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

Abstract

We study the maximum principle and existence of positive solutions for the nonlinear system egin{gather*} -Delta _{p,_{P}}u=a(x)|u|^{p-2}u+b(x)|u|^{alpha }|v|^{eta }v+f quad ext{in } Omega , \ -Delta _{Q,q}v=c(x)|u|^{alpha }|v|^{eta }u+d(x)|v|^{q-2}v+g quad ext{in } Omega , \ u=v=0 quad ext{on }partial Omega , end{gather*} where the degenerate p-Laplacian defined as $Delta _{p,_{P}}u=mathop{m div}[P(x)| abla u|^{p-2} abla u]$. 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.

Keywords

• Maximum principle
• existence of positive solution
• nonlinear elliptic system
• degenerated p-Laplacian.