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.

Show me the PDF file (214 KB), TEX file, and other files for this article.

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

Return to the EJDE web page