Let be a compact Riemannian surface,
be a positive
function on
. The standard mean field equation can be stated as follows
in , where
are given distinct points,
and
denotes the Dirac measure with pole at
. Here the area of
is assumed to be constant
and
stands for the Laplace Beltrami operator with respect to
.
Clearly, the above PDE is invariant under adding a constant. Hence, is normalized to satisfy
.
Let be the Green function with pole at
, that is,
and let
.