I am struggling with figuring out the details of proposition 7.1. in the paper Curvature and Uniformization - R. Mazzeo and M. Taylor.
Setting is as follows.
Let $\Omega$ be a noncompact Riemann surface with compatible Riemannian metric $g_0$, and let take a sequence $(\Omega_\nu)_\nu$ of compact, smoothly bounded subsets $\Omega_0 \subset \subset \Omega_1 \subset \subset \ldots$ exhausting $\Omega$ (i.e. for any compact $K \subset\Omega$ there exists $\nu$ such that $K \subset \Omega_\nu$).
Let $u_\nu \in C^\infty(\Omega_\nu)$ denote the solution to PDE $$\Delta u_\nu = e^{2u_\nu} + k(x) \ \ \ \ \ \ \ \ (\star),$$ where $k(x)$ is a Gaussian curvature associated to the metric $g_0$. (We get the existence of $u_\nu$ by a results established in preceding sections. They satisfy $u_\nu|_{\partial\Omega_\nu} = +\infty$, and moreover these solutions are monotonically decreasing in a following sense: $\forall \nu \ u_\nu \ge u_{\nu+1}$ on $\Omega_\nu$)
Proposition 7.1: For every such $\Omega$ one of the following must happen:
- $u_\nu \searrow u \in C^\infty(\Omega)$, where $u$ satisfies PDE $(\star)$, or
- $u_\nu \searrow -\infty$ on $\Omega$.
Now I will try to go through the proof, and emphasize the parts which I don't understand.
By a results from preceding sections we know that the case 1. holds if there exists a function $v : \Omega \to \mathbb{R}$ which is locally bounded such that $\forall \nu$ we have $u_\nu \ge v$. So we'd like to assume that there is no such $v$ and show that it implies that 2. must hold. There is no such $v$ if we can find a sequence $(p_\nu)_\nu \subset \mathcal{O} \subset \subset \Omega_N$ for some open set $\mathcal{O}$ and $N \in \mathbb{N}$, such that $$u_\nu(p_\nu) \to -\infty \ \ \text{as} \ \ \nu \to \infty.$$
From now on let's assume that $\nu \ge N+1$, then we have a following uniform bound (w.r.t. indices $\nu$) on $u_\nu$'s by a monotonicity: $$u_\nu \le A_N := \sup_{x \in \bar \Omega_N}{u_{N+1}(x)} < \infty \ \ \text{for} \ \ \nu \ge N+1.$$ And similarly we may define a uniform bound $$|e^{2u_\nu}+k| \le A_{2n} := \sup_{x \in \bar \Omega_N}{|e^{2u_{N+1}(x)}+k(x)|} < \infty \ \ \text{for} \ \ \nu \ge N+1.$$
My question is: now the authors claim that hence we can find $v_\nu \in C^1(\bar \Omega_N)$ for $\nu \ge N + 1$, such that $$\Delta v_\nu = e^{2u_\nu} + k \ \ \text{on} \ \Omega_N, \ |v_\nu|_{L^\infty(\Omega_N)} \le A_{3N}.$$ How do we know that? Reference to the result implying this will be enough.
In fact, we just solve the Dirichlet problem$$\nabla v = e^{2u_\mu} + k \text{ on }\Omega_\nu, \quad u = 0 \text{ on } \partial \Omega_\nu,$$then we know that on $\Omega_N$ which is a compact set of $\Omega_\nu$ we have all the estimate we need by standard elliptic theory. Then $v - u_\nu$ is harmonic and satisfies a Harnack inequality which ensures the dichotomy of the behavior. You can also have a look at this paper of Brezis and Merle on that type of result.