If $u\in \mathcal C_c^\infty (\mathbb R^d)$ does $|u|^a\in W^{1,p}$ if $a>0$ for $p\geq 1$?

Question

If $u\in \mathcal C_c^\infty (\mathbb R^d)$ (i.e. compacted supported) does $|u|^a\in W^{1,p}(\mathbb R^d)$ if $a>0$ and $p\geq 1$ ? I don't need a proof, just a confirmation because I used it in my bachelor thesis, but I'm not totally sure that really correct. To me it's yes, but I wanted to be sure.

Answer 1

Unfortunately not.

If you are looking for counterexamples it is best to start with the simplest cases. Take $n=1$ and let $u(x) = x$ in some neighborhood of $0$. Then $|u(x)|^a = |x|^a$ in some neighborhood of $0$, so that $|u'(x)| \sim |x|^{a-1}$ near the origin. If it happens that $p(a-1) \le -1$ you will have $|u'| \notin L^p(\mathbb R)$.