Exact Multiplicity of Positive Solutions for a Quasilinear Boundary Value Problem with Density-Dependent Diffusion and Bistable Nonlinearity
DOI:
https://doi.org/10.22399/ijcesen.5257Keywords:
p-Laplacian, Positive solutions, p-Laplacian Time-map, Nonlinear diffusion, BistabilityAbstract
We investigate the exact number of positive solutions for a quasilinear Dirichlet problem with a density-dependent diffusion coefficient of the form and a bistable nonlinearity , where , , , , . Using the quadrature (time-map) method, we determine the exact multiplicity of positive solutions for all . For , we identify two critical values such that the problem admits zero, one, or exactly two solutions depending on . The paper also includes a physical interpretation of the parameters and a numerical illustration of the theoretical results for .
References
[1] Addou, I., & Benmezai, A. (1999). Boundary value problems for the one-dimensional p-Laplacian with even superlinearity. Electronic Journal of Differential Equations, 1999(9), 1-29.
[2] Addou, I., Bouguima, S. M., Benmezai, A., & Derhab, M. (2000). Exactness results for generalized Ambrosetti-Brezis-Cerami and related one-dimensional elliptic equations. Electronic Journal of Differential Equations, 66, 1-34.
[3] Derhab, M., & Megnafi, M. (2008). Exact number of positive solutions for a class of quasilinear boundary value problems. Communications on Applied Nonlinear Analysis, 15(4), 15-37.
[4] Derhab, M., & Sebbagh, H. (2012). Exact number of positive solutions for a class of quasilinear boundary value problems with a singular nonlinearity. Communications on Applied Nonlinear Analysis, 19(3), 103-125.
[5] Derhab, M., & Sebbagh, H. (2013). Exact number of positive solutions for quasilinear boundary value problems with p-convex nonlinearity. Filomat, 27, 499-513.
[6] Guedda, M., & Veron, L. (1988). Bifurcation phenomena associate to the p-Laplacian operator. Transactions of the American Mathematical Society, 310, 419-431.
[7] Aronson, D., Crandall, M. G., & Peletier, L. A. (1982). Stabilization of solutions of degenerate nonlinear diffusion problem. Nonlinear Analysis, 6, 1001-1022.
[8] Murray, J. D. (2002). Mathematical biology: I. An introduction (3rd ed.). Springer.
[9] Schaaf, R. (1990). Global solution branches of two-point boundary value problems. Lecture Notes in Mathematics 1458. Springer-Verlag.
[10] Bazighifan, O., Alshammari, N., Al-Ghafri, K. S., & Iambor, L. F. (2024). Differential equations of fourth-order with p-Laplacian-like operator: Oscillation theorems. Mathematics, 12(22), 1-10.
[11] Okubo, A., & Levin, S. (2001). Diffusion and ecological problems: modern perspectives (2nd ed., Vol. 14). Springer, New York
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2026 International Journal of Computational and Experimental Science and Engineering
This work is licensed under a Creative Commons Attribution 4.0 International License.
