Classical Sobolev inequality says, $n\geq 3$, we have \begin{equation} \left(\int_{\mathbb{R}^n}|u|^{2 n /(n-2)}\right)^{(n-2) /(2 n) } \leq C(n)\left(\int_{\mathbb{R}^n}|\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 $n$. Here the space $D^{1,2}(\mathbb{R}^n)=\{u\in L^{2}(\mathbb{R}^n):|\nabla u|\in L^{2 n /(n-2)}(\mathbb{R}^n)\}$.

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

Louis Nirenberg. "On elliptic partial differential equations." Annali della Scuola Normale Superiore di Pisa-Scienze Fisiche e Matematiche 13, no. 2 (1959): 115-162.$\endgroup$Alberto Fiorenza, Maria Rosaria Formica, Tomáš G. Roskovec, and Filip Soudský. "Detailed proof of classical Gagliardo–Nirenberg interpolation inequality with historical remarks" Zeitschrift für Analysis und ihre Anwendungen 40, no. 2 (2021): 217-236which is very helpful. $\endgroup$1more comment