Classical Sobolev equality says, $n\geq 3$, we have \begin{equation} \left(\int_{\Omega}|u|^{2 n /(n-2)}\right)^{(n-2) /(2 n) } \leq C(n)\left(\int_{\Omega}|\nabla u|^{2}\right)^{1 / 2}\quad \text{for all}\,u\in D^{1,2}(\mathbb{R}^n). \end{equation} for some constant $C(n)$ depends on $n$.

My question is whether it is ture we have the Sobolev type equality \begin{equation} \left(\int_{\Omega}|u|^{2 n /(n-2)}\right)^{(n-2) /(2 n) } \leq C_{1}(n,\Omega)\left(\int_{\Omega}|\nabla u|^{2}\right)^{1 / 2}, \end{equation} with $\Omega=\mathbb{R}^n\backslash B_1(0)$, $u\in E=\{u \in L^{(n-2) /(2 n)}(\Omega),|\nabla u|\in L^{2}(\Omega)\}$, $C_{1}(n,\Omega)$ is a constant only depends on $n$ and $\Omega$.