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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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