Let $\psi \in C^{\infty}_{c}(\Omega)$ where $\Omega$ is a bounded smooth domain, and $\phi$ the solution to 
\begin{equation*}
-\Delta \phi =\psi, ~\phi|_{\partial \Omega}=0.
 \end{equation*}
My question is how to get the following estimate :
$$\|\phi\|_{C^1( \overline{\Omega})} \leq C \|\psi\|_{L^{\infty}(\Omega)}.$$
The usual elliptic gradient estimate using the Bernstein method shows
$$\|\phi\|_{C^1( \overline{\Omega})} \leq C \sup|\psi|+ C\sup|\phi|+C\sup |\nabla \psi|.$$
The problem comes from [Stable Solutions of Elliptic Partial Differential Equations].

Another question is how to prove:
* Let $u$ be a $L^1$ weak solution of $-\Delta u= f(u)$ with zero boundary condition, if $f \in C^{\alpha}$ for some $\alpha \in (0, 1)$ and $f(u)\in L^p$ for some $p>\frac{n}{2}$, then $u$ is $C^2$, hence a classic solution.

for the definition of $L^1$ weak solution, see
[definition][1].





  [Stable Solutions of Elliptic Partial Differential Equations]: https://i.sstatic.net/DqdAe.png


  [1]: https://i.sstatic.net/IQ5A2.png