Consider $\Omega \subset R^n$ $(n \geq 3) $ a bounded and smooth domain. Let $f: \Omega \times R \rightarrow R$ a Caratheodory function. Supose that exists $0 \leq \alpha < (N+2)/(N-2)$ and $c,d > 0$ such that
$$ |f(x,s)| \leq c|s|^{\alpha} + d$$
$$f(x,s) = o(|s|)\text{as } s \rightarrow 0 , \text{uniformly in x}$$
there exist $\theta >0 $ and $r >0$ such that
$$ 0 < \theta F(x,s) \leq s f(x,s)$$
for $|s| \geq r,$ uniformly in $x.$ (where $F(x,s) = \int_{0}^{s} f(x,t) \ dt$).
Under these hipothesis Ambrosetti and Rabinowitz prove (in this article http://www.sciencedirect.com/science/article/pii/0022123673900517#) that exists a nontrivial weak solution $u \in H^{1}_{0}(\Omega)$ for the problem
$$ -\Delta u = f(x,u), \ x\in \Omega$$ $$ u = 0, \ \ x \in \partial \Omega$$
my book (link for the book : http://books.google.com.br/books/about/An_Invitation_to_Variational_Methods_in.html?id=dzSPXN4q1RYC&redir_esc=y) says(and dont prove) that if $f: \Omega \times R \rightarrow R$ is a locally lipschitz function then by a bootstrap argument can be concluded that the weak solution above is a classical solution.
the article is that i said above is very hard to read. Someone know a good book (every book of pde that i know dont talk about of the regularity of weak solution for this problem)with this boot strap argument (for the problem that i said above)?
thanks in advance