If $u\in H_{0}^{m}(\Omega)$ then $\|D^{m} f(u)\|_{2}<+\infty $?

Question

Suppose that $u\in H_{0}^{m}(\Omega)$ and $f(u)=|u|^{p-2}u$, $(p\geq2)$. Is it possible to say $\|D^{m} f(u)\|_{2}<+\infty $, $(m\geq1)$? My statement is correct for $m=1$ and $p=2$. It seems that it is also true for $m=1$ and $p>2$. But what happens when $m>1$ and $p>2$? ($\Omega$ is considered to be an open bounded domain in $R^{N}, N\geq 1$).