A population biological model with a singular nonlinearity

Abstract

We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form

$\left\{ \begin{gathered} - div(|x|^{ - \alpha p} |\nabla u|^{p - 2} \nabla u) = |x|^{ - (\alpha + 1)p + \beta } \left( {au^{p - 1} - f(u) - \frac{c} {{u^\gamma }}} \right),x \in \Omega , \hfill \\ u = 0,x \in \partial \Omega , \hfill \\ \end{gathered} \right. $

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.