Multiplicity of positive solutions for Minkowski curvature problems arising from population models with constant-yield harvesting

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Abstract

This paper investigates the bifurcation curves and the multiplicity of positive solutions for a one-dimensional Minkowski curvature problem with constant-yield harvesting

$\left\{\begin{array}{l} -\left(u^{\prime}(x) / \sqrt{1-\left[u^{\prime}(x)\right]^{2}}\right)^{\prime}=\lambda g(u(x))-\mu, \text { for } x \in(-L, L), \\ u(-L)=u(L)=0, \end{array}\right.$

where $\lambda, L, \mu>0$ and there exist constants $\sigma>u_{0}>0$ such that $g \in C[0, \sigma] \cap C^{2}(0, \sigma), g(0)=g(\sigma)=0$, $g^{\prime}(u)>0 \text { on }\left(0, u_{0}\right), g^{\prime}\left(u_{0}\right)=0, \text { and } g^{\prime}(u)<0 \text { on }\left(u_{0}, \sigma\right)$. We first show that the bifurcation curve has a $\subset$-like shape and then provide additional sufficient conditions under which the curve is exactly $\subset$-shaped. These results yield the exact multiplicity of positive solutions. As an application to population and ecological models, we further consider the nonlinearity $g(u)=u^{p}\left(1-u^{q} / K\right)^{r}$, where $p, q, r, K>0$.

Keywords

positive solutions / bifurcation curves / Minkowski curvature problem

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References

[1] J. Ali, R. Shivaji, K. Wampler, Population models with diffusion, strong Allee effect and constant yield harvesting, J. Math. Anal. Appl., 352 (2009), 907-913. https://doi.org/10.1016/j.jmaa.2008.11.047

[2] P. Girão, H. Tehrani, Positive solutions to logistic type equations with harvesting, J. Differ. Equations, 247 (2009), 574-595. https://doi.org/10.1016/j.jde.2009.02.008

[3] J. Jiang, J. Shi, Bistability Dynamics in Some Structured Ecological Models, Spatial Ecology, CRC Press, Boca Raton, 2009.

[4] M. R. Myerscough, B. F. Gray, W. L. Hogarth, J. Norbury, An analysis of an ordinary differential equations model for a two species predator-prey system with harvesting and stocking, J. Math. Biol., 30 (1992), 389-411. https://doi.org/10.1007/BF00173294

[5] S. Oruganti, J. Shi, R. Shivaji, Diffusive logistic equation with constant yield harvesting. I. Steady states, Trans. Amer. Math. Soc., 354 (2002), 3601-3619. https://doi.org/10.1090/S0002-9947-02-03005-2

[6] S. Oruganti, J. Shi, R. Shivaji, Logistic equation with the p-Laplacian and constant yield harvesting, Abstr. Appl. Anal., 2004 (2004), 723-727. https://doi.org/10.1155/S1085337504311097

[7] C. Corsato, Mathematical Analysis of Some Differential Models Involving the Euclidean or the Minkowski Mean Curvature Operator, Ph.D thesis, University of Trieste, 2015.

[8] S. Y. Huang, Classification and evolution of bifurcation curves of semipositone problem with Minkowski curvature operator and its applications, J. Differ. Equations, 400 (2024), 278-313. https://doi.org/10.1016/j.jde.2024.04.026

[9] S. Y. Huang, W. H. Lee, Sufficient conditions for exact bifurcation curves in Minkowski curvature problems and their applications, Electron. Res. Arch., 33 (2025), 2325-2351. https://doi.org/10.3934/era.2025103

[10] K. C. Hung, Y. N. Suen, S. H. Wang, Structures and evolution of bifurcation diagrams for a one-dimensional diffusive generalized logistic problem with constant yield harvesting, J. Differ. Equations, 269 (2020), 3456-3488. https://doi.org/10.1016/j.jde.2020.03.001

[11] K. C. Hung, S. H. Wang, Structures and evolution of bifurcation diagrams for a one-dimensional diffusive generalized logistic problem with constant yield harvesting. II. Convex-concave and convex-concave-convex nonlinearities, J. Differ. Equations, 308 (2022), 1-39. https://doi.org/10.1016/j.jde.2021.10.047

[12] S. Y. Huang, Bifurcation curve diagrams for a diffusive generalized Logistic problem with Minkowski curvature operator and constant-yield harvesting, preprint, arXiv:2510.00362v2.

[13] S. Y. Huang, Bifurcation diagrams of positive solutions for one-dimensional Minkowski curvature problem and its applications, Discrete Contin. Dyn. Syst., 39 (2019), 3443-3462. https://doi.org/10.3934/dcds.2019142