Electron. J. Diff. Equ., Vol. 2015 (2015), No. 177, pp. 1-8.

Positive bounded solutions for semilinear elliptic systems with indefinite weights in the half-space

Ramzi Alsaedi, Habib Maagli, Vicentiu Radulescu, Noureddine Zeddini

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.

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Ramzi Alsaedi
Department of Mathematics, Faculty of Sciences
King Abdulaziz University, P.O. Box 80203
Jeddah 21589, Saudi Arabia
email: ramzialsaedi@yahoo.co.uk
Habib Mâaagli
Department of Mathematics, Faculty of Sciences
King Abdulaziz University, P.O. Box 80203
Jeddah 21589, Saudi Arabia
email: abobaker@kau.edu.sa
Vicentiu Radulescu
Department of Mathematics, Faculty of Sciences
King Abdulaziz University, P.O. Box 80203
Jeddah 21589, Saudi Arabia
email: vicentiu.radulescu@imar.ro
Noureddine Zeddini
Department of Mathematics, Faculty of Sciences
King Abdulaziz University, P.O. Box 80203
Jeddah 21589, Saudi Arabia
email: nalzeddini@kau.edu.sa

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