I am now reviewing Perron’s method and it is a fun to go over a similar argument like this. The materials come from the lecture by Prof. Li and T.Aubin’s book.
Theorem On a compact Riemannian manifold without boundary , the equation
with and on , has a solution .
We will use the method of super & subsolution. This equation enjoys a nice maximum principle as well. Compared to Perron’s method to Poisson equation, we actually work on functions instead of harmonic functions and use Azera-Ascoli theorem to get the convergence result. The uniform bound of norm can be achieved by Schauder estimate. And this method doesn’t give uniqueness of the equation.
Proof: First we denote
Note that when is small enough and fixed, we have
Here is the place where we need . Then are supersolution and subsolution respectively. We will produce a solution such that .
Fix a large constant such that
Let , then the equation is equivalent to
Now set and define a sequence by . Indeed, given fixed, by Schauder estimate, we have . We know that since is a constant, and we can do the procedure inductively.
Next we will show
Since and , . Since is compact, there exists such that . Since , we have , which gives . Also, . By the construction of we will have and follows. Repeating this procedure with and , we will have , and so on.
Now we obtain a uniform bound for independent of , since
Now look at the equation
, in which we have a uniform bound for RHS, hence we obtain a uniform bound . Then by Morrey inequality, . Finally by Schauder estimate, we get a uniform bound of norm: . Applying Azela-Ascoli theorem, there exists a subsequence converges to a function in . Note that is a monotone increasing sequence with upper bound, so it has a pointwise limit, and this limit can be nothing but .
Therefore is a solution to
By Schauder estimate again, .