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June  2019, 39(6): 3443-3462. doi: 10.3934/dcds.2019142

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

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

Received  August 2018 Published  February 2019

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.
Citation: Shao-Yuan Huang. Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3443-3462. doi: 10.3934/dcds.2019142
References:

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References:
ⅰ) monotone increasing. (ⅱ) $\subset$-shaped. (ⅲ) S-shaped
Graphs of bifurcation curve $S_{L}$ of (1) with varying $L>0$. (ⅰ) (C2) and (H2) hold. (ⅱ) either (C4) or (C6) holds
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
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 $\Phi (K, \phi _{1}(K)) = \Phi (K, \phi _{2}(K)) = 0.$
Graphs of bifurcation curve SL of (1), (6)
Graphs of bifurcation curve SL of (1), (7). (ⅰ) c = 0. (ⅱ) c > 0
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