It is well-known that if $k\in\mathbb{N}$
\begin{equation} \|u^k\|_{H^m(\mathbb{R}^d)}\leq \|u\|_{H^m(\mathbb{R}^d)}^k \ \ \text{for all} \ \ m> \frac{d}{2}+1 \end{equation} by the property of Banach algebra of Sobolev space. Then I have a question. We assume that $u\leq0$. If $s\geq 1$ \begin{equation} \|u^s\|_{H^m(\mathbb{R}^d)}\leq \|u\|_{H^m(\mathbb{R}^d)}^s \ \ \text{for all} \ \ m>\frac{d}{2}+1 \end{equation} Is the above statement also true? If this is true, show me the proof, please.
Assume that $\frac{d}{2} < m < s$. Then, for any $u \in H^m(\mathbb{R}^d)$, there holds: $$ \| |u|^s \|_{H^m} \lesssim \| u \|_{H^m} \| u \|_{L^\infty}^{s-1}. $$ This is a particular case of [Runst and Sickel - Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations. de Gruyter, 1996, Section 5.4.3, Theorem 1], which is itself a particular case of more general question: when is $G(u) \in H^m$ for some arbitrary function $G$? Here $G(\cdot) = |\cdot|^s$.
When $s < m$, there might be some difficulties. Take for example $d = 1$ and $u(t) := t \chi(t)$ where $\chi$ is some smooth compactly supported function, with $\chi \equiv 1$ near 0. When $s = \frac{3}{2}$, near $t = 0$, $|u(t)|^s = t^{3/2}$, so in particular $u''(t) = \frac{3}{4} t^{-1/2}$ for $0 < t \ll 1$, so $u \notin H^2$ locally.
In my counter example, $u$ changes sign. It is possible that, in your particular setting with the a priori assumption that $u \geq 0$, you can avoid this pathological case. I have not given it much thought.