Existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditions - Shalmali Bandyopadhyay; Thomas Lewis; Nsoki Mavinga (Electronic Journal of Differential Equations)
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.