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Let $f:H^s\to H^s$ at least continuous and not necesarily linear. Is there some kind of criterion or condition over $f$ that lets to ensure that $f({H^{s+k}})\subseteq H^{s+k}$?

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    $\begingroup$ There are whole books on this. Give a try at Runst and Sickel, "Sobolev spaces of fractional order etc" $\endgroup$
    – Piero D'Ancona
    Commented Sep 10, 2010 at 18:16

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Let $f:\mathbb C\rightarrow\mathbb C$ be a $C^\infty$ function such that $f(0)=0$. Let $1\le n\in \mathbb N$ and $s>n/2$. Then $$\forall u\in H^s(\mathbb R^n),\quad f(u)\in H^s(\mathbb R^n). $$ In particular, for this range of values, $H^s(\mathbb R^n)$ is an algebra included in $L^\infty(\mathbb R^n)$.

Bazin.

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