Sufficient conditions for exact bifurcation curves in Minkowski curvature problems and their applications

Shao-Yuan Huang; Wei-Hsun Lee; Shao-Yuan Huang; Wei-Hsun Lee

doi:10.3934/era.2025103

Electronic Research Archive

2025, Volume 33, Issue 4: 2325-2351. doi: 10.3934/era.2025103

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Research article

Sufficient conditions for exact bifurcation curves in Minkowski curvature problems and their applications

Shao-Yuan Huang ^,,
Wei-Hsun Lee

Department of Mathematics and Information Education, National Taipei University of Education, Taipei 106, Taiwan

Received: 08 February 2025 Revised: 09 April 2025 Accepted: 14 April 2025 Published: 21 April 2025

In this paper, we continued and extended the work in ^[1,2,3] by establishing sufficient conditions to determine the exact shape of the bifurcation curve of positive solutions for the Minkowski curvature problem
$ \left\{ \begin{array}{l} -\left( u^{\prime }/\sqrt{1-{u^{\prime }}^{2}}\right) ^{\prime } = \lambda f(u), \text{ in }\left( -L, L\right) , \\ u(-L) = u(L) = 0, \end{array} \right. $
where $ \lambda, L > 0 $ and $ f\in C^{2}(0, \infty) $. Notice that we allow $ f(0^{+}) = -\infty $. To illustrate the applicability of these results, we presented some examples, such as the diffusive logistic equation with a Holling type-Ⅱ functional response.
- positive solution,
- bifurcation curve,
- Minkowski curvature problem,
- diffusive logistic equation
Citation: Shao-Yuan Huang, Wei-Hsun Lee. Sufficient conditions for exact bifurcation curves in Minkowski curvature problems and their applications[J]. Electronic Research Archive, 2025, 33(4): 2325-2351. doi: 10.3934/era.2025103

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Abstract

In this paper, we continued and extended the work in ^[1,2,3] by establishing sufficient conditions to determine the exact shape of the bifurcation curve of positive solutions for the Minkowski curvature problem

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

where $ \lambda, L > 0 $ and $ f\in C^{2}(0, \infty) $. Notice that we allow $ f(0^{+}) = -\infty $. To illustrate the applicability of these results, we presented some examples, such as the diffusive logistic equation with a Holling type-Ⅱ functional response.

References

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