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 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 , ) 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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  1. Atkinson, C., El-Ali, K., 10.1016/0377-0257(92)87006-W, J. Non-Newtonian Fluid Mech. 41 (1992), 339-363. (1992) Zbl0747.76012DOI10.1016/0377-0257(92)87006-W
  2. Bueno, H., Ercole, G., Ferreira, W., Zumpano, A., 10.1016/j.jmaa.2008.01.001, J. Math. Anal. Appl. 343 (2008), 151-158. (2008) Zbl1141.35029MR2409464DOI10.1016/j.jmaa.2008.01.001
  3. Caffarelli, L., Kohn, R., Nirenberg, L., First order interpolation inequalities with weights, Compos. Math. 53 (1984), 259-275. (1984) Zbl0563.46024MR0768824
  4. Cañada, A., Drábek, P., Gámez, J. L., 10.1090/S0002-9947-97-01947-8, Trans. Am. Math. Soc. 349 (1997), 4231-4249. (1997) Zbl0884.35039MR1422596DOI10.1090/S0002-9947-97-01947-8
  5. Cantrell, R. S., Cosner, C., Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology Wiley, Chichester (2003). (2003) Zbl1059.92051MR2191264
  6. Cîrstea, F., Motreanu, D., Rădulescu, V., 10.1016/S0362-546X(99)00224-2, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 43 (2001), 623-636. (2001) Zbl0972.35038MR1804861DOI10.1016/S0362-546X(99)00224-2
  7. Drábek, P., Hernández, J., 10.1016/S0362-546X(99)00258-8, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 44 (2001), 189-204. (2001) Zbl0991.35035MR1816658DOI10.1016/S0362-546X(99)00258-8
  8. Drábek, P., Krejčí, P., (eds.), P. Takáč, Nonlinear Differential Equations. Proceedings of talks given at the seminar in differential equations, Chvalatice, Czech Republic, June 29--July 3, 1998, Chapman & Hall/CRC Research Notes in Mathematics 404 Chapman & Hall/CRC, Boca Raton (1999). (1999) Zbl0919.00053
  9. Drábek, P., Rasouli, S. H., 10.4171/ZAA/1419, Z. Anal. Anwend. 29 (2010), 469-485. (2010) Zbl1202.35149MR2735484DOI10.4171/ZAA/1419
  10. Fang, F., Liu, S., 10.1016/j.jmaa.2008.09.064, J. Math. Anal. Appl. 351 (2009), 138-146. (2009) Zbl1161.35016MR2472927DOI10.1016/j.jmaa.2008.09.064
  11. Lee, E. K., Shivaji, R., Ye, J., 10.1016/j.na.2010.02.022, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72 (2010), 4475-4479. (2010) Zbl1190.35095MR2639195DOI10.1016/j.na.2010.02.022
  12. Miyagaki, O. H., Rodrigues, R. S., 10.1016/j.jmaa.2007.01.018, J. Math. Anal. Appl. 334 (2007), 818-833. (2007) Zbl1155.35024MR2338630DOI10.1016/j.jmaa.2007.01.018
  13. Murray, J. D., Mathematical Biology, Vol. 1: An Introduction. 3rd ed, Interdisciplinary Applied Mathematics 17 Springer, New York (2002). (2002) Zbl1006.92001MR1908418
  14. Rasouli, S. H., Afrouzi, G. A., 10.1016/j.na.2010.07.021, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73 (2010), 3390-3401. (2010) Zbl1200.35103MR2680032DOI10.1016/j.na.2010.07.021
  15. Smoller, J., Wasserman, A., 10.1016/0022-0396(81)90077-2, J. Differ. Equations 39 (1981), 269-290. (1981) Zbl0425.34028MR0607786DOI10.1016/0022-0396(81)90077-2
  16. Xuan, B., The eigenvalue problem for a singular quasilinear elliptic equation, Electron. J. Differ. Equ. (electronic only) 2004 (2004), Paper No. 16. (2004) Zbl1217.35131MR2036200
  17. Xuan, B., 10.1016/j.na.2005.03.095, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 62 (2005), 703-725. (2005) Zbl1130.35061MR2149911DOI10.1016/j.na.2005.03.095

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