Nonlinear elliptic boundary value problem

A nonlinear elliptic boundary value problem contains a nonlinear dependence on the unknown function or its derivatives while retaining an elliptic principal part. Small-data existence can often be proved by applying a fixed-point theorem to the inverse of the linear elliptic part.

In three dimensions the map $N(u)=|Du{|}^{2}u$ sends $H^2$ into $L^2$, because $H^2\hookrightarrow W^{1,4}\cap L^{\infty }$. On an $H^2$ ball of radius $R$ it is locally Lipschitz with constant $O(R^2)$.