A population biological model with a singular nonlinearity

We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form $\{ {gathered} - div(|x|- α p |∇ u|p - 2 ∇ u) = |x|- (α + 1)p + β ( {aup - 1 - f(u) - {c} {{u^γ }}} ),x ∈ Ω , \\ u = 0,x ∈ ∂ Ω , \\ {gathered} . $ where Ω is a bounded smooth domain of ℝ N with 0 ∈ Ω, 1 < p < N, 0 ⩽ α < (N − p)/p, γ ∈ (0, 1), and a, β, c and λ are positive parameters. Here f: [0,∞) → ℝ is a continuous function. This model arises in the studies of population biology of one species with u representing the concentration of the species. We discuss the existence of a positive solution when f satisfies certain additional conditions. We use the method of sub-supersolutions to establish our results.