Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 81, pp. 1-10.

Title: Existence of weak solutions for nonlinear systems involving 
       several p-Laplacian operators

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

Abstract:
 In this article, we study nonlinear systems involving several  p-Laplacian
 operators with variable coefficients. We consider  the system  
 $$
 -\Delta _{p_i}u_i=a_{ii}(x)|u_i|^{p_i-2}u_i  
 -\sum_{j\neq i}^{n}a_{ij}(x)|u_i|^{\alpha _i}|u_j|^{\alpha_j}u_j+f_i(x),  
 $$
 where $\Delta _p$ denotes the p-Laplacian defined by  
 $\Delta_{p}u\equiv \mathop{\rm div} [|\nabla u|^{p-2}\nabla u]$  
 with $p>1$, $p\neq 2$; $\alpha _i\geq 0$; $f_i$ are  given functions; 
 and the coefficients $a_{ij}(x)$  ($1\leq i,j\leq n$) are bounded 
 smooth positive functions. We prove the existence of  
 weak solutions defined on bounded and unbounded domains 
 using the theory of nonlinear monotone operators.
 
Submitted  December 9, 2008. Published July 10, 2009.
Math Subject Classifications: 74H20, 35J65.
Key Words: Existence of weak solution; nonlinear system, p-Laplacian.