Exact multiplicity of positive solutions of a semipositone problem with concave–convex nonlinearity

We study the exact multiplicity of positive solutions and bifurcation diagrams of the Dirichlet boundary value problem{u″(x)+λf(u)=0,−10λ>0 is a bifurcation parameter, f∈C[0,∞)∩C2(0,∞)f∈C[0,∞)∩C2(0,∞) satisfies f(0)<0f(0)<0 (semipositone), and f   is concave–convex on (0,∞)(0,∞) and is asymptotic superlinear. Assuming additional suitable conditions on f  , on the (λ,‖u‖∞)(λ,‖u‖∞)-plane, we give a classification of totally three qualitatively different bifurcation diagrams: a reversed S-shaped curve, a broken reversed S-shaped curve, and a monotone decreasing curve. Our results improve those in [J. Shi, R. Shivaji, Exact multiplicity of solutions for classes of semipositone problems with concave–convex nonlinearity, Discrete Contin. Dyn. Syst. 7 (2002) 559–571]. We also give an application to determine completely the exact multiplicity of positive solutions and bifurcation diagrams of the problem with cubic nonlinearity{u″(x)+λ(u−a)(u−b)(u−c)=0,−1