Let $W^{1,p}(\Omega)$ be the usual Sobolev space, $f(x,\eta,\xi)$ a continuous function, $1<p<\infty$, $\Omega$ an open and bounded subset of $\mathbb R^n$, and $F: W^{1,p}(\Omega) \longrightarrow \mathbb{ \bar R}$ defined as $$F(u)=\int_\Omega f(x,u(x),\nabla u(x)) \, dx.$$ It is known that, if
$f$ is convex in $\xi$ and
$f(x,\eta,\xi) \geq \alpha(x)+\beta|\eta|^p+\langle \gamma(x), \xi \rangle$ for some $\alpha \in L^1, \,\beta \in \mathbb R$ and $\gamma \in L^{p'}$,
then $F$ is weak lower semicontinuous.
My question is: is this theorem true also for $f$ quasiconvex instead of convex with respect to $\xi$ only?