Asymptotically linear system of three equations near resonance

This paper deals with the asymptotically linear system−Δu1=λθ1u3+f1(λ,x,u1,u2,u3)in Ω−Δu2=λθ2u2+f2(λ,x,u1,u2,u3)in Ω−Δu3=λθ3u1+f3(λ,x,u1,u2,u3)in Ωu1=u2=u3=0on ∂Ω,} where θi>0θi>0 for i=1,2,3i=1,2,3 with θ2≠θ1θ3, λ   is a real parameter and Ω⊂RNΩ⊂RN is a bounded domain with smooth boundary. The linear part of the system has two simple eigenvalues with nonnegative eigenfunctions each with at least one zero component. We provide sufficient conditions which guarantee bifurcation from infinity of positive solutions from both, one or none of the two simple eigenvalues. Under additional assumptions on the nonlinear perturbations, we determine the λ-direction of bifurcation as well. We use bifurcation theory to establish our results.