On a Riemannian surface , let consider the following PDE
naturally arising from the prescribed Gaussian curvature problem. A simple variable change, one can assume that is a negative constant, see this. It follows from a very well-known result due to Kazdan and Warner that it is necessary to have
In addition, Kazdan and Warner also showed that if and
changes sign, then there exists a number
such that the above PDE is solvable for all
but not if
. In fact, the number
can be characterized as follows
This can be easily seen from the the following comprising property: If the PDE is solvable for some , it is also solvable for any
.
However, Kazdan and Warner did not tell us what happens when . In an attempt to see what really happens when
, Chen and Li made use of the Brezis-Li-Shafrir estimate to answer in the following way: The PDE is also solvable even when
. The purpose of this note is to talk about the beautiful Chen-Li argument, see this.
The idea is to approximate the equation for by a sequence
of negative real numbers in the following sense
as
. Their proof consists of three steps as follows: