$$ \bar{U}_{x, \mu}(y)=\frac{C_0 \mu^{\frac{N-2}{2}}}{\left(1+\mu^2|y-x|^2\right)^{\frac{N-2}{2}}}, \quad x \in \mathbb{R}^N, \mu \in \mathbb{R}_{+}, $$ and Let $\delta>0$ be small enough such that $B_{2 \delta}(0) \subset \Omega$. Let $\xi \in$ $C_0^{\infty}\left(B_{2 \delta}(0)\right)$, which satisfies $\xi=1$ in $B_\delta(0), 0 \leq \xi \leq 1$ and $|\nabla \xi| \leq \frac{C}{\delta}$. $$\int_{\mathbb{R}^N}\left|\nabla \bar{U}_{0, \mu}\right|^2 = S$$ I want to prove that for sufficiently large $\mu>0$, the equation
$$ \begin{aligned} \int_{\Omega}\left|\nabla\left(\xi \bar{U}_{0, \mu}\right)\right|^2 & =\int_{\Omega}\left(\xi^2\left|\nabla \bar{U}_{0, \mu}\right|^2+2 \xi \bar{U}_{0, \mu} \nabla \xi \nabla \bar{U}_{0, \mu}+|\nabla \xi|^2 \bar{U}_{0, \mu}^2\right) \\ &{\color{red} =\int_{\mathbb{R}^N}\left|\nabla \bar{U}_{0, \mu}\right|^2+O\left(\int_{\mathbb{R}^N \backslash B_\delta(0)}\left|\nabla \bar{U}_{0, \mu}\right|^2+\int_{\Omega \backslash B_\delta(0)} \bar{U}_{0, \mu}^2\right)} \\ & =S+O\left(\frac{1}{\mu^{N-2}}\right), \end{aligned} $$