there is a generalization of pohozaev identity. That is
$$\int_N \mathscr{L}_X R \, dv=\frac{2n}{n-2}\int_{\partial N}(\operatorname{Ric} - n^{-1}Rg)(X,\nu) \, d\sigma.$$ Here $\operatorname{Ric}$ denotes the Ricci tensor of $(N,g)$, $\mathscr{L}_X$ denotes the Lie derivative, $\nu$ denotes the unit outward derivative.
Could you tell me if I take $N$ as a bounded domain in euclidean space, how can i get the pohozaev identity? Thanks a lot.