Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 12, pp. 1-9.

Title: Existence of positive solutions for some
       nonlinear elliptic systems on the half space

Author: Noureddine Zeddini (King Abdulaziz Univ., Rabigh, Saudi Arabia)

Abstract:
 We prove some existence of positive solutions to the semilinear
 elliptic system
 $$\displaylines{
 \Delta u =\lambda p(x)g(v)\cr
 \Delta v =\mu q(x)f(u)
 }$$
 in the half space ${\mathbb{R}}^n_+$, $n\geq 2$,  subject to some
 Dirichlet conditions, where $\lambda$ and $\mu$ are nonnegative
 parameters. The functions $f, g$ are nonnegative continuous
 monotone on $(0,\infty)$ and the potentials  $p, q$ are nonnegative
 and satisfy some hypotheses related to the Kato class
 $K^\infty({\mathbb{R}}^n_+)$.

Submitted March 16, 2010. Published January 21, 2011.
Math Subject Classifications: 35J55, 35J60, 35J65.
Key Words: Green function; Kato class; elliptic systems; positive solutions.