What I'm trying to prove is the following:
Let $B$ be a ball with center in the origin and $u \in H_0^1(B) \cap C^0(\overline B)$ be a solution of $$ -\Delta u = f(|x|, u), u > 0 \quad \text{ in } B, \quad u = 0 \quad \text{ on } \partial B $$ where $f$ is continuous and locally Lipschitz continuous in the second variable, uniformly with respect to $x$. Then $u \in C^1(B) \cap C^0(\overline B)$.
This should be a standard elliptic regularity argument, but I must be missing something. My idea would be to use the Agmon-Douglis-Nirenberg Theorem to obtain, in a first step, that $u \in H^2(B)$ (since $f(u)$, being continuous up to the boundary, is $L^2$), and then to iterate this until $k$ is so big that we have $u \in C^1(B)$ by the Sobolev embedding. However, I'm not sure if I can do this, since I don't know if the composition $f(u)$ has Sobolev regularity.
My question is, as one can read in the title:
If $f$ is continuous and $u \in C \cap H^k$, then is it true that $f(u) \in H^{k}$?
Thanks in advance.
EDIT
Is this just a bootstrap argument?