It is stated in Evans's Partial Differential Equations (2nd edition) that
I don't understand the "In particular" part. The case $1\leq p<n$ is obvious since in this case $p\in [1,p^*]$. How is the case $n\leq p\leq\infty$ implied by the former statement?
If $p\geq n$ is finite, then use the fact that $W^{1,p}_0(U) \subset W^{1,p'}_0(U)$ for all $p' \leq p$ (as $U$ is bounded). So fix a value of $p'<n$ so that $(p')^* \geq p$. This is possible since $(p')^* \to \infty$ as $p'\to n$. Then use $q=p \leq (p')^*$ in the theorem to get
$$\|u\|_{L^p}\leq C\|Du\|_{L^{p'}} \leq C\|Du\|_{L^p}.$$
The constant $C$ depends on $p'$ and $p$, but really just on $p$ since we can choose $p'$ as a function of $p$.
This gets you the $n \leq p < \infty$ cases. The $p=\infty$ case is easy to prove by hand.