Turkish Journal of Mathematics

Abstract: We study the Maximum Principle and existence of positive weak solutions for the n \times n nonlinear elliptic system -D_P,pu_i=\sum_j=1ⁿa_ij(x)|u_j|^p-2u_j+f_i(x,u₁,u₂, ... ,u_n) in W, u_i=0,\ i=1,2,. n on \partial W \} where the degenerated p-Laplacian defined as D _P,pu=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''.

Key Words: Maximum principle, existence of positive weak solution, nonlinear elliptic system, degenerated p-Laplacian.