Sobolev Inequality with powers

Question

The following is one of the Sobolev Inequalities as stated in Gilbarg & Trudinger's text on elliptic PDEs:

$\displaystyle W_{0}^{1,p}(\Omega)\subset L^{\frac{np}{(n-p)}}(\Omega)$ for $p<n$.

Furthermore, there exists a constant $C=C(n,p)$ such that, for any $u \in W_{0}^{1,p}(\Omega)$,

(1). $||u||_{\frac{np}{(n-p)}}\leq C||Du||_{p}$ for $p<n$.

My question is, does Inequality (1) still hold if both sides are raised to an integer power, say $||u||_{\frac{np}{(n-p)}}^{p}\leq C||Du||_{p}^{p}$?

If so, why (sketch of a proof), and if not, why not?

Thank you in advance!!

Answer 1

Let be $t_1=||u||_{\frac{np}{(n-p)}}\le C||Du||_{p}=t_2$. As $t\mapsto t^p$ is increasing, $$||u||_{\frac{np}{(n-p)}}^p=t_1^p\le t_2^p=C^p||Du||_{p}^p.$$ And $p$ can be $\ne p$ (joke but true).