Shalmali Bandyopadhyay, Thomas Lewis, Nsoki Mavinga
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.
DOI: 10.58997/ejde.2025.43
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Shalmali Bandyopadhyay Department of Mathematics and Statistics University of Tennessee at Martin Martin, TN 38238, USA email: sbandyo5@utm.edu |
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Thomas Lewis Department of Mathematics and Statistics The University of North Carolina at Greensboro Greensboro, NC 27412, USA email: tllewis3@uncg.edu |
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Nsoki Mavinga Department of Mathematics and Statistics Swarthmore College Swarthmore, PA 19081, USA email: nmaving1@swarthmore.edu |
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