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

Abstract

We study the Maximum Principle and existence of positive weak solutions for the n×n nonlinear elliptic system$left. begin{array}{ll} -Delta_{P_{,p}} u_i = sum_{j=1}^{n}a_{ij}(x)|u_j|^{p-2}u_j+f_i(x,u_1,u_2,...,u_n) & in hspace{5 mm} Omega , hspace{15 mm} u_i = 0, i=1,2,...n hspace{20 mm} & on hspace{2 mm} partialOmega, end{array} right }$where the degenerated p-Laplacian defined as $Delta_{P_{,p}}u = div [P(x)| Delta u|^{p-2} Delta u$ 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”.