Uniqueness of global positive solution branches of nonlinear elliptic problems

has been studied in many publications (the first global description was given in [19]). Nonetheless there were open questions some of which are answered in this paper. We motivate some of these questions now: For a suitable class of (smooth) fimctions f : IR+ ~ IR problem (1.1) has infinitely many solutions for 2 = 1. This was shown in [15] via the method of suband supersolutions which also gives an order of these solutions: 0 0 (which is not excluded in [15]) there is also a global branch ( = continuum) C + of solutions (2 ,u) bifurcating from the trivial solution line {(2, 0)} at 2 = 21 where #1 = 21 f ' ( 0 ) is the principal eigenvalue of the linearization