I would like to prove the following result: Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function and $F(u)=\int_{0}^uf(s)ds$. Let $u\in C^2(\Omega)\cap C^1(\overline{\Omega})$ be such that $-\Delta(u)=f(u)$ in $\Omega$ and $u=0$ on $\partial \Omega$, where $\Omega \subset \mathbb{R}^n$ is a domain. Then, the following equality holds:
$$\frac{n-2}{2}\int_{\Omega}|\nabla u|^2 \, dx-n\int_{\Omega}F(u)\, dx+\frac{1}{2}\int _{\partial \Omega}\left|\frac{\partial u}{\partial \nu}\right| ^2 x\cdot\nu \, d\sigma=0,$$
where $\nu$ is the normal pointing outwards $\partial\Omega$.
The proof I'm trying to understand on Struwe, "Variational Methods" states the following result:
$0=(\Delta u+f(u))(x\cdot \nabla u)=\operatorname{div} (\nabla u(x\cdot\nabla u))-|\nabla u|^2-x\cdot\nabla(\left|\frac{\nabla u}{2}\right|^2)+x\cdot\nabla F(u)$.
My question is: how can I use the Divergence in this way, in particular how can I pass from $f$ to $F$ in this result?
Moreover, how can I deduce the equality I want to show from this result? (I know that Divergence Theorem must be involved).