Existence results for quasimonotone semilinear coupled elliptic systems via sub-supersolution method

TL;DR

This paper proves the existence of solutions for coupled elliptic PDE systems with quasimonotone nonlinearities, using sub-supersolution methods and monotone iteration, with applications demonstrated through concrete examples.

Contribution

It introduces new existence results for coupled elliptic systems with quasimonotone nonlinearities, extending previous methods to non-monotone cases under growth conditions.

Findings

Existence of minimal and maximal weak solutions established.

Applicability demonstrated through concrete examples.

Results cover both monotone and certain non-monotone nonlinearities.

Abstract

We establish the existence of weak solutions of coupled systems of elliptic partial differential equations with quasimonotone nonlinearities in the domain interior and on the boundary. When the nonlinearities satisfy some monotonicity conditions, we employ monotone iteration techniques to establish the existence of minimal and maximal weak solutions between an ordered pair of sub- and supersolution. In the absence of monotonicity, we prove an existence result when the nonlinearities satisfy certain growth conditions. In addition, we provide concrete examples that illustrate the applicability of our theoretical results.