An Elliptic Nonlinear System of Multiple Functions with Application

Document Type

Article

Publication Date

2022

Keywords

competition system, coexistence state, maximum principles, second order elliptic systems, variational methods for eigenvalues of operators

Abstract

The purpose of this paper is to give a sufficient conditions for the existence and uniqueness of positive solutions to a rather general type of elliptic system of the Dirichlet problem on a bounded domain Ω" role="presentation" style="display: inline; line-height: normal; font-size: 17.3333px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(64, 64, 64); font-family: "Times New Roman", Times, serif; position: relative;">ΩΩ in Rn" role="presentation" style="display: inline; line-height: normal; font-size: 17.3333px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(64, 64, 64); font-family: "Times New Roman", Times, serif; position: relative;">RnRn. Also considered are the effects of perturbations on the coexistence state and uniqueness. The techniques used in this paper are super‑sub solutions method, eigenvalues of operators, maximum principles, spectrum estimates, inverse function theory, and general elliptic theory. The arguments also rely on some detailed properties for the solution of logistic equations. These results yield an algebraically computable criterion for the positive coexistence of competing species of animals in many biological models.

Journal Title

Dynamics of Partial Differential Equations

Volume

19

Issue

2

First Page

141

Last Page

162

DOI

https://doi.org/https://dx.doi.org/10.4310/DPDE.2022.v19.n2.a3

First Department

Mathematics

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