On maximum principle and existence of positive weak solutions for n × n nonlinear elliptic systems involving degenerated p-Laplacian operators

We study the Maximum Principle and existence of positive weak solutions for the n×n nonlinear elliptic system-Delta_{P,p}u_i=sum_{j=1}^na_{ij}(x)|u_j|^{p-2}u_j+f_i(x,u_1,u_2, ... ,u_n) in Omega, u_i=0, i=1,2,. n on partial Omega } where the degenerated p-Laplacian defined as Delta _{P,p}u=div [P(x)|nabla u|^{p-2}nabla u] with p 1,p neq 2 and P(x) is a weight function. We give some conditions for having the Maximum Principle for this system and then we prove the existence of positive weak solutions for the quasilinear system by using ``sub-super solutions method .