Exact Multiplicity of Solutions for Classes of Semipositone Problems with Concave-convex Nonlinearity

where λ is a positive parameter. The nonlinearity f(u) is called semipositone if f(0) < 0. In this paper we will only consider the positive solutions of (1.1). Semipositone problems were introduced by Castro and Shivaji in [CS1], and they arise from various disciplines, like astrophysics and population dynamics. (see [CMS] for more details.) It is possible that (1.1) has non-negative solutions with interior zeros (see [CS1]). This is not the case when f(0) ≥ 0, where any non-negative solution of (1.1) is strictly positive in (0, 1). Note that any solution of (1.1) is symmetric with respect to any point x0 ∈ (−1, 1) such that u′(x0) = 0, so any positive solution of (1.1) is a reflection extension of a monotone decreasing solution of   u′′ + λf(u) = 0, x ∈ (0, 1), u′(0) = u(1) = 0, u′(x) < 0, x ∈ (0, 1). (1.2)