Simplifying an expression with $\Delta$

Question

Let $\sum_j\chi_j(x)^2=1$ be a partition of unity with respect to the variable $x\in \mathbb{R}^{2d}$. Please help me to simplify as possible

$$\sum_j\langle \chi_ju,-\Delta(\chi_ju)\rangle-\langle u,-\Delta u\rangle$$

for all $u\in\mathcal{C}_0^{\infty}(\mathbb{R}^{2d})$

Thanks in advance