The existence of solutions to
\Delta u + 8\pi e^u / (\int e^u) = 0 in \Omega
u = 0 on \partial \Omega
where \Omega is a bounded domain of R^2 depends on the geometry of \Omega. For example, if \Omega is a ball, then the equation has no solutions, but for a long and thin ellipse, solutions exist. In this talk I will give a sufficient and necessary condition for the existence of solutions. This condition is expressed in terms of the regular part of the Green function of \Omega.
I will also talk about the uniqueness problem for this equation.