# On the Stability of Positive Weak Solutions for Weighted p-Laplacian Nonlinear System

### From NZJM

**New Zealand Journal of Mathematics**

**Vol.** 45, (2015), **Pages** 39-43

**S. A. Khafagy**

Mathematics Department,

Faculty of Science in Zulfi,

Majmaah University,

Zulfi 11932, P.O. Box 1712,

Saudi Arabia.

Mathematics Department,

Faculty of Science,

Al-Azhar University,

Nasr City (11884), Cairo,

Egypt.

**Abstract** In this paper we study the stbility and instability of positive weak
solution for the weighted *p*-Laplacian nonlinear system − Δ_{P,p}*u* + *a*(*x*) | *u* | ^{p − 2}*u* = λ*b*(*x*) *u*^{α} in Ω, *B**u* = 0 on , where Δ_{P,p} with *p* > 1 and *P* = *P*(*x*) is a weight function, denotes the weighted *p*-Laplacian defined by , *a*(*x*) is a weight
function, λ is a positive parameter, the continuous function satisfies either *b*(*x*) > 0 or *b*(*x*) < 0 for all , 0 < α < *p* − 1 and is a bounded
domain with smooth boundary where , with *h* = 1 when δ = 1.

**Keywords** Stability, weak solution, p-Laplacian.

**Classification** (MSC2000) 34D20, 35D30, 35J92.