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


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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,



Abstract In this paper we study the stbility and instability of positive weak solution for the weighted p-Laplacian nonlinear system − ΔP,pu + a(x) | u | p − 2u = λb(x) uα in Ω, Bu = 0 on \partial \Omega, where ΔP,p with p > 1 and P = P(x) is a weight function, denotes the weighted p-Laplacian defined by \Delta _{P,p}u\equiv div[P(x)|\nabla u|^{p-2}\nabla u], a(x) is a weight function, λ is a positive parameter, the continuous function b(x):\Omega \rightarrow R satisfies either b(x) > 0 or b(x) < 0 for all x\in \Omega, 0 < α < p − 1 and \Omega \subset R^{N} is a bounded domain with smooth boundary Bu=\delta h(x)u+(1-\delta )\frac{\partial u}{\partial n} where \delta \in \lbrack 0,1], h:\partial \Omega \rightarrow R^{+} with h = 1 when δ = 1.

Keywords Stability, weak solution, p-Laplacian.

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

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