The Sobolev embedding $H_{0, rad}^{1}(\Omega) \hookrightarrow L^{p+1}(\Omega)$ is compact for all $p>1$ where $\Omega= \{x\in \mathbb R^N: 0<a<|x|<b\}.$

Is it possible to determine the best Sobolev constant of the embedding when $N \geq 2$.

Or is it possible to show that the $C_p>0$ is uniformly bounded below by a positive constant when $$\displaystyle\int_{\Omega } |\nabla u|^2 dx\geq C_p \displaystyle(\int_{\Omega } |u|^{p+1} dx)^{2/p+1} .$$