Nonzero positive solutions of elliptic systems with gradient dependence and functional BCs

TL;DR

This paper investigates the existence and non-existence of non-negative solutions for elliptic systems with gradient-dependent nonlinearities and functional boundary conditions, using topological methods.

Contribution

It introduces new topological techniques to establish conditions for solutions and non-solutions in elliptic systems with gradient dependence and functional boundary conditions.

Findings

Proves existence of non-negative solutions under certain conditions

Establishes non-existence results for specific cases

Provides examples illustrating theoretical results

Abstract

We discuss, by topological methods, the solvability of systems of second-order elliptic differential equations subject to functional boundary conditions under the presence of gradient terms in the nonlinearities. We prove the existence of non-negative solutions and provide a non-existence result. We present some examples to illustrate the applicability of the existence and non-existence results.