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.



A proof for the existence of smooth solution of PDE in form $\Delta u=f(x, u)$ in Michael E. Taylor's book Partial Differential Equations III
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.)


