Solvability of Quasilinear Elliptic Equations with Nonlinear Boundary Conditions

On an $n$-dimensional domain $\Omega$, we consider the boundary value problem \[ (\ast )\quad Qu = 0\;{\text {in}}\Omega {\text {,}}\quad Nu = 0\;{\text {on}}\;\partial \Omega \] where $Q$ is a quasilinear elliptic second-order differential operator and $N$ is a nonlinear first order differential operator satisfying an Agmon-Douglis-Nirenberg consistency condition. If the coefficients of $Q$ and $N$ satisfy additional hypotheses (such as sufficient smoothness), Fiorenza was able to reduce the solvability of $(\ast )$ to the establishment of a priori bounds for solutions of a related family of boundary value problems. We simplify Fiorenza’s argument, obtaining the reduction under more general hypotheses and requiring a priori bounds only for solutions of $Qu = f$, $Nu = g$ where $f$ and $g$ range over suitable function spaces. As an example, classical solutions of the capillary problem are shown to exist without using the method of elliptic regularization.