Electron. J. Diff. Eqns., Vol. 2005(2005), No. 85, pp. 1-12.

Maximum principle and existence of positive solutions for nonlinear systems on $\mathbb{R}^N$

Hassan M. Serag, Eada A. El-Zahrani

Abstract:
In this paper, we study the following non-linear system on $\mathbb{R}^N$
$$\displaylines{ -\Delta_pu=a(x)|u|^{p-2}u+b(x)|u|^{\alpha}|v|^{\beta}v+f\quad x\in \mathbb{R}^N\cr -\Delta_qv=c(x)|u|^{\alpha}|v|^{\beta}u+d(x)|v|^{q-2}v+g \quad x\in \mathbb{R}^N\cr \lim_{|x|\to\infty}u(x)=\lim_{|x|\to\infty}v(x)=0,\quad u,v>0\quad \hbox{in }\mathbb{R}^N\cr }$$
where $\Delta_pu=\hbox{\rm div}|\nabla u|^{p-2}\nabla u)$ with $ p$ greater than 1 and $p\neq 2$ is the ``p-Laplacian", $\alpha,\beta$ greater than 0, $p,q$ greater than 1, and $f,g$ are given functions. We obtain necessary and sufficient conditions for having a maximum principle; then we use an approximation method to prove the existence of positive solution for this system.

Submitted May 19, 2005. Published July 27, 2005.
Math Subject Classifications:
Key Words: aximum principle; nonlinear elliptic systems; $p$-Laplacian; sub and super solutions.

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Hassan M. Serag
Mathematics Department, Faculty of Science
Al-Azhar University
Nasr City (11884), Cairo, Egypt
email: serraghm@yahoo.com
  Eada A. El-Zahrani
Mathematics Department, Faculty of Science for Girls
Dammam, P. O. Box 838, Pincode 31113, Saudi Arabia
email: eada00@hotmail.com

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