we are facing the following problem: Let $\Omega \subset \mathbb{R}^d$ open bounded. Consider the elliptic operator $L = -div(a\cdot \nabla u)$, where $a_{i,j}$, where $a_{i,j}\in C^2(\Omega)$ are symmetric and uniformly elliptic.
Assume that $u\in H^1$ is a weak solution to $Lu = f(u)$ in $\Omega$.
Assume further that $a_{i,j}\in C^3(\Omega)$ and that $f\in C^2(\mathbb{R})$ with $||f‘‘||_{L^\infty}, ||f‘||_{L^\infty} < \infty$. Show that $u\in H^4_{loc}$, provided that the space dimension $d$ is not too large.
In the lecture we had the following (higher regularity) theorem:
Let $m \in \mathbb{N}$, $L$ as above, $f\in H^m, u\in H^1$ weak solution of $Lu = f$. Then $u\in H^{m+2}_{loc}$.
As a hint we got that we should use sobolev embeddings.
Unfortunately, we have no clue what to do. May someone help please?