Article
MSC:
35A01,
35B09,
35J60,
35J62,
35J65,
35J75,
92D25 | MR 3232629 | Zbl 06362225 | DOI: 10.1007/s10492-014-0053-7
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Keywords:
population biology; infinite semipositone; sub-supersolution
population biology; infinite semipositone; sub-supersolution
Summary:
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 $, $1<p<N$, $0\leq \alpha < {(N-p)}/{p}$, $\gamma \in (0,1)$, 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.
References:
[1] Atkinson, C., El-Ali, K.: Some boundary value problems for the Bingham model. J. Non-Newtonian Fluid Mech. 41 (1992), 339-363. DOI 10.1016/0377-0257(92)87006-W | Zbl 0747.76012
[2] Bueno, H., Ercole, G., Ferreira, W., Zumpano, A.: Existence and multiplicity of positive solutions for the $p$-Laplacian with nonlocal coefficient. J. Math. Anal. Appl. 343 (2008), 151-158. DOI 10.1016/j.jmaa.2008.01.001 | MR 2409464 | Zbl 1141.35029
[3] Caffarelli, L., Kohn, R., Nirenberg, L.: First order interpolation inequalities with weights. Compos. Math. 53 (1984), 259-275. MR 0768824 | Zbl 0563.46024
[4] Cañada, A., Drábek, P., Gámez, J. L.: Existence of positive solutions for some problems with nonlinear diffusion. Trans. Am. Math. Soc. 349 (1997), 4231-4249. DOI 10.1090/S0002-9947-97-01947-8 | MR 1422596 | Zbl 0884.35039
[5] Cantrell, R. S., Cosner, C.: Spatial Ecology via Reaction-Diffusion Equations. Wiley Series in Mathematical and Computational Biology Wiley, Chichester (2003). MR 2191264 | Zbl 1059.92051
[6] Cîrstea, F., Motreanu, D., Rădulescu, V.: Weak solutions of quasilinear problems with nonlinear boundary condition. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 43 (2001), 623-636. DOI 10.1016/S0362-546X(99)00224-2 | MR 1804861 | Zbl 0972.35038
[7] Drábek, P., Hernández, J.: Existence and uniqueness of positive solutions for some quasilinear elliptic problems. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 44 (2001), 189-204. DOI 10.1016/S0362-546X(99)00258-8 | MR 1816658 | Zbl 0991.35035
[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). Zbl 0919.00053
[9] Drábek, P., Rasouli, S. H.: A quasilinear eigenvalue problem with Robin conditions on the non-smooth domain of finite measure. Z. Anal. Anwend. 29 (2010), 469-485. DOI 10.4171/ZAA/1419 | MR 2735484 | Zbl 1202.35149
[10] Fang, F., Liu, S.: Nontrivial solutions of superlinear $p$-Laplacian equations. J. Math. Anal. Appl. 351 (2009), 138-146. DOI 10.1016/j.jmaa.2008.09.064 | MR 2472927 | Zbl 1161.35016
[11] Lee, E. K., Shivaji, R., Ye, J.: Positive solutions for infinite semipositone problems with falling zeros. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72 (2010), 4475-4479. DOI 10.1016/j.na.2010.02.022 | MR 2639195 | Zbl 1190.35095
[12] Miyagaki, O. H., Rodrigues, R. S.: On positive solutions for a class of singular quasilinear elliptic systems. J. Math. Anal. Appl. 334 (2007), 818-833. DOI 10.1016/j.jmaa.2007.01.018 | MR 2338630 | Zbl 1155.35024
[13] Murray, J. D.: Mathematical Biology, Vol. 1: An Introduction. 3rd ed. Interdisciplinary Applied Mathematics 17 Springer, New York (2002). MR 1908418 | Zbl 1006.92001
[14] Rasouli, S. H., Afrouzi, G. A.: The Nehari manifold for a class of concave-convex elliptic systems involving the $p$-Laplacian and nonlinear boundary condition. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73 (2010), 3390-3401. DOI 10.1016/j.na.2010.07.021 | MR 2680032 | Zbl 1200.35103
[15] Smoller, J., Wasserman, A.: Global bifurcation of steady-state solutions. J. Differ. Equations 39 (1981), 269-290. DOI 10.1016/0022-0396(81)90077-2 | MR 0607786 | Zbl 0425.34028
[16] Xuan, B.: The eigenvalue problem for a singular quasilinear elliptic equation. Electron. J. Differ. Equ. (electronic only) 2004 (2004), Paper No. 16. MR 2036200 | Zbl 1217.35131
[17] Xuan, B.: The solvability of quasilinear Brezis-Nirenberg-type problems with singular weights. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 62 (2005), 703-725. DOI 10.1016/j.na.2005.03.095 | MR 2149911 | Zbl 1130.35061
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