I have a question that it seems simple but I can not solve it.

Let $\Omega$ be the unit ball centered at zero in $\mathbb{R}^N$, $N>4$. Assume that $C_{0,rad}^{\infty}(\Omega)$ is the space of all radial symmetric smooth functions with compact support.

What is the completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the following norm

$$\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $$

Is it $W_0^{2,2}(\Omega)$ or $W_{0,rad}^{2,2}(\Omega)$.

I think that the answer is $W_0^{2,2}(\Omega)$ but I have no idea to prove it.

I will be thanked to any useful hint or answer.