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 problemwhere 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 problemwhere 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,

Classification and evolution of bifurcation curves for a porous-medium combustion problem with large activation energy

“…To prove Lemmas 3.1-3.7, we modify the time-map techniques used in [2,8] and develop some new time-map techniques. For any d > 0 and a ≥ 0, the time map formula which we apply to study (1.1) takes the form as follows:…”

Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps

“…Using bifurcation theory, in [21], Shi and Shivaji obtained the exact multiplicity of positive solutions of problem (1.2) with f (0) < 0, which is called semipositone problem, and proved that the bifurcation diagram of (1.2) looks exactly like one of the following two graphs: Another way is time map method. In [8], Hung and Wang studied the bifurcation curve and exact multiplicity of positive solutions of problem (1.2) by using time map analysis, and gave an application to the perturbed Gelfand problem…”

Classification and Evolution of Bifurcation Curves for a Dirichlet-Neumann Boundary Value Problem and its Application

“…the shapes of the bifurcation curve of positive solutions for (1.14) on the (λ, u ∞ )-plane have been studied exuberantly; see, e.g., [1,5,6,8,10] and references therein. While the shapes of the bifurcation curves with the mixed boundary conditions such as (1.1) are much less studied; see [2-4, 7, 11].…”

“…For one-dimensional zero Dirichlet boundary value problem (1.14), under (H1)-(H3) on f , Hung and Wang [6] proved that the bifurcation curve of positive solutions is exactly type 1 S-shaped on the (λ, u ∞ )-plane. They gave an application to the one-dimensional perturbed Gelfand problem.…”

“…Moreover, it is exactly type 1 S-shaped. In particular, f (u) = exp au a+u satisfies (H1)-(H3) for a ≥ a * ≈ 4.166 for some a * defined in [6,Eq. (3.22)].…”