# Maximum Principle and Existence of Weak Solutions for Nonlinear System Involving Different Degenerated P-Laplacian

### New Zealand Journal of Mathematics

Vol. 39, (2009), Pages 151-163

Salah A. Khafagy

Mathematics Department

Faculty of Science

Al-Azhar University

Nasr City

11884

Cairo

Egypt

Abstract In this paper, we study the maximum principle and existence of weak solutions for the following nonlinear system on $\Re ^{N}$

where ΔP,p with 1 < p < N, $p\neq 2$ and P(x) is a weight function, denotes the degenerate p-Laplacian defined by $\Delta_{P,p}u\equiv div\ [P(x)|\nabla u|^{p-2}\nabla u]$. We give necessary and sufficient conditions to have a maximum principle for this system and then we prove the existence of weak solutions for the same system by using an approximation method.

Keywords maximum principle, existence of weak solution, nonlinear elliptic system, degenerated p-Laplacian.

Classification (MSC2000) Primary: 35B50, 35J67, 35J55.