A global bifurcation result for a class of semipositone elliptic systems

We consider a system of the form−Δu=λ(θ1v++f(v))inΩ;−Δv=λ(θ2u++g(u))inΩ;u=0=von∂Ω,} where s+=defmax⁡{s,0}, θ1 and θ2 are fixed positive constants, λ∈R is the bifurcation parameter, and Ω⊂RN (N>1) is a bounded domain with smooth boundary ∂Ω (a bounded open interval if N=1). The nonlinearities f,g:R→R are continuous functions that are bounded from below, sublinear at infinity and have semipositone structure at the origin (f(0),g(0)<0). We show that there are two disjoint unbounded connected components of the solution set and discuss the nodal properties of solutions on these components. Finally, as a consequence of these results, we infer the existence and multiplicity of solutions for λ in a neighborhood containing the simple eigenvalue of the associated eigenvalue problem.