# Existence of Weak Solutions for n x n Nonlinear Systems Involving Different Degenerated p-Laplacian Operators

### New Zealand Journal of Mathematics

Vol. 38, (2008), Pages 75-86

H. M. Serag

Mathematics Department

Faculty of Science

Al-Azhar University

Nasr City (11884)

Cairo

EGYPT

S. A. Khafagy

Mathematics Department

Faculty of Science

Al-Azhar University

Nasr City (11884)

Cairo

EGYPT

Abstract In this paper, we consider n$\times$n nonlinear systems involving different degenerated p-Laplacian operators with variable coefficients. We study the existence of weak solutions for the following nonlinear systems:

-Δ$P_{i, p_{i}}$ ui = aii (x) |ui| pi − 2 ui - $\sum_{j \ne i}^{n}$ aij (x) |ui|αi |uj|αj uj + fi (x),

whereΔP,p denotes the degenerated p-Laplacian defined by

ΔP,p u = div [P (x) |$\nabla u$|p − 2 $\nabla u$]

with p > 1, p$\neq$2, P(x) is a weight function, αi$\ge$0, fi are given functions and the coefficients aij(x) (1 $\le$i, j$\le$n) are bounded smooth positive functions. We prove the existence of weak solutions for these systems defined on bounded domains using the theory of monotone operators. We also discuss the case of unbounded domains.

Keywords Weak solutions, nonlinear system, degenerated p-Laplacian, monotone operators

Classification (MSC2000) 35B45, 35J55