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

Abstract

We study the bifurcation curve and exact multiplicity of positive solutions of the positone problem { u ″ ( x ) + λ f ( u ) = 0 , − 1 < x < 1 , u ( − 1 ) = u ( 1 ) = 0 , where λ > 0 is a bifurcation parameter, f ∈ C 2 [ 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 , ‖ u ‖ ∞ ) -plane. We give an application to the perturbed Gelfand problem { u ″ ( x ) + λ exp ( a u a + u ) = 0 , − 1 < x < 1 , u ( − 1 ) = u ( 1 ) = 0 , 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 , ‖ 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].