Geodesic
A population biological model with a singular nonlinearity
Applications of Mathematics, Tome 59 (2014) no. 3, pp. 257-264

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We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form $$ \begin {cases} -{\rm div}(|x|^{-\alpha p}|\nabla u|^{p-2}\nabla u)=|x|^{-(\alpha +1)p+\beta } \Big (a u^{p-1}-f(u)-\dfrac {c}{u^{\gamma }}\Big ), \quad x\in \Omega ,\\ u=0, \quad x\in \partial \Omega , \end {cases} $$ where $\Omega $ is a bounded smooth domain of ${\mathbb R}^N$ with $0\in \Omega $, $1$, $0\leq \alpha {(N-p)}/{p}$, $\gamma \in (0,1)$, and $a$, $\beta $, $c$ and $\lambda $ are positive parameters. Here $f\colon [0,\infty )\to {\mathbb R}$ 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 to establish our results.
We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form $$ \begin {cases} -{\rm div}(|x|^{-\alpha p}|\nabla u|^{p-2}\nabla u)=|x|^{-(\alpha +1)p+\beta } \Big (a u^{p-1}-f(u)-\dfrac {c}{u^{\gamma }}\Big ), \quad x\in \Omega ,\\ u=0, \quad x\in \partial \Omega , \end {cases} $$ where $\Omega $ is a bounded smooth domain of ${\mathbb R}^N$ with $0\in \Omega $, $1$, $0\leq \alpha {(N-p)}/{p}$, $\gamma \in (0,1)$, and $a$, $\beta $, $c$ and $\lambda $ are positive parameters. Here $f\colon [0,\infty )\to {\mathbb R}$ 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 to establish our results.
DOI : 10.1007/s10492-014-0053-7
Classification : 35A01, 35B09, 35J60, 35J62, 35J65, 35J75, 92D25
Keywords: population biology; infinite semipositone; sub-supersolution 
Rasouli, Sayyed Hashem. A population biological model with a singular nonlinearity. Applications of Mathematics, Tome 59 (2014) no. 3, pp. 257-264. doi: 10.1007/s10492-014-0053-7
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     title = {A population biological model with a singular nonlinearity},
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     pages = {257--264},
     year = {2014},
     volume = {59},
     number = {3},
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     zbl = {06362225},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10492-014-0053-7/}
}
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