Abstract
We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form
{−div(|x|−αp|∇u|p−2∇u)=|x|−(α+1)p+β(aup−1−f(u)−cuγ),x∈Ω,u=0,x∈∂Ω,{−div(|x|−αp|∇u|p−2∇u)=|x|−(α+1)p+β(aup−1−f(u)−cuγ),x∈Ω,u=0,x∈∂Ω,
where Ω is a bounded smooth domain of ℝN 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