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 $$\displaystyle\int_{\Omega } |\nabla u|^2 dx\geq C_p \displaystyle(\int_{\Omega } |u|^{p+1} dx)^{2/p+1}; u\in H_{0, rad}^{1}(\Omega) .$$
Your first question seems hard to me. Using calculus of variations, we can probably find an ODE for $u$ (in terms of $r$) which maximizes the Sobolev constant. Maybe this will give you a closed form for the best constant, but it seems tricky.
However, your second question is true (at least when $a>0$). To see this, write $u = u(r ,\theta)$ where $\theta \in S^{N-1}$. Then we have the following estimates.
\begin{eqnarray} \int_\Omega |\nabla u|^2 &= & \int_{S^{N-1}} \int_a^b \left( \frac{\partial u}{\partial r} \right)^2 r^{N-1} dr ~d\theta \\ & = & Vol(S^{N-1}) \int_a^b \left( \frac{\partial u}{\partial r} \right)^2 r^{N-1} dr \\ &\geq & Vol(S^{N-1}) a^{N-1} \int_a^b \left( \frac{\partial u}{\partial r} \right)^2 dr \\ & \geq & Vol(S^{N-1}) a^{N-1} \frac{ \|u\|_{L^\infty}^2 }{b-a} \end{eqnarray}
From this, you can estimate all of the other $L^p$ norms as well.
