You have the compact embedding of radial Sobolev functions in $X_0(A)$ to $L^p(A)$ for all $1\leq p<\infty$ if and only if $s\geq 1/2$.

The proof goes as follows.

A radial function $F(x)$ on the annulus $A=\{ a<|x|<b\}$ is a function of the form $F(x)=f(|x|)$ for some $f$ defined on the interval $(a,b)$. Now $F\in L^p$ iff $f\in L^p(a,b)$ and $F_k\to F$ in $L^p$ iff $f_k\to f$ in $L^p(a,b)$ by Fubini's theorem. Also  $F\in X_0(A)$ iff $f\in H^s_0(a,b)$,
Therefore compact embedding of radial functions in $X_0(A)$ to all $L^p$, $1\leq p<\infty$ is equivalent to compact embedding of $H^s_0(a,b)$ to $L^p(a,b)$ for all $1\leq p<\infty$. This however, is true when $s\geq 1/2$. Just check the textbooks that discuss compactness of embedding of $H^s$. I do not have a right reference on top of my head.