Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 66, pp. 1-14.

Title: Maximum principle and existence of positive solutions for
       nonlinear systems involving degenerate p-Laplacian operators

Authors: Salah A. Khafagy (Al-Azhar Univ., Cairo, Egypt) 
         Hassan M. Serag  (Al-Azhar Univ., Cairo, Egypt)

Abstract:
 We study the maximum principle and existence of positive solutions for the
 nonlinear system 
 \begin{gather*}
 -\Delta _{p,_{P}}u=a(x)|u|^{p-2}u+b(x)|u|^{\alpha }|v|^{\beta }v+f \quad 
 \text{in } \Omega , \\
 -\Delta _{Q,q}v=c(x)|u|^{\alpha }|v|^{\beta }u+d(x)|v|^{q-2}v+g \quad 
 \text{in } \Omega , \\
 u=v=0 \quad \text{on }\partial \Omega ,
 \end{gather*}
 where the degenerate p-Laplacian defined as 
 $\Delta _{p,_{P}}u=\mathop{\rm div}[P(x)|\nabla u|^{p-2}\nabla u]$. 
 We give necessary and sufficient
 conditions for having the maximum principle for this system and then we
 prove the existence of positive solutions for the same system by using an
 approximation method.
  
Submitted February 2, 2007. Published May 9, 2007.
Math Subject Classifications: 35B50, 35J67, 35J55.
Key Words: Maximum principle; existence of positive solution;
           nonlinear elliptic system; degenerated p-Laplacian.