , then the equation has no even solutions for any flat torus E τ E_{\tau } . Our second result might suggest the symmetric solution of the above mean field equation does not hold in general.