Cubic gradient nonlinearity ($N(u)=|Du{|}^{2}u$)

On a fixed bounded smooth three-dimensional domain, the nonlinear mapsends $H^2$ into $L^2$. The Sobolev embedding theorem gives $H^2\hookrightarrow W^{1,4}\cap L^{\infty }$, and Holder inequality yieldsThe domain regularity is part of the embedding hypotheses.

For real $u,v$, write $w=u-v$. The difference identityand the same embeddings giveThus 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.