Theorem 6.2 in Gilbarg-Trudinger (Clarification)

Question

In Theorem 6.2 of G-T's book we have in the hypothesis of the theorem that $\Omega$ is an open subset of $\mathbb{R}^n$ and that $u\in C^{2, \alpha}(\Omega)$ (here $\alpha\in (0,1)$) is a bounded solution of the equation:

\begin{equation} Lu=a^{i,j}D_{i,j}u+b^{i}D_{i}u+cu=f \end{equation}

with $f\in C^{\alpha}(\Omega)$ and certain bounds on the coefficients.

I am not sure why they needed to specify that $u$ is bounded on $\Omega$ and that it is in $C^{2, \alpha}(\Omega)$.

Is it possible to have $u$ not bounded but still in $C^{2, \alpha}(\Omega)$?

Answer 1

$\sqrt{x}$ is Hölder continuous on $(0,\infty)$