Let $\Omega \subset R^n$ be an open and bounded set with smooth boundary and $f \in C^2(\Omega)\cap C(\overline{\Omega})$. Let $ a \geq 0$ be a constant.
Consider the following problem:
$$ \left\{ \begin{array}{ccccccc} -\Delta u +au = f , \ in \ \Omega \\ \ u=0 , \ in \ \partial\Omega \\ \end{array} \right. $$
Can someone give me a reference for the proof of the existence of classical solution for this problem?
In this text http://www3.nd.edu/~qhan/PDE.pdf, it discusses maximum principles about the problem that I said above. but I haven't found a book with the proof of the existence.
$f\in C(\overline{\Omega})$ implies that $f\in L^2(\Omega)$ which implies that there exist unique weak solution $u\in H_0^1(\Omega)\cap H^2(\Omega)$.
Moreover, $f\in C(\overline{\Omega})$ implies that $f\in L^p(\Omega)$ for all $p$, which implies that $u\in W_0^{1,p}(\Omega)\cap W^{2,p}(\Omega)$ for all $p$ (see Theorem 9.32 of Brezis book).
We conclude from the last paragraph that $u\in C^{1,\alpha}(\overline{\Omega})$ for some $\alpha\in (0,1)$. To conclude note that $f\in C^2(\Omega)$ implies interior regularity (see section 9.6 of Brezis book) and therefore $u\in C^2(\Omega)$.