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

  • * Corresponding author: Shao-Yuan Huang

    * Corresponding author: Shao-Yuan Huang
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  • 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.

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

    Citation:

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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.

    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.

    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.

    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.

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

    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.

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

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