Uniqueness and nondegeneracy of positive solutions to an elliptic system in ecology

Zaizheng Li; Zhitao Zhang; Zaizheng Li; Zhitao Zhang

doi:10.3934/era.2021060

Electronic Research Archive

2021, Volume 29, Issue 6: 3761-3774. doi: 10.3934/era.2021060

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Uniqueness and nondegeneracy of positive solutions to an elliptic system in ecology

Zaizheng Li ^{1,2
,},
Zhitao Zhang ^{3,4,5
,
,}

1.
School of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050024, China
2.
Hebei Key Laboratory of Computational Mathematics and Applications
3.
School of Mathematical Science, Jiangsu University, Zhenjiang 212013, China
4.
HLM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
5.
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received: 01 March 2021 Revised: 01 June 2021 Published: 13 August 2021
Primary: 35J15, 35J47; Secondary: 35J57

In this paper, we study the follwing important elliptic system which arises from the Lotka-Volterra ecological model in $ \mathbb{R}^N $
$ \begin{equation*} \begin{cases} -\Delta u+\lambda u = \mu_1u^2+\beta uv, & x\in\mathbb{R}^N,\\ -\Delta v+\lambda v = \mu_2v^2+\beta uv, & x\in \mathbb{R}^N,\\ u, v>0, u, v\in H^1(\mathbb{R}^N), \end{cases} \end{equation*} $
where $ N\leq 5, $ $ \lambda, \mu_1, \mu_2 $ are positive constants, $ \beta\geq 0 $ is a coupling constant. Firstly, we prove the uniqueness of positive solutions under general conditions, then we show the nondegeneracy of the positive solution and the degeneracy of semi-trivial solutions. Finally, we give a complete classification of positive solutions when $ \mu_1 = \mu_2 = \beta. $
- Lotka-Volterra system,
- positive solution,
- nondegeneracy
Citation: Zaizheng Li, Zhitao Zhang. Uniqueness and nondegeneracy of positive solutions to an elliptic system in ecology[J]. Electronic Research Archive, 2021, 29(6): 3761-3774. doi: 10.3934/era.2021060

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Abstract

In this paper, we study the follwing important elliptic system which arises from the Lotka-Volterra ecological model in $ \mathbb{R}^N $

$ \begin{equation*} \begin{cases} -\Delta u+\lambda u = \mu_1u^2+\beta uv, & x\in\mathbb{R}^N,\\ -\Delta v+\lambda v = \mu_2v^2+\beta uv, & x\in \mathbb{R}^N,\\ u, v>0, u, v\in H^1(\mathbb{R}^N), \end{cases} \end{equation*} $

where $ N\leq 5, $ $ \lambda, \mu_1, \mu_2 $ are positive constants, $ \beta\geq 0 $ is a coupling constant. Firstly, we prove the uniqueness of positive solutions under general conditions, then we show the nondegeneracy of the positive solution and the degeneracy of semi-trivial solutions. Finally, we give a complete classification of positive solutions when $ \mu_1 = \mu_2 = \beta. $

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