# American Institute of Mathematical Sciences

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

## A unique positive solution to a system of semilinear elliptic equations

 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 $\kappa>0$.
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), 331-364.  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), 759-778.  Google Scholar [3] W. Zhou and X. Wei, Uniqueness of positive solutions for an elliptic system, Electronic Journal of Differential Equations, 2011 (2011), 1-6.  Google Scholar

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##### References:
 [1] Y.H. Du and S.B. Hsu, A diffusive predator-prey model in heterogeneous environment, Journal of Differential Equations, 203 (2004), 331-364.  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), 759-778.  Google Scholar [3] W. Zhou and X. Wei, Uniqueness of positive solutions for an elliptic system, Electronic Journal of Differential Equations, 2011 (2011), 1-6.  Google Scholar
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