Electronic Journal of Differential Equations,
Vol. 2015 (2015), No. 177, pp. 1-8.
Title: Positive bounded solutions for semilinear elliptic systems
with indefinite weights in the half-space
Authors: Ramzi Alsaed (King Abdulaziz Univ., Jeddah, Saudi Arabia)
Habib Maagli (King Abdulaziz Univ., Jeddah, Saudi Arabia)
Vicentiu Radulescu (King Abdulaziz Univ., Jeddah, Saudi Arabia)
Noureddine Zeddini (King Abdulaziz Univ., Jeddah, Saudi Arabia)
Abstract:
In this article, we study the existence and nonexistence of positive bounded
solutions of the Dirichlet problem
$$\displaylines{
-\Delta u=\lambda p(x)f(u,v),\quad \text{in } {\mathbb{R}}_+^n,\cr
-\Delta v=\lambda q(x)g(u,v), \quad \text{in } {\mathbb{R}}_+^n,\cr
u=v=0\quad \text{on }\partial {\mathbb{R}}_+^n,\cr
\lim_{|x|\to \infty}u(x)=\lim_{|x|\to \infty}v(x)=0,
}$$
where ${\mathbb{R}}_+^n=\{x=(x_1,x_2,\dots, x_n)\in {\mathbb{R}}^n: x_n>0\}$
($n\geq 3$) is the upper half-space and $\lambda$ is a positive parameter.
The potential functions p,q are not necessarily bounded, they may change
sign and the functions $f,g:\mathbb{R}^2 \to \mathbb{R}$ are continuous.
By applying the Leray-Schauder fixed point theorem, we establish the
existence of positive solutions for $\lambda$ sufficiently small when
$f(0,0)>0$ and $g(0,0)>0$. Some nonexistence results of positive bounded
solutions are also given either if $\lambda$ is sufficiently small
or if $\lambda$ is large enough.
Submitted April 13, 2015. Published June 26, 2015.
Math Subject Classifications: 35B09, 35J47, 35J57, 58C30.
Key Words: Semilinear elliptic system; Leray-Schauder fixed point theorem;
positive bounded solution; indefinite potential.