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?

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

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

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

Do you have some references?