An existence result in annular regions times conical shells and its application to nonlinear Poisson systems

TL;DR

This paper establishes a new topological existence theorem for nonlinear operator systems in normed spaces, demonstrating its applicability to nonlinear Poisson equations and reaction-diffusion systems, with explicit examples and numerical validation.

Contribution

It introduces a novel existence result using topological methods for systems in annular and conical regions, applicable to nonlinear Poisson and reaction-diffusion equations.

Findings

Existence and localization of solutions with all components nontrivial

Explicit example with numerical approximation confirming theoretical results

Application to reaction-diffusion Lotka-Volterra systems

Abstract

We provide a new existence result for abstract nonlinear operator systems in normed spaces, by means of topological methods. The solution is located within the product of annular regions and conical shells. The theoretical result possesses a wide range of applicability, which, for concreteness, we illustrate in the context of systems of nonlinear Poisson equations subject to homogeneous Dirichlet boundary conditions. For the latter problem we obtain existence and localization of solutions having all components nontrivial. This is also illustrated with an explicit example in which we also furnish a numerically approximated solution, consistent with the theoretical results. We conclude with an application of our results to a reaction--diffusion Lotka--Volterra system with source terms for competing species.