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  • Existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditions
Title: Existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditions
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.  For more information see https://ejde.math.txstate.edu/Volumes/2025/43/abstr.html  more » « less
Award ID(s):
2111059
PAR ID:
10621147
Author(s) / Creator(s):
; ;
Publisher / Repository:
Department of Mathematics, Texas State University
Date Published:
Journal Name:
Electronic Journal of Differential Equations
Volume:
2025
Issue:
01-??
ISSN:
1072-6691
Page Range / eLocation ID:
43; 1-21
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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