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

Serag, Hassan M. ; El-Zahrani, Eada A.

Electronic Journal of Differential Equations, Tome 2005 (2005).

Résumé

Summary: In this paper, we study the following non-linear system on $$\displaylines{ -\Delta_pu=a(x)|u|^{p-2}u+b(x)|u|^{\alpha}|v|^{\beta}v+f\quad x\in \mathbb{R}^N\cr -\Delta_qv=c(x)|u|^{\alpha}|v|^{\beta}u+d(x)|v|^{q-2}v+g \quad x\in \mathbb{R}^N\cr \lim_{|x|\to\infty}u(x)=\lim_{|x|\to\infty}v(x)=0,\quad u,v>0\quad \hbox{in }\mathbb{R}^N\cr }$$ where $\Delta_pu=\hbox{\rm div}|\nabla u|^{p-2}\nabla u)$ with $ p$ and $p\neq 2$ is the "p-Laplacian", $\alpha,\beta, p,q$, 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.

Détail

Classification : 35B45, 35J55, 35P65
Keywords: aximum principle, nonlinear elliptic systems, $p$-Laplacian, sub and super solutions