we denote by $\overline{u}$ a positif fuction "radially symmetric about the origin" that realize $$\inf\{\int_{\mathbb{R}^N} (|\nabla u|^2+\lambda u^2) dx, u\in H^1_0({\mathbb{R}^N}), \int_{\mathbb{R}^N} |u|^p=1\},$$ Then consider the function $w_{\rho}\in H^1_0(B_{\rho}(0))$ defined by $$w_{\rho}(x)=c_{\rho}\xi_{\rho}(x) \overline{u}(x)$$ where $\xi_{\rho}(x):\mathbb{R}^N\rightarrow [0,1]$ is a $C^{\infty}-$function defined by $$\xi_{\rho}(x)=\tilde{\xi}(\frac{|x|}{\rho})$$ with $\tilde{\xi}: \mathbb{R}^+\cup\{0\}\rightarrow [0,1]$ being a decreasing $C^{\infty}-$function such that $$\tilde{\xi}(t)=\begin{cases} 1,~t\leq\frac12,\\0, t\geq 1\end{cases}$$ $c_{\rho}=|\xi_{\rho} \overline{u}|_{L^p(B_{\rho})}$
My question is How to prove that $\int_{\mathbb{R}^N}|(\overline{u}-w_{\rho})(x)|^p dx\leq\int_{\mathbb{R}^N\setminus {B_{\rho/2}(0)}}|\overline{u(x)}|^p dx$
I try to prove it but i only proved that
$c_{\rho}=|\overline{u}|_{L^p(B_{\rho/2})}$ and
$\int_{\mathbb{R}^N} |(\overline{u}-w_{\rho})(x)|^p=\int_{\mathbb{R}^N\setminus B_{\rho}}|\overline{u}|^p dx+\int_{B_{\frac{\rho}{2}}} |1-|\overline{u}|_{L^p(B_{\rho/2})}|^p |\overline{u}|^p dx$
Thank you