Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application

Shao-Yuan Huang

doi:10.3934/cpaa.2018061

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2018, Volume 17, Issue 3: 1271-1294. Doi: 10.3934/cpaa.2018061

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Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application

Shao-Yuan Huang^,

Center for General Education, National Formosa University, Yunlin 632, Taiwan

* Corresponding author: Shao-Yuan Huang
* Corresponding author: Shao-Yuan Huang

Received: June 30, 2016

Revised: April 30, 2017

Published: May 2018

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Abstract

In this paper, we discuss exact multiplicity and bifurcation curves of positive solutions of the one-dimensional Minkowski-curvature problem

$\begin{equation*}\left\{\begin{array}{l}\left[ -u^{\prime }/\sqrt{1-u^{\prime 2}}\right] ^{\prime }=\lambda f(u),\,\,\,\,\,\,-L < x < L, \\u(-L)=u(L)=0,\end{array}\right.\end{equation*} $

where $λ >0$ is a bifurcation parameter, $L>0$ is an evolution parameter, $f∈ C[0, ∞)\cap C^{2}(0, ∞), $ $f(u)>0$ for $u>0$ , and $f^{\prime \prime }(u)$ is not sign-changing on $\left( 0,\infty \right)$ .We find that if $f^{\prime \prime }(u)≤q 0$ for $u>0$ , the shapes of bifurcation curves are monotone increasing for $L>0$, and if $f^{\prime \prime }(u)>0$ for $u>0$ and $f(u)$ satisfies some suitable hypotheses, the shapes of bifurcation curves has three possibilities. Furthermore, we study, in the $(\lambda ,L,{\left\| u \right\|_\infty })$ -space, the shapes and structures of the bifurcation surfaces. Finally, we give an application for this problem with a nonlinear term $f(u) = u^{p}+u^{q}$ where $q≥p>0$ satisfy some conditions.

Keywords:

Mathematics Subject Classification: Primary: 34B15; Secondary: 34B18.

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Figure 1. Graphs of bifurcation curves $ S_{L}$ of (1). (ⅰ) hypotheses of Theorem 2.1 (ⅰ) hold. (ⅱ) hypotheses of Theorem 2.1(ⅱ) hold. (ⅲ) hypotheses of Theorem 2.1(ⅲ) hold. (ⅳ) hypotheses of Theorem 2.1(ⅳ) hold. (ⅴ) hypotheses of Theorem 2.2 hold.

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Figure 2. Graphs of bifurcation surface $\Gamma $ of (1). (ⅰ) hypotheses of Theorem 2.1(ⅰ) hold. (ⅱ) hypotheses of Theorem 2.1(ⅱ) hold. (ⅲ) hypotheses of Theorem 2.1 (ⅲ) hold. (ⅳ) hypotheses of Theorem 2.1(ⅳ) hold. (ⅴ) hypotheses of Theorem 2.2 hold.

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Figure 3. Graphs of bifurcation set $\tilde{\Sigma}$ and $\bar{\Sigma}\equiv \{(\lambda, \frac{\pi }{2\sqrt{2\lambda \eta }}):\lambda >0\}$. (ⅰ) hypotheses of Theorem 2.1(ⅳ) hold. (ⅱ) hypotheses of Theorem 2.1(ⅲ) hold. (ⅲ) hypotheses of Theorem 2.2 hold.

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Figure 4. Numerical simulations of bifurcation surfaces $\Gamma $ of ( 4). (ⅰ) $0<p<q\leq 1$ or $0<p = q<1.$ (ⅱ) $p = q = 1$. (ⅲ) $1 = p<q\leq 2$. (ⅳ) $1<p\leq q\leq \left( 2+\sqrt{3}\right) p-1-\sqrt{3}.$ (ⅴ) $p = 1, $ $q = 3, $ and $\hat{\lambda} = 2.$ The red curve $S_{\hat{L}}$ is monotone increasing.

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Figure 5. The conjecture of global bifurcation curves $S_{L}$ for ( 4).

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Figure 6. Graphs of $T_{\lambda }(\alpha )$ on $\left( 0, \infty \right) $. (ⅰ) ((C5) and (H)), or ((C6) and (H)), or ((C7) and (H)) holds. (ⅱ) (C4) and (H) hold. (ⅲ) (C9) holds and there exists $\check{ \lambda}>\hat{\lambda}$ such that (H) holds under $0< \lambda \leq \check{\lambda}$ where $\hat{\lambda}$ is defined in Theorem 2.2.

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Figure 7. The sets $\Theta _{1}, \Theta _{2}, ..., \Theta _{6}$ in $\left( 0, \infty \right) \times (0, 2.01].$

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