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:
Step 1. Minimizing the associated energy functional, for each fixed ,
to obtain a minimizer solving the equation.
Step 2. Show that the sequence of minimizers is bounded in the region where is positively bounded away from .
Step 3. Prove that is bounded in and hence converges to a desired solution.
Since a proof for Step 1 is probably well-known, we omit it. However, the key point in this prove is to show that any solution for the PDE is bounded from below, say by a negative constant . To show that it is also bounded from above, we consider the region where is positively bounded away from . We will use a inequality of Brezis, Li, and Shafrir.
Theorem (Brezis, Li, and Shafrir). Assume that is a Lipschitz function satisfying
and let be a compact subset of a domain in . Then any solution of the equation
To do so, we pick a point at which . Then we select sufficiently small in such a way that
Then we let solve the following
Then we let
which then implies that solves
in . By the Maximum Principle, the sequence is bounded from above in and from below in the small ball .
This can be seen as follows: Assume that achieves its maximum value at some . Then there holds which is impossible since and . Thus, which gives an uniformly upper bound for in . To obtain a lower bound for the sequence , let consider
Since , the function achieves its minimum value at some point on the boundary . Therefore, for all which gives a lower bound as desired.
Since the metric is locally point-wise conformal to the Euclidean metric, we can apply the Brezis-Li-Shafrir estimate to conclude that is also uniformly bounded from above in the smaller ball . This tells us that is uniformly bounded in the region where is positively bounded away from , so is .
In the final step, we prove that the sequence is bounded in . To do so, we first select small enough such that the set
is non-empty. From the previous step, we know that is uniformly bounded in . We now introduce a function given as follows
Then, it is well-known that for each , there exists a unique solution to the following equation
xThen since and , the sequence is bounded. Let , we then know that
It then suffices to prove that the sequence is bounded in . Multiplying both sides of the preceding equation by and integrating the resulting equation over , we arrive at
On the other hand, since each is a minimizer of the functional , the second order derivative is positive definite. That is to saying
for any , or equivalently,
Chosing , we obtain
Hence, we have just shown that
Noticing that is uniformly bounded in the region and is bounded from below in , the preceding inequality tells us that is bounded, so does . Therefore, it is necessary to control from above. From now on, there is nothing to do with our PDE, the only thing we need is the fact that is bounded in . Suppose that
as . Making use of the Friedrichs inequality
we conclude that, as ,
In view of the Poincare inequality and the fact that is bounded, the sequence is bounded in . Using the following elementary inequality
and the fact that is bounded in , we know that
as . This contradicts to the fact that the sequence is bounded in .