Let $\Omega\subset \mathbb R^N$ be a bounded smooth domain, and $f:\Omega\times \mathbb R\to \mathbb R$ be a smooth function (let's say $C^2$).

Let $X=W^{1,p}(\Omega)$ p>1 the Sobolev space of functions with first derivatives in $L^p$.

Which conditions on f, and p guarantee that the Nemitskii map F(u)=f(x,u(x)) is of class $C^2$ from $X$ to $L^1(\Omega)$ (or $L^q(\Omega)$) ?

Do you have some references?