Positive solutions and global bifurcation of strongly coupled elliptic systems

Abstract

In this article, we study the existence of positive solutions for the coupled elliptic system egin{gather*} -Delta u= lambda (f(u, v)+ h_{1}(x) ) quad ext{in }Omega, \ -Delta v= lambda (g(u, v)+ h_{2}(x))quad ext{in }Omega, \ u =v=0 quad ext{on }partial Omega, end{gather*} under certain conditions on $f,g$ and allowing $h_1, h_2$ to be singular. We also consider the system egin{gather*} -Delta u= lambda ( a(x) u + b(x)v+ f_{1}(v)+ f_{2}(u) ) quad ext{in }Ome ga, \ -Delta v= lambda ( b(x)u+ c(x)v+ g_{1}(u)+ g_{2}(v) )quad ext{in }Omega , \ u =v=0 quad ext{on }partial Omega, end{gather*} and prove a Rabinowitz global bifurcation type theorem to this system.

Keywords

• Elliptic system
• bifurcation
• positive solutions