Dynamics of Partial Differential Equations

Volume 19 (2022)

Number 2

An elliptic nonlinear system of multiple functions with application

Pages: 141 – 162

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


Joon Hyuk Kang (Department of Mathematics, Andrews University, Berrien Springs, Michigan, U.S.A.)

Timothy Robertson (Department of Mathematics, University of Tennessee, Knoxville, Tn., U.S.A.)


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 $\Omega$ in $R^n$. 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.


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

2010 Mathematics Subject Classification

35A01, 35A02

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Received 4 February 2019

Published 19 May 2022