Iterative Methods for Globally Lipschitz Nonlinear Laplace Equations

We introduce an iterative method to prove the existence and uniqueness of the complex-valued nonlinear elliptic PDE of the form $-\Delta u + F(u) = f$ with Dirichlet or Neumann boundary conditions on a precompact domain $\Omega \subset \mathbb{R}^{n}$, where $F : \mathbb{C} \rightarrow \mathbb{C}$ is Lipschitz. The same method gives a solution to $- \Delta_{g} u + F(u) = f$ for these boundary conditions on a smooth, compact Riemannian manifold $(M, g)$ with $\mathcal{C}^{1}$ boundary, where $- \Delta_{g}$ is the Laplace-Beltrami operator. We also apply parametrix methods to discuss an integral version of these PDEs.