Composition of a compacted supported function with a Holder continuous function is a $H^1_0$ function.

Question

Let $u \in C^\infty_0(B)$, where $B$ denotes que unit open ball in $\mathbb{R}^N$. I would like to know if $v(x) := u (|x|^{\beta})$, with $\beta \in (0,1)$ is in $H^1_0(B)$. Clearly $v$ is not differentiable in whole $B$, specifically ate $x = 0$. I know when $G : \mathbb{R} \rightarrow \mathbb{R}$ is a Lipschitz continuous function, then $G \circ u \in H^1_0(B)$. However, if we take $G: \mathbb{R}^N \rightarrow \mathbb{R}, G(x) = |x|^\beta$, I would like to conclude that $u \circ G \in H^1_0(B)$.