A note on the Maximum Principle and the Iteration Method for elliptic equations

TL;DR

This paper presents an iteration method based on the maximum principle to prove the existence of solutions for nonlinear Poisson equations with Dirichlet boundary conditions, applicable to unbounded domains and systems.

Contribution

It introduces a novel iteration procedure leveraging the maximum principle for nonlinear elliptic equations, extending to unbounded domains and systems.

Findings

Existence of solutions for nonlinear Poisson equations established.

Method applicable to special unbounded domains.

Extension to systems demonstrated with examples.

Abstract

We use an iteration procedure propped up by a a classical form of the maximum principle to show the existence of solutions to a nonlinear Poisson equation with Dirichlet boundary conditions. These methods can be applied to the case of special unbounded domains, and can be adapted to show the existence of nontrivial solutions to systems, which we show via some examples.