Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications

Shao-Yuan Huang

doi:10.3934/dcds.2019142

Article Contents

2019, Volume 39, Issue 6: 3443-3462. Doi: 10.3934/dcds.2019142

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Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications

Shao-Yuan Huang

Center for General Education, Ming Chi University of Technology, New Taipei City 24301, Taiwan

Received: July 31, 2018

Published: February 2019

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Abstract

In this paper, we study the classification and evolution of bifurcation curves of positive solutions for one-dimensional Minkowski-curvature problem

$\left\{ \begin{array}{*{35}{l}} \begin{align} & -{{\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{align} \\\end{array} \right. $

where $ \lambda >0 $ is a bifurcation parameter, $ L>0 $ is an evolution parameter, $ f\in C[0, \infty )\cap C^{2}(0, \infty ) $ and there exists $ \beta >0 $ such that $ \left( \beta -z\right) f(z)>0 $ for $ z\neq \beta $. In particular, we find that the bifurcation curve $ S_{L} $ is monotone increasing for all $ L>0 $ when $ f(u)/u $ is of Logistic type, and is either $ \subset $-shaped or S-shaped for large $ L>0 $ when $ f(u)/u $ is of weak Allee effect type. Finally, we can apply these results to obtain the global bifurcation diagrams in some important applications including ecosystem model.

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Mathematics Subject Classification: 34B15, 34B18, 34C23, 74G35.

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Figure 1. ⅰ) monotone increasing. (ⅱ) $ \subset $-shaped. (ⅲ) S-shaped

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Figure 2. Graphs of bifurcation curve $ S_{L} $ of (1) with varying $ L>0 $. (ⅰ) (C2) and (H2) hold. (ⅱ) either (C4) or (C6) holds

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Figure 3. Graphs of bifurcation curve $S_{L}$ of (1) with varying $L>0$. (ⅰ) $S_{\mathring{L}}$ is monotone increasing. (ⅱ) $S_{% \mathring{L}}$ is not monotone increasing

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Figure 4. Graphs of bifurcation curve $ S_{L} $ of (1), (5). $ \phi _{1}, \phi _{2}\in C(\sqrt{27} , \infty ) $ satisfy $ \phi _{1}(K)>\phi _{2}(K) $ and $ \Phi (K, \phi _{1}(K)) = \Phi (K, \phi _{2}(K)) = 0. $

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Figure 5. Graphs of bifurcation curve S_L of (1), (6)

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Figure 6. Graphs of bifurcation curve S_L of (1), (7). (ⅰ) c = 0. (ⅱ) c > 0

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