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 . 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.

Show me the PDF file (264K), TEX file, and other files for this article.

 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