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.