Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 85, pp. 1-12.
Title: Maximum principle and existence of positive solutions for
nonlinear systems on $\mathbb{R}^N$
Authors: Hassan M. Serag (Al-Azhar Univ., Cairo, Egypt)
Eada A. El-Zahrani (Faculty of Science for Girls, Saudi Arabia)
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 $\Delta_pu=\mathop{\rm div}|\nabla u|^{p-2}\nabla u)$ with
$ p>1$ and $p\neq 2$ is the ``p-Laplacian", $\alpha,\beta>0$,
$p,q>1$, 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.