Suppose I have a function $u\in L^p(\Omega)$, with $1\leq p <\infty$. Clearly, for $r\geq 1$,
$$ \|u^r\|_p = \|u\|_{rp}^r. $$
Now, for $r>1$, I would like to be able to write $\|u^r\|_p\leq C \|u\|_s$, for some $s\geq 1$, $C>0$ (independent on $u$). My intuition says that this is not possible, but I was wondering if there were any additional requirement on $u$, and/or $p$ and/or $r$ that would make it possible. We can assume $\Omega$ bounded and smooth, and, if it helps, $u\in H^1(\Omega)$.
Edit: to give a concrete example, I'd like to find (if any) conditions on $u$ and $s$ that make the following valid:
$$ \|u\|_8^2 = \|u^2\|_4\leq C(s)\|u\|_s. $$
Note: if $|u|<1$ pointwise, then this is clearly true, since $|u|^p<|u|$. However, I do not have such information. But, if it helps, I have the following inequality
$$ \|u\|_4^4 \leq A \|u\|_8^2. $$
What if I also have $u\in L^\infty(\Omega)$? Would that help?