A Population biological model with a singular nonlinearity and Caffarelli-Kohn-Nirenberg exponents

Document Type : Research Articles

Author

Ayatollah Amoli Branch, Islamic Azad University

Abstract

We consider the existence of positive solutions of singular nonlinear semipositone problem of the form \[\left\{\begin{array}{ll} -div(|x|^{-\alpha p}|\nabla u|^{p-2}\nabla u)=|x|^{-(\alpha+1)p+\beta}(au^{p-1}-bu^{r}-f(u)-\frac{c}{u^{\gamma}}), & x\in\Omega, \\ u=0, & x\in\partial\Omega, \end{array}\right.\]
where $\Omega$ is a bounded domain in ${\mathbb{R}}^{N}$ with smooth boundary $\partial\Omega$, $1<p<N$, $0\leq\alpha<\frac{N-p}{p}$,$ r>p-1 $,$\gamma\in (0,1)$, $a,b,c,\beta$ are positive parameters, and $f:[0,+\infty)\to{\mathbb{R}}$ 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 obtain our results via the method of sub and supersolutions.

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