American Institute of Mathematical Sciences

Advanced
2013, 2013(special): 193-195. doi: 10.3934/proc.2013.2013.193

A unique positive solution to a system of semilinear elliptic equations

Diane Denny 1,

1. 

Department of Mathematics and Statistics, Texas A&M University - Corpus Christi, Corpus Christi, Texas 78412, United States

Received  July 2012 Revised  November 2012 Published  November 2013

We study a system of semilinear elliptic equations that arises from a predator-prey model. Previous related work proved the existence of a unique positive solution to this system of equations in the special case in which the parameter $\alpha=0$ in this system of equations, provided that a positive parameter $\kappa$ in this system of equations is sufficiently large. We prove the existence of a unique positive solution to this system of equations for any $\alpha \geq 0$ and for any $\kappa>0$.
Keywords: uniqueness, existence, Elliptic, semilinear..
Mathematics Subject Classification: Primary: 35A0.
Citation: Diane Denny. A unique positive solution to a system of semilinear elliptic equations. Conference Publications, 2013, 2013 (special) : 193-195. doi: 10.3934/proc.2013.2013.193
References:
[1]

Y.H. Du and S.B. Hsu, A diffusive predator-prey model in heterogeneous environment,, Journal of Differential Equations, 203 (2004), 331.   Google Scholar

[2]

Y.H. Du and M.X. Wang, Asymptotic behaviour of positive steady states to a predator-prey model,, Proceedings of the Royal Society of Edinburgh, 136A (2006), 759.   Google Scholar

[3]

W. Zhou and X. Wei, Uniqueness of positive solutions for an elliptic system,, Electronic Journal of Differential Equations, 2011 (2011), 1.   Google Scholar

show all references

References:
[1]

Y.H. Du and S.B. Hsu, A diffusive predator-prey model in heterogeneous environment,, Journal of Differential Equations, 203 (2004), 331.   Google Scholar

[2]

Y.H. Du and M.X. Wang, Asymptotic behaviour of positive steady states to a predator-prey model,, Proceedings of the Royal Society of Edinburgh, 136A (2006), 759.   Google Scholar

[3]

W. Zhou and X. Wei, Uniqueness of positive solutions for an elliptic system,, Electronic Journal of Differential Equations, 2011 (2011), 1.   Google Scholar

[1]

Dongho Chae. Existence of a semilinear elliptic system with exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 709-718. doi: 10.3934/dcds.2007.18.709
[2]

Ruofei Yao, Yi Li, Hongbin Chen. Uniqueness of positive radial solutions of a semilinear elliptic equation in an annulus. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1585-1594. doi: 10.3934/dcds.2018122
[3]

Luiz F. O. Faria. Existence and uniqueness of positive solutions for singular biharmonic elliptic systems. Conference Publications, 2015, 2015 (special) : 400-408. doi: 10.3934/proc.2015.0400
[4]

Robert Jensen, Andrzej Świech. Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE. Communications on Pure & Applied Analysis, 2005, 4 (1) : 199-207. doi: 10.3934/cpaa.2005.4.187
[5]

Yen-Lin Wu, Zhi-You Chen, Jann-Long Chern, Y. Kabeya. Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 949-960. doi: 10.3934/cpaa.2014.13.949
[6]

Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control & Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018
[7]

Zhijun Zhang. Large solutions of semilinear elliptic equations with a gradient term: existence and boundary behavior. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1381-1392. doi: 10.3934/cpaa.2013.12.1381
[8]

Yinbin Deng, Shuangjie Peng, Li Wang. Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 795-826. doi: 10.3934/dcds.2012.32.795
[9]

Yajing Zhang, Jianghao Hao. Existence of positive entire solutions for semilinear elliptic systems in the whole space. Communications on Pure & Applied Analysis, 2009, 8 (2) : 719-724. doi: 10.3934/cpaa.2009.8.719
[10]

Haitao Yang. On the existence and asymptotic behavior of large solutions for a semilinear elliptic problem in $R^n$. Communications on Pure & Applied Analysis, 2005, 4 (1) : 187-198. doi: 10.3934/cpaa.2005.4.197
[11]

Piero Montecchiari, Paul H. Rabinowitz. A nondegeneracy condition for a semilinear elliptic system and the existence of 1- bump solutions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6995-7012. doi: 10.3934/dcds.2019241
[12]

Jorge García-Melián, Julio D. Rossi, José C. Sabina de Lis. Elliptic systems with boundary blow-up: existence, uniqueness and applications to removability of singularities. Communications on Pure & Applied Analysis, 2016, 15 (2) : 549-562. doi: 10.3934/cpaa.2016.15.549
[13]

Alexander Quaas, Aliang Xia. Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2653-2668. doi: 10.3934/dcds.2017113
[14]

Zhihua Huang, Xiaochun Liu. Existence theorem for a class of semilinear totally characteristic elliptic equations involving supercritical cone sobolev exponents. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3201-3216. doi: 10.3934/cpaa.2019144
[15]

Philip Korman. On uniqueness of positive solutions for a class of semilinear equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 865-871. doi: 10.3934/dcds.2002.8.865
[16]

Carmen Cortázar, Marta García-Huidobro, Pilar Herreros. On the uniqueness of bound state solutions of a semilinear equation with weights. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6761-6784. doi: 10.3934/dcds.2019294
[17]

Rafael Ortega, James R. Ward Jr. A semilinear elliptic system with vanishing nonlinearities. Conference Publications, 2003, 2003 (Special) : 688-693. doi: 10.3934/proc.2003.2003.688
[18]

Lei Wei, Zhaosheng Feng. Isolated singularity for semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3239-3252. doi: 10.3934/dcds.2015.35.3239
[19]

Maria Francesca Betta, Olivier Guibé, Anna Mercaldo. Uniqueness for Neumann problems for nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1023-1048. doi: 10.3934/cpaa.2019050
[20]

Miao Chen, Youyan Wan, Chang-Lin Xiang. Local uniqueness problem for a nonlinear elliptic equation. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1037-1055. doi: 10.3934/cpaa.2020048

 Impact Factor: 

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences