Electron. J. Diff. Equ., Vol. 2011 (2011), No. 153, pp. 1-9.

Existence of non-negative solutions for predator-prey elliptic systems with a sign-changing nonlinearity

Jagmohan Tyagi

Abstract:
By the method of monotone iteration and Schauder fixed point theorem, we prove the existence of non-negative solutions to the system
$$\displaylines{ -\Delta u= \lambda a(x) f(v)\quad \hbox{in }\Omega,\cr -\Delta v= \lambda b(x) g(u)\quad \hbox{in } \Omega,\cr u =v=0\quad \hbox{on }\partial \Omega, }$$
for $\lambda$ sufficiently small, where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary $\partial \Omega$ and $\lambda$ is a positive parameter. In this work, we allow the sign changing nature of a and b with $a(x) b(x)\leq 0, \forall x\in \bar{\Omega}$.

Submitted July 19, 2011. Published November 10, 2011.
Math Subject Classifications: 35J45, 35J55.
Key Words: Elliptic system; non-negative solution; existence.

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Jagmohan Tyagi
Indian Institute of Technology Gandhinagar
Vishwakarma Government Engineering College Complex
Chandkheda, Visat-Gandhinagar Highway, Ahmedabad
Gujarat, India - 382424
email: jtyagi1@gmail.com, jtyagi@iitgn.ac.in

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