Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 153, pp. 1-9.

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

Author: Jagmohan Tyagi (Indian Inst. of Tech., Gandhinagar, India)

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.