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

Authors: H. M. SERAG, S. A. KHAFAGY

Abstract: We study the Maximum Principle and existence of positive weak solutions for the n \times 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''.

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

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