I'm trying to understand how it integrates by parts the following integral. This belong to the academic paper Stable solutions of $-\Delta u=f(u)$ in $\mathbb{R}^N$ with DOI 10.4171/JEMS/217. Specifically, page 863.
Proof of Theorem 1.1. For $R>0$, let $B_R$ denote the ball of radius $R$ centred at the origin. We begin by proving that there exists a constant $C>0$ independent of $R>0$ such that $$ \int_{B_R}|\nabla u|^2 d x \leq C R^{N-2} . $$
Let $M \geq\|u\|_{\infty}, \varphi \in C_c^2\left(\mathbb{R}^N\right)$ and multiply (1) by $(u-M) \varphi$ : $$ \int_{\mathbb{R}^N}-\Delta u(u-M) \varphi d x=\int_{\mathbb{R}^N} f(u)(u-M) \varphi d x . $$
Integrating by parts and recalling that $f \geq 0$, it follows that $$ \int_{\mathbb{R}^N}|\nabla u|^2 \varphi d x+\int_{\mathbb{R}^N}(u-M) \nabla u \nabla \varphi d x=\int_{\mathbb{R}^N} f(u)(u-M) \varphi d x \leq 0, $$
whence $$ \begin{aligned} \int_{\mathbb{R}^N}|\nabla u|^2 \varphi d x & \leq-\int_{\mathbb{R}^N} \frac{1}{2} \nabla(u-M)^2 \nabla \varphi d x=\int_{\mathbb{R}^N} \frac{(u-M)^2}{2} \Delta \varphi d x \\ & \leq 2 M^2 \int_{\mathbb{R}^N}|\Delta \varphi| d x . \end{aligned} $$
