Let $\Omega \subset \mathbb{R}^n $ be open and bounded, $f\in L^{2}(\Omega)$ and let $a : \mathbb{R} \to \mathbb{R}$ be continuous such that $\alpha_{1} \le a(s) \le \alpha_2$ for every $s\in \mathbb{R}$, where $0<\alpha_1 <\alpha_2<\infty $. Prove that there exists a weak solution $u \in H_{0}^{1}(\Omega)$ to the boundary value problem $$\begin{cases} -\text{div}\left(a\left(u\right)\nabla u\right)=f & \text{in}\ \Omega\\ u=0 & \text{on}\ \partial\Omega \end{cases}$$
I know that there is a solution using Schauder's fixed point theorem , but I cannot fill all the details at the moment .