Let $f:\mathbb{R} \to \mathbb{R}$ be a smooth function with bounded derivative. Define the Nemytskii map $F:H^1(\Omega) \to H^1(\Omega)$ by $F(u)(x) := f(u(x))$. Here $\Omega$ is a bounded smooth domain.

There exists work where we can deduce continuity and differentiability of $F$ under some assumptions on $f$.

What conditions on $f$ will ensure that $F$ satisfies the following compactness criterion: if $u_n \rightharpoonup u$ in $H^1(\Omega)$ (weakly), then there is a subsequence of the $u_n$ such that $F(u_{n_j}) \to F(u)$ strongly in $H^1(\Omega)$?

If convergence is $H^1$ is too difficult, $L^2$ (in both domain and range of $F$) can be ok. Does anyone come across this before?