American Institute of Mathematical Sciences

December  2021, 29(6): 3761-3774. doi: 10.3934/era.2021060

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

 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

* Corresponding author: Zhitao Zhang

Received  March 2021 Revised  June 2021 Published  December 2021 Early access  August 2021

Fund Project: Dedicated to Professor E. N. Dancer's 75th Birthday. Supported by National Natural Science Foundation of China (11771428, 12031015, 12026217)

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. Electronic Research Archive, 2021, 29 (6) : 3761-3774. doi: 10.3934/era.2021060
References:
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References:
 [1] T. Bartsch, Bifurcation in a multicomponent system of nonlinear Schrödinger equations,, J. Fixed Point Theory Appl., 13 (2013), 37-50.  doi: 10.1007/s11784-013-0109-4.  Google Scholar [2] T. Bartsch, R. Tian and Z.-Q. Wang, Bifurcations for a coupled Schrödinger system with multiple components,, Z. Angew. Math. Phys., 66 (2015), 2109-2123.  doi: 10.1007/s00033-015-0498-x.  Google Scholar [3] J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space,, J. Differential Equations, 163 (2000), 41-56.  doi: 10.1006/jdeq.1999.3701.  Google Scholar [4] C. Cosner and A. C. Lazer, Stable coexistence states in the Volterra-Lotka competition model with diffusion,, SIAM J. Appl. Math., 44 (1984), 1112-1132.  doi: 10.1137/0144080.  Google Scholar [5] E. N. Dancer, K. Wang and Z. Zhang, Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species,, J. Differential Equations, 251 (2011), 2737-2769.  doi: 10.1016/j.jde.2011.06.015.  Google Scholar [6] E. N. Dancer, K. Wang and Z. Zhang, Dynamics of strongly competing systems with many species,, Trans. Amer. Math. Soc., 364 (2012), 961-1005.  doi: 10.1090/S0002-9947-2011-05488-7.  Google Scholar [7] E. N. Dancer and J. Wei, Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction,, Trans. Amer. Math. Soc., 361 (2009), 1189-1208.  doi: 10.1090/S0002-9947-08-04735-1.  Google Scholar [8] E. N. Dancer and Z. Zhang, Dynamics of Lotka-Volterra competition systems with large interaction,, Journal of Differential Equations, 182 (2002), 470-489.  doi: 10.1006/jdeq.2001.4102.  Google Scholar [9] C. Gui and Y. Lou, Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model,, Comm. Pure Appl. Math., 47 (1994), 1571-1594.  doi: 10.1002/cpa.3160471203.  Google Scholar [10] X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations,, Adv. Differential Equations, 5 (2000), 899-928.   Google Scholar [11] K. I. Kim and Z. Lin, Blow-up in a three-species cooperating model,, Appl. Math. Lett., 17 (2004), 89-94.  doi: 10.1016/S0893-9659(04)90017-1.  Google Scholar [12] P. Korman and A. W. Leung, A general monotone scheme for elliptic systems with applications to ecological models,, Proc. Roy. Soc. Edinburgh Sect. A, 102 (1986), 315-325.  doi: 10.1017/S0308210500026391.  Google Scholar [13] P. Korman and A. Leung, On the existence and uniqueness of positive steady states in the Volterra-Lotka ecological models with diffusion,, Appl. Anal., 26 (1987), 145-160.  doi: 10.1080/00036818708839706.  Google Scholar [14] M. K. Kwong, Uniqueness of positive solutions of {$\Delta u- u+ u^p = 0$} in {$R^n$},, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar [15] A. Leung and P. H. Rabinowitz, Monotone schemes for semilinear elliptic systems related to ecology,, Math. Methods Appl. Sci., 4 (1982), 272-285.  doi: 10.1002/mma.1670040118.  Google Scholar [16] H. Li and M. Wang, Critical exponents and lower bounds of blow-up rate for a reaction-diffusion system,, Nonlinear Anal., 63 (2005), 1083-1093.  doi: 10.1016/j.na.2005.05.037.  Google Scholar [17] Z. Lin, Blowup estimates for a mutualistic model in ecology,, Electron. J. Qual. Theory Differ. Equ., (2002), no. 8, 14 pp. doi: 10.14232/ejqtde.2002.1.8.  Google Scholar [18] J. López-Gómez, R. Ortega and A. Tineo, The periodic predator-prey Lotka-Volterra model,, Adv. Differential Equations, 1 (1996), 403-423.   Google Scholar [19] J. López-Gómez and R. Pardo San Gil, Coexistence regions in Lotka-Volterra models with diffusion,, Nonlinear Anal., 19 (1992), 11-28.  doi: 10.1016/0362-546X(92)90027-C.  Google Scholar [20] Y. Lou, Necessary and sufficient condition for the existence of positive solutions of certain cooperative system,, Nonlinear Anal., 26 (1996), 1079-1095.  doi: 10.1016/0362-546X(94)00265-J.  Google Scholar [21] Y. Lou, T. Nagylaki and W.-M. Ni, On diffusion-induced blowups in a mutualistic model,, Nonlinear Anal., 45 (2001), 329-342.  doi: 10.1016/S0362-546X(99)00346-6.  Google Scholar [22] P. Quittner, Liouville theorems, universal estimates and periodic solutions for cooperative parabolic Lotka-Volterra systems,, J. Differential Equations, 260 (2016), 3524-3537.  doi: 10.1016/j.jde.2015.10.035.  Google Scholar [23] P. Quittner and P. Souplet, Superlinear Parabolic Problems, , 2$^nd$ edition, Birkhäuser/Springer, Cham, 2019. doi: 10.1007/978-3-030-18222-9.  Google Scholar [24] R. Tian and Z. Zhang, Existence and bifurcation of solutions for a double coupled system of Schrödinger equations,, Sci. China Math., 58 (2015), 1607-1620.  doi: 10.1007/s11425-015-5028-y.  Google Scholar [25] J. Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem,, J. Differential Equations, 129 (1996), 315-333.  doi: 10.1006/jdeq.1996.0120.  Google Scholar [26] J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations,, Commun. Pure Appl. Anal., 11 (2012), 1003-1011.  doi: 10.3934/cpaa.2012.11.1003.  Google Scholar [27] Z. Zhang, Variational, Topological, and Partial Order Methods with Their Applications, , Springer, 2013. doi: 10.1007/978-3-642-30709-6.  Google Scholar [28] Z. Zhang and W. Wang, Structure of positive solutions to a Schrödinger system,, J. Fixed Point Theory Appl., 19 (2017), 877-887.  doi: 10.1007/s11784-016-0383-z.  Google Scholar
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