Exact Multiplicity of Positive Solutions for a Class of Second-Order Two-Point Boundary Problems with Weight Function

An exact multiplicity result of positive solutions for the boundary value problems u′′ λa t f u 0, t ∈ 0, 1 , u′ 0 0, u 1 0 is achieved, where λ is a positive parameter. Here the function f : 0,∞ → 0,∞ is C2 and satisfies f 0 f s 0, f u > 0 for u ∈ 0, s ∪ s,∞ for some s ∈ 0,∞ . Moreover, f is asymptotically linear and f ′′ can change sign only once. The weight function a : 0, 1 → 0,∞ is C2 and satisfies a′ t < 0, 3 a′ t 2 < 2a t a′′ t for t ∈ 0, 1 . Using bifurcation techniques, we obtain the exact number of positive solutions of the problem under consideration for λ lying in various intervals in R. Moreover, we indicate how to extend the result to the general case.