Abstract
We study the bifurcation curve and exact multiplicity of positive solutions of the positone problem. where λ>0 is a bifurcation parameter, fεC <sup>2</sup> [0,∞) satisfies f(0)>0 and f(u)>0 for u>0, and f is convex-concave on (0,∞). Under a mild condition, we prove that the bifurcation curve is S-shaped on the (λ,{double pipe}u{double pipe}∞)-plane. We give an application to the perturbed Gelfand problem. where a>0 is the activation energy parameter. We prove that, if a≥a*≈4.166, the bifurcation curve is S-shaped on the (λ,{double pipe}u{double pipe}∞)-plane. Our results improve those in [S.-H. Wang, On S-shaped bifurcation curves, Nonlinear Anal. 22 (1994) 1475-1485] and [P. Korman, Y. Li, On the exactness of an S-shaped bifurcation curve, Proc. Amer. Math. Soc. 127 (1999) 1011-1020]. © 2011 Elsevier Inc.