Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 43, pp. 1-21.

Title: Existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditions

Authors: Shalmali Bandyopadhyay (Univ. of Tennessee, Martin, TN, USA)
         Thomas Lewis (The Univ. of North Carolina, Greensboro, NC, USA)
         Nsoki Mavinga (Swarthmore College, Swarthmore, PA, USA)         

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.

Submitted April 17, 2024. Published April 23, 2025.
Math Subject Classifications: 35J60, 35J67, 65N06, 65N22.
Key Words: Weak solutions; quasimonotone; subsolution; supersolution;
Zorn's lemma; finite difference method; Kato's inequality.