Multiple Positive Solutions for a Class of Semipositone Neumann Two Point Boundary Value Problems

We consider the two point Neumann boundary value problem −u″ = (x) = λƒ(u(x)); x ∈ (0, 1)u′(0) = 0 = u′(l) where λ is a positive parameter, ƒ ∈ C2[0, ∞), ƒ′(u) > 0 for u > 0, and for some β > 0, ƒ(u) < 0 for u ∈ [0, β) (semipositone) and ƒ′(u) > 0 for u > β. We discuss existence and multiplicity results for positive solutions. In particular, we prove that if the set S = (π2n2/ƒ′(β), θ2/−2F(β)), where n ∈ N, F(u) = ∫u0ƒ(s) ds and θ is the unique positive zero of F, is nonempty, then there exist at least 2n + 1 positive solutions for each λ ∈ S. Furthermore, if ƒ″ > 0 on [0, β) and ƒ″ < 0 on (β, ∞), then we prove that there are exactly 2n + 1 positive solutions for each λ ∈ S. We also discuss examples to which our results apply