UNIQUENESS OF STEADY STATE POSITIVE SOLUTIONS TO A GENERAL ELLIPTIC SYSTEM WITH DIRICHLET BOUNDARY CONDITIONS

Joon Hyuk Kang

doi:10.11948/20210500

2022 Volume 12 Issue 6

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Joon Hyuk Kang. UNIQUENESS OF STEADY STATE POSITIVE SOLUTIONS TO A GENERAL ELLIPTIC SYSTEM WITH DIRICHLET BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2022, 12(6): 2370-2385. doi: 10.11948/20210500

Citation:

Joon Hyuk Kang. UNIQUENESS OF STEADY STATE POSITIVE SOLUTIONS TO A GENERAL ELLIPTIC SYSTEM WITH DIRICHLET BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2022, 12(6): 2370-2385. doi: 10.11948/20210500

UNIQUENESS OF STEADY STATE POSITIVE SOLUTIONS TO A GENERAL ELLIPTIC SYSTEM WITH DIRICHLET BOUNDARY CONDITIONS

Joon Hyuk Kang^1, ,

1.
Department of Mathematics, Andrews University, Berrien Springs, MI. 49104

Corresponding author: Email: kang@andrews.edu(J. Kang)

Abstract

Abstract

The purpose of this paper is to give conditions for the uniqueness of positive solution 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.
- Coexistence /
- cooperating species of animals
MSC: 35J66, 35J67

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References

[1]	A. Alhasanat and C. Ou, Minimal-speed selection of traveling waves to the Lotka-Volterra competition model, J. Diff Eqs., 2019, 266(11), 7357–7378. doi: 10.1016/j.jde.2018.12.003 CrossRef Google Scholar
[2]	R. S. Cantrell and C. Cosner, On the uniqueness and stability of positive solutions in the Volterra-Lotka competition model with diffusion, Houston J. Math., 1989, 15, 341–361. Google Scholar
[3]	C. Cosner and A. C. Lazer, Stable coexistence states in the Volterra-Lotka competition model with diffusion, Siam J. Appl. Math., 1984, 44, 1112–1132. Google Scholar
[4]	D. Dunninger, Lecture note for applied analysis in Michigan State University. Google Scholar
[5]	F. Dong, B. Li and W. Li, Forced waves in a Lotka-Volterra competition-diffusion model with a shifting habitat, J. Diff. Eqs., 2021, 276, 433–459. doi: 10.1016/j.jde.2020.12.022 CrossRef Google Scholar
[6]	C. Gui and Y. Lou, Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model, Comm. Pure and Appl. Math., 1994, 2(12), 1571–1594. Google Scholar
[7]	J. Kang, Smooth positive solutions to an elliptic model with C² functions, International J. Pure and Appl. Math., 2015, 105(4), 653–667. Google Scholar
[8]	P. Korman and A. Leung, A general monotone scheme for elliptic systems with applications to ecological models, Proceedings of the Royal Society of Edinburgh, 1986, 102A, 315–325. Google Scholar
[9]	P. Korman and A. Leung, On the existence and uniqueness of positive steady states in the Volterra-Lotka ecological models with diffusion, Applicable Analysis, 26, 145–160. Google Scholar
[10]	A. Leung, Equilibria and stabilities for competing-species, reaction-diffusion equations with Dirichlet boundary data, J. Math. Anal. Appl., 1980, 73, 204–218. Google Scholar
[11]	L. Li and A. Ghoreishi, On positive solutions of general nonlinear elliptic symbiotic interacting systems, Appl. Anal., 1991, 40(4), 281–295. Google Scholar
[12]	L. Li and R. Logan, Positive solutions to general elliptic competition models, Diff. and Integral Eqs., 1991, 4, 817–834. Google Scholar
[13]	J. Lopez-Gomez and R. Pardo San Gil, Coexistence regions in Lotka-Volterra Models with diffusion, Nonlinear Anal. Theory, Methods and Appl., 1992, 19(1), 11–28. Google Scholar
[14]	Y. Lou, Necessary and sufficient condition for the existence of positive solutions of certain cooperative system, Nonlinear Anal. Theory, Methods and Appl., 1996, 26(6), 1079–1095. Google Scholar
[15]	X. Lu, H. Hui, F. Liu and Y. Bai, Stability and optimal control strategies for a novel epidemic model of COVID-19, Nonlinear Dynamics, 2021, 25. http://doi.org/10.1007/s11071-021-06524-x. doi: 10.1007/s11071-021-06524-x CrossRef Google Scholar
[16]	X. Lü and S. Chen, Interaction solutions to nonlinear partial differential equations via Hirota bilinear forms: One-lump-multi-stripe and one-lump-multi-soliton types, Nonlinear Dynamics, 2021, 103, 947–977. Google Scholar
[17]	P. Mckenna and W. Walter, On the Dirichlet problem for elliptic systems, Applicable Anal., 1986, 21, 207–224. Google Scholar
[18]	A. Slavik, Lotka-Volterra competition model on graphs, Siam J. on Applied Dynamical Systems, 19(2), 725–762. Google Scholar
[19]	F. Xu and W. Gan, On a Lotka-Volterra type competition model from river ecology, Nonlinear Anal., 2019, 47, 373–384. Google Scholar
[20]	M. Yin, Q. Zhu and X. Lü, Parameter estimation of the incubation period of COVID-19 based on the doubly interval-censored data model, Nonlinear Dynamics, 2021. https://doi.org/10.1007/s11071-021-06587-w. doi: 10.1007/s11071-021-06587-w CrossRef Google Scholar
[21]	Y. Yin, X. Lü and W. Ma, Bäcklund transformation, exact solutions and diverse interaction phenomena to a $ (3 + 1)$-dimensional nonlinear evolution equation, Nonlinear Dynamics, 2021. https://doi.org/10.1007/s11071-021-06531-y. doi: 10.1007/s11071-021-06531-y CrossRef $ (3 + 1)$-dimensional nonlinear evolution equation" target="_blank">Google Scholar
[22]	L. Zhengyuan and P. De Mottoni, Bifurcation for some systems of cooperative and predator-prey type, J. Partial Diff. Eqs., 1992, 25–36. Google Scholar
[23]	P. Zhou and D. Xiao, Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system, J. Functional Anal., 2018, 275(2), 356–380. Google Scholar