IJPAM: Volume 105, No. 4 (2015)

SMOOTH POSITIVE SOLUTIONS TO
AN ELLIPTIC MODEL WITH $C^{2}$ FUNCTIONS

Joon Hyuk Kang
Department of Mathematics
Andrews University
Berrien Springs, MI 49104, USA


Abstract. Two species of animals are competing or cooperating in the same environment. Under what conditions do they coexist peacefully? Or under what conditions does either one of the two species become extinct, that is, is either one of the two species excluded by the other? We investigate this phenomena in mathematical point of view.

In this paper we concentrate on coexistence solutions of the competition or cooperation model

\begin{displaymath}\left\{ \begin{array}{l}
\left.\begin{array}{l}
\Delta u + ...
...ial\Omega} = v\vert _{\partial\Omega} = 0.
\end{array} \right.\end{displaymath}

This system is the general model for the steady state of a competitive or cooperative interacting system depending on growth conditions for $g$ and $h$. The techniques used in this paper are elliptic theory, super-sub solutions, maximum principles, and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.

Received: August 4, 2015

AMS Subject Classification: 35A05, 35A07

Key Words and Phrases: competition and cooperation system, coexistence state

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DOI: 10.12732/ijpam.v105i4.7 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2015
Volume: 105
Issue: 4
Pages: 653 - 667


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