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.