A remark on global bifurcations of solutions of Ginzburg–Landau equation

We have proved that all the closed connected sets of solutions of the complex Ginzburg–Landau equation {−Δu(x)+2i〈A(x),∇u(x)〉+‖A(x)‖2u(x)=λ(1−|u(x)|2)u(x)in Ω,u=0on ∂Ω, bifurcating from the set of normal solutions {0}×(0,+∞)⊂H01(Ω,C)×(0,+∞) are unbounded, where Ω⊂R2Ω⊂R2 is an open, bounded domain with smooth boundary, A(x1,x2)=(−x2,x1)A(x1,x2)=(−x2,x1) and ‖⋅‖‖⋅‖ is the usual norm in R2R2.