### A population biological model with a singular nonlinearity

We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form $$\left\{\begin{array}{c}-{\mathrm{div}\left(\right|x|}^{-\alpha p}{\left|\nabla u\right|}^{p-2}{\nabla u)=|x|}^{-(\alpha +1)p+\beta}\left(a{u}^{p-1}-f\left(u\right)-{\displaystyle \frac{c}{{u}^{\gamma}}}\right),\phantom{\rule{1.0em}{0ex}}x\in \Omega ,\hfill \\ u=0,\phantom{\rule{1.0em}{0ex}}x\in \partial \Omega ,\hfill \end{array}\right.$$ where $\Omega $ is a bounded smooth domain of ${\mathbb{R}}^{N}$ with $0\in \Omega $, $1<p<N$, $0\le \alpha <(N-p)/p$, $\gamma \in (0,1)$, and $a$, $\beta $, $c$ and $\lambda $ are positive parameters. Here $f:[0,\infty )\to \mathbb{R}$ 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...