Integration: Step in paper unclear I've seen in a paper the following step:
$$2\operatorname{Re}\int_{\mathbb{R}^n} r \partial_r \bar u \Delta u \, dx=(n-2)\int_{\mathbb R^n} |\nabla u|^2$$
This is not clear to me as I calculated:
\begin{align}2\operatorname{Re}\int_{\mathbb{R}^n} r \partial_r \bar u \Delta u \, dx = 2\operatorname{Re}\int_{\mathbb{R}^n} x \nabla \bar u \Delta u \, dx=\int_{\mathbb{R}^n} x (\nabla \bar u \Delta u + \nabla u \Delta \bar u)\, dx = \int_{\mathbb{R}^n} x\nabla|\nabla u|^2\, dx \\=-n \int_{\mathbb{R}^n}|\nabla u|^2\end{align}
Reference: On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, R. T. Glassey, page 1795 left side integral II.
Edit: I think one doesn't need it here but $u$ solves $iu_t = -\Delta u - u |u|^{p-1}$.
 A: A careful use of integration by parts will derive the formula. 
\begin{align}
2\operatorname{Re}\int_{\Bbb R^n} r\partial_r \bar{u} \Delta u\, dx &= 2\operatorname{Re}\int_{\Bbb R^n} (x\cdot \nabla \bar{u})\Delta u\, dx \\
&= \int_{\Bbb R^n} [(x\cdot \nabla \bar{u})\Delta u + (x\cdot \nabla u)\Delta \bar{u}]\, dx\\
&= \int_{\Bbb R^n} (x_i \bar{u}_{x_i}u_{x_ix_i} + x_i u_{x_i}\bar{u}_{x_jx_j})\, dx\\
&= \int_{\Bbb R^n} [x_i(\bar{u}_{x_i}u_{x_j})_{x_j} - x_i\bar{u}_{x_jx_i}u_{x_i} + x_i(u_{x_i}\bar{u}_{x_j})_{x_j} - x_iu_{x_jx_i}\bar{u}_{x_j}]\, dx\\
&= \int_{\Bbb R^n} [x_i(\bar{u}_{x_i}u_{x_j} + u_{x_i}\bar{u}_{x_j})_{x_j}\, dx - \int_{\Bbb R^n} (x_i\bar{u}_{x_jx_i}u_{x_j} + x_iu_{x_jx_i}\bar{u}_{x_i})\, dx\\
&= -\int_{\Bbb R^n} \delta_{ij}(\bar{u}_{x_i}u_{x_j} + u_{x_i}\bar{u}_{x_j})\, dx - \int_{\Bbb R^n} x_i (\bar{u}_{x_j}u_{x_j})_{x_i}\, dx\\
&= -\int_{\Bbb R^n} 2|\nabla u|^2\, dx - \left(-n\int_{\Bbb R^n} |\nabla u|^2\, dx\right)\\
&= (n - 2)\int_{\Bbb R^n} |\nabla u|^2\, dx.
\end{align}
