### Electronic Research Archive

2021, Issue 6: 3761-3774. doi: 10.3934/era.2021060
Special Issues

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

• 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 $\mu_1 = \mu_2 = \beta.$

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

### Related Papers:

• 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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