Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 69, pp. 1-10.

Title: Existence of weak solutions for nonlinear
elliptic systems on $\mathbb{R}^N$

Authors: Eada A. El-Zahrani (Faculty of Science for Girls, Saudi Arabia)
         Hassan M. Serag (Al-Azhar Univ., Cairo, Egypt)

Abstract: 
 In this paper, we consider the nonlinear elliptic system
 $$\displaylines{
 -\Delta_pu=a(x)|u|^{p-2}u-b(x)|u|^\alpha|v|^\beta v+f,\cr
 -\Delta_qv=-c(x)|u|^\alpha |v|^\beta u + d(x) |v|^{q-2}v +g ,\cr
 \lim_{|x|\to\infty}u=\lim_{|x|\to\infty}v=0\quad u,v>0
 }$$
on a bounded and unbounded domains of \mathbb{R}^N,
where $\Delta_p$ denotes the p-Laplacian.
The existence of weak solutions for these systems is proved using
the theory of monotone operators
 
Submitted February 9, 2006. Published July 6, 2006.
Math Subject Classifications: 35B45, 35J55.
Key Words: Weak solutions; nonlinear elliptic systems; p-Laplacian;
           monotone operators.