This part is from page 107 in Michael E. Taylor's book Partial Differential Equations III.
In this part, we want a proof for the existence of smooth solution of the PDE
$\Delta u=f(x, u)$ on $U$ with boudary condition $\left.u\right|_{\partial U}=g$ where $g$ is smooth
under the assumption that $\frac{\partial f}{\partial u} \geq 0$.
After making the temporary assumption that for $|u| \geq K, \partial_{u} f(x, u)=0$ we have that
there exists soomth solution of the PDE
$\Delta u=f(x, u)$ on $U$ with boudary condition $\left.u\right|_{\partial U}=g$ when $g$ is smooth.
we construct a sequence $f_{j}(x, u)=f(x, u)$, for $|u| \leq j，f_{j}(x, u)=f(x, u)$ and when $|u| \geq K_{j}, \partial_{u} f_{j}(x, u)=0$
so we have solutions $u_{j} \in C^{\infty}(\bar{U})$ to
$\Delta u_{j}=f_{j}\left(x, u_{j}\right),\left.\quad u_{j}\right|_{\partial U}=g$
And we have $\sup _{U}\left|u_{j}\right| \leq \sup _{U} 2|\Phi|$
where $f(x, 0)=\varphi(x) \in C^{\infty}(\bar{U})$, take $g \in C^{\infty}(\partial U)$, and let $\Phi \in C^{\infty}(\bar{U})$ be the solution to
$\Delta \Phi=\varphi$ on $U, \quad \Phi=g$ on $\partial U$
Then we can see that the sequence $\left(u_{j}\right)$ stabilizes for large $j$, then we finish the proof.
How does this proof finish the proof, by Arzela-Ascoli? But this does not meet the conditions of Arzela-Ascoli theorem, actually I don't even know how to use Arzela-Ascoli here.
Best Answer
It is much simpler than Arzela-Ascoli.
Since you have the uniform bound $$ \sup_U |u_j| \leq \sup_U 2|\Phi| $$ For simplicity I will assume your $K_j = j$. Take $J = \sup_U 2|\Phi|$, then for every $j,k \geq J$ you have that $$ f_j(x,u_j) = f_k(x,u_j) = f(x,u_j).$$ This tells you that for every $j,k \geq J$ you have $$ \Delta u_j = f_k(x,u_j). $$ By the uniqueness of the solution to the problem with the temporary truncation, you find therefore that $u_k = u_j$ for every $j,k\geq J$.
(This is what the author means by "stabilizes for large $j$"; that after some $J$ the sequence $\{u_j\}$ becomes the constant sequence.)