The Relationships Between Equilibria and Positive Solutions of Certain Nonlinear Elliptic Systems

AbstractLetM(u,v)=0,N(u,v)=0 define two distinct phase curves Γ1,Γ2in the (u,v)-phase plane. This paper presents results on the relationships among the positive equilibria, the phase curves, and the existence of positive solutions to the PDE systemΔu+uM(u,v)=0,Δv+vN(u,v)=0in Ω⊂Rnunder Dirichlet boundary conditions, where Ω is a bounded domain andM,Nare monotone functions