Nonzero radial solutions for elliptic systems with coupled functional BCs in exterior domains

TL;DR

This paper establishes new fixed point index results for the existence and multiplicity of nonzero radial solutions to elliptic systems with coupled functional boundary conditions in exterior domains, answering an open question in the field.

Contribution

It introduces novel fixed point index techniques to analyze elliptic systems with nonlocal boundary conditions, extending previous results and providing explicit examples.

Findings

Proved existence of nonzero radial solutions under certain conditions

Established non-existence results for specific boundary conditions

Provided examples illustrating the applicability of the theory

Abstract

We prove new results on the existence, non-existence, localization and multiplicity of nontrivial radial solutions of a system of elliptic boundary value problems on exterior domains subject to nonlocal, nonlinear, functional boundary conditions. Our approach relies on fixed point index theory. As a by-product of our theory we provide an answer to an open question posed by do {\'O}, Lorca, S\'{a}nchez and Ubilla. We include some examples with explicit nonlinearities in order to illustrate our theory.