Suppose that $f:\Bbb R^2\to\Bbb R$ is a *continuous non-linearity* and consider the following semi-linear elliptic PDE given by:
$$-\Delta u=f(x,u),\;\;x\in\Omega\subset\Bbb R^n,\tag{1}\label{1}$$
To avoid the mention of critical Sobolev exponents and to narrow down the scope of the answer, let us assume that $n=2$ and $\Omega$ is bounded. Assume that $u\in H^1_0(\Omega)$ is a weak solution of \eqref{1}.

**Q1.***Under what assumptions on $f$ can we show that* (i) $u\in H^2(\Omega)$, (ii) *recover a classical solution of \eqref{1}*?

**Remark 1**. There are numerous conditions on $f$ that guarantee the existence of $u$. For $n=2$, see this question. For $n\geq 3$, see the the book Semi-linear elliptic equations for beginners by Badiale and Serra.

**Remark 2.** In Evan's Partial Differential Equtions book, Chpater 6 exercise 4, if $f(x,u)=g(x)+h(u)$, where $g\in L^2(\mathbb{R}^n)$, $h(0)=0$, $h$ is smooth, and $h'\geq 0$, then a solution $u\in H^1(\Bbb R^n)$ to \eqref{1} is in fact in $H^2(\Bbb R^n)$.

**Remark 3.** From this paper, where $\Omega$ is the unit ball in $\Bbb R^n$ with $n\geq 2$, and $f$ is Lipschitz with $\partial_uf\geq -M$ for some $M\geq 0$, I quote the following sentence in the introduction:

We remark that solutions to \eqref{1} are already in $W^{2,p}(\Omega)\cap C^{1,\alpha}(\Omega)$ for $p<\infty$ and $\alpha<1$ (since $f$ is bounded).

I think I can workout a proof as to why $u\in H^2(\Omega)$, but not $u\in C^{1,\alpha}(\Omega)$ for $\alpha<1$, which brings us to the second question.

**Question 2.***How does one obtain $u\in C^{1,\alpha}(\Omega)$*?

**Remark 4.** In the answer, assume *any* needed regularity on $\Omega$ to obtain regularity of $u$.