| Abstract: |
| We analyze weak solutions for a coupled system of elliptic equations with quasimonotone nonlinearity on the boundary.
We also formulate a finite difference method to approximate the PDE solutions.
We establish the existence of maximal and minimal weak solutions in between ordered pairs of weak sub and supersolutions as well as the existence of maximal and minimal finite difference approximations in between ordered pairs of discrete sub and supersolutions.
The analysis employs monotone iteration methods to construct the maximal and minimal solutions when the nonlinearity is monotone. We explore existence, nonexistence, uniqueness and nonuniqueness properties of positive solutions by analyzing particular examples with numerical simulations. 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 for systems up to the boundary. |
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