I often come across instances in texts where people calculate the weak derivative of $|u|^s$ for $s>1$ as $s|u|^{s-1} \operatorname{sign}(u) \partial_x u$ for some $u\in W^{1,s}(\Omega)$.
However, as far as I understand, to apply such a chain rule, one needs $|u|^s$ to be globally Lipschitz on $\mathbb{R}$, which is not the case.
I am relatively new to the field of Sobolev spaces and PDEs and am not yet familiar with common techniques.
Rather than spending too much time pondering this problem without guidance, I thought I would ask this question here.