Electron. J. Diff. Eqns., Vol. 2007(2007), No. 66, pp. 1-14.

Maximum principle and existence of positive solutions for nonlinear systems involving degenerate p-Laplacian operators

Salah A. Khafagy, Hassan M. Serag

Abstract:
We study the maximum principle and existence of positive solutions for the nonlinear system \begin{gather*} -\Delta _{p,_{P}}u=a(x)|u|^{p-2}u+b(x)|u|^{\alpha }|v|^{\beta }v+f \quad \text{in } \Omega , \\ -\Delta _{Q,q}v=c(x)|u|^{\alpha }|v|^{\beta }u+d(x)|v|^{q-2}v+g \quad \text{in } \Omega , \\ u=v=0 \quad \text{on }\partial \Omega , \end{gather*} where the degenerate p-Laplacian defined as $\Delta _{p,_{P}}u=\hbox{\rm div}[P(x)|\nabla u|^{p-2}\nabla u]$. We give necessary and sufficient conditions for having the maximum principle for this system and then we prove the existence of positive solutions for the same system by using an approximation method.

Submitted February 2, 2007. Published May 9, 2007.
Math Subject Classifications: 35B50, 35J67, 35J55.
Key Words: Maximum principle; existence of positive solution; nonlinear elliptic system; degenerated p-Laplacian.

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Salah A. Khafagy
Mathematics Department
Faculty of Science, Al-Azhar University
Nasr City (11884), Cairo, Egypt
email: el_gharieb@hotmail.com
Hassan M. Serag
Mathematics Department
Faculty of Science, Al-Azhar University
Nasr City (11884), Cairo, Egypt
email: serraghm@yahoo.com

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