A mean field equation with symmetry on torus

Let T2 be a flat two-dimensional torus with fundamental cell domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ { - \tfrac{1} {2},\tfrac{1} {2}} \right] \times \left[ { - \tfrac{1} {2},\tfrac{1} {2}} \right]$$\end{document}, h(x) a positive smooth function satisfying the symmetric property (8) on T2. In this paper we give some sufficient condition under which the mean field equation Δu = 16π − 16πheu, has a smooth solution.