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 $-\mathrm{\Delta }u+F(u)=f$ 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 $-{\mathrm{\Delta }}_{g}u+F(u)=f$ for these boundary conditions on a smooth, compact Riemannian manifold $(M,g)$ with ${\mathcal{C}}^1$ boundary, where $-{\mathrm{\Delta }}_g$ is the Laplace-Beltrami operator. We also apply parametrix methods to discuss an integral version of these PDEs.