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

In this paper, we consider the mean field equation \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}\in \mathbb {Z}_{\geq 0}, E_{\tau } is the flat torus with periods \omega _{1}=1, \omega _{2}=\tau and \operatorname {Im}\tau >0. Assuming 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\} there is only one odd integer, then there is always an even solution. Furthermore, if n_{0} = 0 and n_{3} is odd, then up to SL_{2}(\mathbb {Z}) action, except for finitely many E_{\tau }, there are exactly \frac {n_{3}+1}{2} even solutions. (ii) If there are exactly three odd integers in \{n_{i}|i=0,1,2,3\}, then the equation has no even solutions for any flat torus E_{\tau }. Our second result might suggest the symmetric solution of the above mean field equation does not hold in general. [ABSTRACT FROM AUTHOR]