Document Type
Article
Publication Date
4-23-2025
Published In
Electronic Journal Of Differential Equations
Abstract
We establish the existence of maximal and minimal weak solutions between ordered pairs of weak sub- and super-solutions for a coupled system of elliptic equations with quasimonotone nonlinearities on the boundary. We also formulate a finite difference method to approximate the solutions and establish the existence of maximal and minimal approximations between ordered pairs of discrete sub- and super-solutions. Monotone iterations are formulated for constructing the maximal and minimal solutions when the nonlinearity is monotone. Numerical simulations are used to explore existence, nonexistence, uniqueness and non-uniqueness properties of positive solutions. When the nonlinearities do not satisfy the monotonicity condition, we prove the existence of weak maximal and minimal solutions using Zorn’s lemma and a version of Kato’s inequality up to the boundary.
Keywords
Weak solutions, quasimonotone, subsolution, supersolution, Zorn's lemma, finite difference method, Kato's inequality
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Recommended Citation
S. Bandyopadhyay, T. Lewis, and Nsoki Mavinga.
(2025).
"Existence Of Maximal And Minimal Weak Solutions And Dinite Difference Approximations For Elliptic Systems With Nonlinear Boundary Conditions".
Electronic Journal Of Differential Equations.
Volume 2025,
Issue 1.
DOI: 10.58997/ejde.2025.43
https://works.swarthmore.edu/fac-math-stat/339
Comments
This work is freely available under a Creative Commons license.