A theorem on S-shaped bifurcation curve for a positone problem with convex–concave nonlinearity and its applications to the perturbed Gelfand problem

We study the bifurcation curve and exact multiplicity of positive solutions of the positone problem{u″(x)+λf(u)=0,−10λ>0 is a bifurcation parameter, f∈C2[0,∞)f∈C2[0,∞) satisfies f(0)>0f(0)>0 and f(u)>0f(u)>0 for u>0u>0, and f   is convex–concave on (0,∞)(0,∞). Under a mild condition, we prove that the bifurcation curve is S-shaped on the (λ,‖u‖∞)(λ,‖u‖∞)-plane. We give an application to the perturbed Gelfand problem{u″(x)+λexp(aua+u)=0,−10a>0 is the activation energy parameter. We prove that, if a⩾a⁎≈4.166a⩾a⁎≈4.166, the bifurcation curve is S-shaped on the (λ,‖u‖∞)(λ,‖u‖∞)-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].