$\int_{\Omega} D^{m} (|u|^{p-2}u) D^{m} u' dx\leq \|D^{m} u\|_{2}^{p-1}. \| D^{m} u'\|_{2}$?

Question

I am very confused about the following inequality which is written during of a proof in a paper. I don't know how it is obtained: $$ \int_{\Omega} D^{m} (|u|^{p-2}u) D^{m} u' dx\leq \|D^{m} u\|_{2}^{p-1}. \| D^{m} u'\|_{2} $$ and in special case ($m=2$) $$ \int_{\Omega}\Delta (|u|^{p-2}u)\Delta u' dx\leq \|\Delta u\|_{2}^{p-1}. \| \Delta u'\|_{2}. $$ Note that $u\in H_{0}^{m}(\Omega)$. (Above inequality has been used in http://www.sciencedirect.com/science/article/pii/S0362546X14002831, the inequalities (4.25) and (4.26)). How it can be true ? Thanks a lot for your guidance.