The Sobolev embedding  $H_{0, rad}^{1}(\Omega)  \hookrightarrow L^{p}(\Omega)$ 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 $N \geq 2$.