A population biological model with a singular nonlinearity

Sayyed Hashem Rasouli

Applications of Mathematics (2014)

  • Volume: 59, Issue: 3, page 257-264
  • ISSN: 0862-7940

Abstract

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We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form - div ( | x | - α p | ∇ u | p - 2 ∇ u ) = | x | - ( α + 1 ) p + β a u p - 1 - f ( u ) - c u γ , x ∈ Ω , u = 0 , x ∈ ∂ Ω , 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.

How to cite

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Rasouli, Sayyed Hashem. "A population biological model with a singular nonlinearity." Applications of Mathematics 59.3 (2014): 257-264. <http://eudml.org/doc/261135>.

@article{Rasouli2014,
abstract = {We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form \[ \{\left\lbrace \begin\{array\}\{ll\} -\{\rm div\}(|x|^\{-\alpha p\}|\nabla u|^\{p-2\}\nabla u)=|x|^\{-(\alpha +1)p+\beta \} \Big (a u^\{p-1\}-f(u)-\dfrac\{c\}\{u^\{\gamma \}\}\Big ), \quad x\in \Omega ,\\ u=0, \quad x\in \partial \Omega , \end\{array\}\right.\} \] 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 $f\colon [0,\infty )\rightarrow \{\mathbb \{R\}\}$ 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.},
author = {Rasouli, Sayyed Hashem},
journal = {Applications of Mathematics},
keywords = {population biology; infinite semipositone; sub-supersolution; quasilinear elliptic problem; singularities; upper and lower solution method; population biology},
language = {eng},
number = {3},
pages = {257-264},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A population biological model with a singular nonlinearity},
url = {http://eudml.org/doc/261135},
volume = {59},
year = {2014},
}

TY - JOUR
AU - Rasouli, Sayyed Hashem
TI - A population biological model with a singular nonlinearity
JO - Applications of Mathematics
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 3
SP - 257
EP - 264
AB - We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form \[ {\left\lbrace \begin{array}{ll} -{\rm div}(|x|^{-\alpha p}|\nabla u|^{p-2}\nabla u)=|x|^{-(\alpha +1)p+\beta } \Big (a u^{p-1}-f(u)-\dfrac{c}{u^{\gamma }}\Big ), \quad x\in \Omega ,\\ u=0, \quad x\in \partial \Omega , \end{array}\right.} \] 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 $f\colon [0,\infty )\rightarrow {\mathbb {R}}$ 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.
LA - eng
KW - population biology; infinite semipositone; sub-supersolution; quasilinear elliptic problem; singularities; upper and lower solution method; population biology
UR - http://eudml.org/doc/261135
ER -

References

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