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Thm 4.9 in Gilbarg-Trudinger's book states that :

  1. if $B$ is a ball in $\mathbb{R}^n$ centred at $x_0$ and
  2. $f\in C^{\alpha}(B): \sup_{x\in B} (\text{dist}(x, \partial B))^{2-\beta}\vert f(x)\vert \leq N<\infty$ for some $\beta\in (0, 1)$.

Then there is a unique function $u\in C^{2}(B)\cap C(\overline{B})$, satisfying $\Delta u=f$ in $B$ and $u=0$ on $\partial B$. Furthermore $u$ satisfies an estimate: \begin{equation} \sup_{x\in B} \ \text{dist}(x, \partial B)^{-\beta}\vert u(x)\vert\leq CN, \end{equation} where $C=C(\beta)$.

I am following the proof of this theorem and I am happy with everything upto the point they start to prove the existence of $u$.

To show the existence of $u$ they first let \begin{equation} f_m=\left\{\begin{aligned} m &\quad\text{if } f\geq m\\ f &\quad\text{if }\vert f\vert\leq m\\ -m &\quad\text{if } f\leq -m\end{aligned}\right. \end{equation}

and let $\{B_k\}_{k=n}^{\infty}$, $n=\lceil \inf_{x\in B} \vert f\vert\rceil$, be a collection of concentric balls about $x_0$ such that: \begin{equation} B_k\subset\subset B_{k+1}, \quad x\in B_{k}\Rightarrow \vert f(x)\vert\leq k\quad\text{and} \quad \bigcup_k B_{k}=B. \end{equation}

With this setup they define the set $\{u_m\}$ to be the set of functions that satisfy: \begin{align*} \Delta u_m &=f_m\quad\text{in } B\\ u_m &=0\quad\text{on } \partial B. \end{align*}

By the estimate provided above (this was proved first in their argument) we know that $u_m$ is uniformly bounded and that for $m\geq k$ \begin{equation} \Delta u_m=f\quad\text{in } B_k. \end{equation}

Now they say that if we apply Corollary 4.7 successively to the balls $B_k$ then we get a subsequence that converges to a function $u\in C^{2}(B)$ that satisfies $\Delta u=f$ in $B$.

Corollary 4.7 says that if $u_n$ is a uniformly bounded sequence of solution to Poisson's equation, $\Delta u=f$ in a domain $\Omega$ with $f\in C^{\alpha}(\Omega)$ then for any $\Omega'\subset\subset \Omega$ there is a subsequence that converges uniformly to a solution.

I tried to apply this Corollary 4.7 in this way, however, I find myself taking subsequences infinitely often as follows:

If $B_k\subset\subset B_{k+1}$ then: \begin{equation} \Delta u_m=f\quad\text{in } B_{k+1}\ \forall m\geq k+1\Rightarrow \exists\ u_{m_j}\rightrightarrows u: \Delta u=f \ \text{in } B_{k}. \end{equation}

So certainly this subsequence converges to a $u\in C^{2}(B_k)$ that satisfies Poisson's equation in $B_{k}$ and if I repeat the process for $B_{k+1}\subset\subset B_{k+2}$ then I take a subsequence of $u_{m_j}$ to get the result for $B_{k+1}$ and so on.

I don't understand how this process will stop such that I find one subsequence that works throughout $B$ as claimed by [G-T]. If we are to use Corollary 4.7 in this way then shouldn't we find ourselves stopping this procedure once $B$ becomes compactly contained in some bigger set in which Poisson's equation is satisfied by $u_m$? This is clearly not the case.

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We have that for $n\leq m$, $\Delta u_m=f$ in $B_n$. Hence, you can extract a subsequence $u_{1,j}$ of $u_m$, such that $u_{1,j}$ converges to some $u_1$ in $B_n$.

Now, you hve that for big $j$, $\Delta u_{1,j}=f$ in $B_{n+1}$. Hence, you can extract a subsequence $u_{2,j}$ of $u_{1,j}$, such that $u_{2,j}$ converges to some $u_2$ in $B_{n+1}$.

...

Now, you hve that for big $j$, $\Delta u_{k,j}=f$ in $B_{n+k}$. Hence, you can extract a subsequence $u_{k+1,j}$ of $u_{k,j}$, such that $u_{k+1,j}$ converges to some $u_{k+1}$ in $B_{n+k}$.

Now try to use a "Cantor Diagonal" argument.

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