Let be a compact Riemannian surface with the volume . The simplest form of the mean field equation studied in the contexts of the prescribing Gaussian curvature, statistical mechanics of many vortex points in the perfect fluid and self-dual gauss theories is given by
where is a real number.
The mean field equation has a variational structure, and is a solution if and only if it is a critical point of
defined for with
It is worth noticing from this entry that so far our Moser-Trudinger’s inequality is just for
Theorem (Moser-Trudinger’s inequality for ). There are constants and such that for each
for all .
Thank to a work due to Luigi Fontana [here] we actually have Moser-Trudinger’s inequality for general manifold of a higher order gradient of function. Presicely,
Theorem (Moser-Trudinger’s inequality for general manifolds). Let be a compact Riemannian manifold of dimension and a positive integer strictly smaller than . There exists a constant such that for all with
the following uniform inequality holds
where the constant is given below
Consequently, for a general Riemannian surface, i.e. , one gets and thus
Corollary. There holds
For a general function , we have
which after integrating implies
Hence, if we deduce
Consequently, we get
The above inequality is about to say if , the functional is bounded from below, and thus has global minimizer which turns out to be a critical point, hence, a solution. When , the functional is still bounded from below (by a different approach) but it is not when . The case of is not completely solved. We refer the reader to works of Ohtsuka, Malchiodi, Jost, etc.