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

satisfies .

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, .

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