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

### From NZJM

**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**n* nonlinear systems involving different degenerated *p*-Laplacian operators with variable coefficients. We study the existence of weak solutions for the following nonlinear systems:

*u*

_{i}=

*a*

_{ii}(

*x*) |

*u*

_{i}|

^{pi − 2}

*u*

_{i}-

*a*

_{ij}(

*x*) |

*u*

_{i}|

^{αi}|

*u*

_{j}|

^{αj}

*u*

_{j}+

*f*

_{i}(

*x*),

whereΔ*P*,_{p} denotes the degenerated *p*-Laplacian defined by

*P*,

_{p}

*u*=

*d*

*i*

*v*[

*P*(

*x*) ||

^{p − 2}]

with *p* > 1, *p*2, *P*(*x*) is a weight function, α_{i}0, *f*_{i} are given functions and the coefficients *a*_{ij}(*x*) (1 *i*, *j**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