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A global bifurcation theorem for a positone multiparameter problem and its application

  • * Corresponding author: Shao-Yuan Huang

    * Corresponding author: Shao-Yuan Huang 
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  • We study the global bifurcation and exact multiplicity of positive solutions for the positone multiparameter problem

    $\left\{ \begin{align} &{{u}^{\prime \prime }}(x)+\lambda {{f}_{\varepsilon }}(u)=0\text{,}\ \ -1<x<1\text{,} \\ &u(-1)=u(1)=0\text{,} \\ \end{align} \right.$

    where λ > 0 is a bifurcation parameter and $\varepsilon >0$ is an evolution parameter. Under some suitable hypotheses on $f_{\varepsilon }(u)$, we prove that there exists $\tilde{\varepsilon}>0$ such that, on the $(λ ,||u||_{∞ })$-plane, the bifurcation curve is S-shaped for $0<\varepsilon <\tilde{\varepsilon}$ and is monotone increasing for $\varepsilon ≥ \tilde{\varepsilon}$. We give an application for this problem with a class of polynomial nonlinearities $f_{\varepsilon}(u)=-\varepsilon u^{p}+bu^{2}+cu+d\ $ of degree p≥ 3 and coefficients $\varepsilon ,b,d>0,$ c ≥ 0. Our results generalize those in Hung and Wang (Trans. Amer. Math. Soc. 365 (2013) 1933-1956.)

    Mathematics Subject Classification: Primary: 34B18; Secondary: 74G35.

    Citation:

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  • Figure 1.  Global bifurcation of bifurcation curves $S_{\varepsilon }$ of (1) with varying $\varepsilon >0$

    Figure 2.  The bifurcation surface $\Gamma $ with the fold curve $C_{\Gamma }$, and the projection of $C_{\Gamma }$ onto the $(\varepsilon ,\lambda )$-parameter plane. $B_{\Gamma }=B_{1}\cup B_{2}\cup \left\{ (\tilde{\varepsilon},\tilde{\lambda})\right\} $ is the bifurcation set

    Figure 3.  The projection of the bifurcation surface $C_{\Gamma }$ onto the $( \varepsilon ,\lambda )$-parameter plane. $B_{\Gamma }=B_{1}\cup B_{2}\cup \left\{ (\tilde{\varepsilon},\tilde{ \lambda})\right\} $ is the bifurcation set.

    Figure 4.  The evolution of time maps $T_{\varepsilon }(\alpha )$ for $\alpha \in (0,\beta _{\varepsilon })$ with varying $\varepsilon >0.$

    Figure 5.  The graph of $\phi _{\varepsilon }(u)$.

    Figure 6.  Three possible graphs of $\theta _{\varepsilon }(u)$. (ⅰ) $\theta _{\varepsilon }(\gamma _{ \varepsilon })\leq 0$. (ⅱ) $\theta _{\varepsilon }(\gamma _{\varepsilon })>0>\theta _{\varepsilon }(p_{2}(\varepsilon ))$. (ⅲ) $\theta _{\varepsilon }(\gamma _{\varepsilon })>\theta _{\varepsilon }(p_{2}(\varepsilon ))\geq 0$.

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