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: