Sobolev inequality with holes

Question

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 $\Omega$.

Answer 1

It looks like you are looking for the extreme case of a Gagliardo-Nirenberg interpolation inequality, in the case of an exterior domain (that is, an unbounded domain with compact boundary). Such inequalities are derived for example in the following paper:

Francesca Crispo and Paolo Maremonti. "An interpolation inequality in exterior domains." Rendiconti del Seminario Matematico della Università di Padova 112 (2004): 11-39.

This paper focuses on the case where no vanishing condition is imposed on $\partial\Omega$.

The case you mention follows from Theorem 2.1 with $m=1$, $p=2$, $k=0$, $r = \frac{2n}{n-2}$. Since $n\geq 3$, beware that you are in the "particular case" ($k = 0$ and $m p < n$) which requires the additionnal assumption that $w$ tends to 0 at infinity, or that $w \in L^{q'}$ for some $q'<\infty$ (which is your case with $q' = 2$).