= Cubic gradient nonlinearity
{title2=$N(u)=|Du|^2u$}
On a fixed bounded smooth three-dimensional <domain>, the nonlinear map
$$
N(u)=|\nabla u|^2u
$$
sends $H^2$ into $L^2$. The <Sobolev embedding theorem> gives $H^2\hookrightarrow W^{1,4}\cap L^\infty$, and <Holder inequality> yields
$$
\|N(u)\|_2\le\|\nabla u\|_4^2\|u\|_\infty\le C\|u\|_{H^2}^3.
$$
The domain regularity is part of the embedding hypotheses.
For real $u,v$, write $w=u-v$. The difference identity
$$
N(u)-N(v)=|\nabla u|^2w+v(\nabla u+\nabla v)\cdot\nabla w
$$
and the same embeddings give
$$
\|N(u)-N(v)\|_2\le C(\|u\|_{H^2}+\|v\|_{H^2})^2\|u-v\|_{H^2}.
$$
Thus its local <Lipschitz continuity> constant on a ball of radius $R$ is $O(R^2)$. Composing this small nonlinear bound with a bounded linear elliptic inverse gives a <contraction mapping> on a sufficiently small ball, which is the mechanism of <small-data existence for a nonlinear elliptic Dirichlet problem>.
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