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An exact multiplicity result of positive solutions for the boundary value problems u + λ a (t) f (u) = 0, t ∈ (0,1), u (0) = 0, u (1) = 0 is achieved, where is a positive parameter. Here the function f: [0, ∞) → [0, ∞) is C 2 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 C 2 and satisfies a′ (t) 0, 3 (a′ (t)) 2 < 2 a (t) a′ (t) for t ∈[ 0,1 ]. Using bifurcation techniques, we obtain the exact number of positive solutions of the problem under consideration for lying in various intervals in R. Moreover, we indicate how to extend the result to the general case. © 2010 Yulian An and Hua Luo.

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