A global bifurcation theorem for a multiparameter positone problem and its application to the one-dimensional perturbed Gelfand problem
We study the global bifurcation and exact multiplicity of positive solutions for \begin{equation*} \begin{cases} u^{\prime \prime }(x)+\lambda f_{\varepsilon }(u)=0\text{,}\; \;-1<x<1\text{,} \\ u(-1)=u(1)=0\text{,} \end{cases} \end{equation*} where $\lambda >0$ is a bifurcation parameter,...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2019-12-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=7633 |
| _version_ | 1827951126910074880 |
|---|---|
| author | Shao-Yuan Huang Kuo-Chih Hung Shin-Hwa Wang |
| author_facet | Shao-Yuan Huang Kuo-Chih Hung Shin-Hwa Wang |
| author_sort | Shao-Yuan Huang |
| collection | DOAJ |
| description | We study the global bifurcation and exact multiplicity of positive solutions for
\begin{equation*}
\begin{cases}
u^{\prime \prime }(x)+\lambda f_{\varepsilon }(u)=0\text{,}\; \;-1<x<1\text{,} \\
u(-1)=u(1)=0\text{,}
\end{cases}
\end{equation*}
where $\lambda >0$ is a bifurcation parameter, $\varepsilon \in \Theta $ is an evolution parameter, and $\Theta \equiv \left( \sigma _{1},\sigma_{2}\right) $ is an open interval with $0\leq \sigma _{1}<\sigma _{2}\leq \infty $. Under some suitable hypotheses on $f_{\varepsilon }$, we prove that there exists $\varepsilon _{0}\in \Theta $ such that, on the $(\lambda,\|u\|_{\infty })$-plane, the bifurcation curve is S-shaped for $\sigma_{1}<\varepsilon <\varepsilon _{0}$ and is monotone increasing for $\varepsilon _{0}\leq \varepsilon <\sigma _{2}$. We give an application to prove global bifurcation of bifurcation curves for the one-dimensional perturbed Gelfand problem. |
| first_indexed | 2024-04-09T13:36:55Z |
| format | Article |
| id | doaj.art-16c137d32fd64b86b6c5a0098b4a861e |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:36:55Z |
| publishDate | 2019-12-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-16c137d32fd64b86b6c5a0098b4a861e2023-05-09T07:53:10ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752019-12-0120199912510.14232/ejqtde.2019.1.997633A global bifurcation theorem for a multiparameter positone problem and its application to the one-dimensional perturbed Gelfand problemShao-Yuan Huang0Kuo-Chih Hung1Shin-Hwa Wang2Department of Mathematics and Information Education, National Taipei University of Education, Taipei 106, TaiwanFundamental General Education Center, National Chin-Yi University of Technology, TaiwanDepartment of Mathematics, National Tsing Hua University, Hsinchu, TaiwanWe study the global bifurcation and exact multiplicity of positive solutions for \begin{equation*} \begin{cases} u^{\prime \prime }(x)+\lambda f_{\varepsilon }(u)=0\text{,}\; \;-1<x<1\text{,} \\ u(-1)=u(1)=0\text{,} \end{cases} \end{equation*} where $\lambda >0$ is a bifurcation parameter, $\varepsilon \in \Theta $ is an evolution parameter, and $\Theta \equiv \left( \sigma _{1},\sigma_{2}\right) $ is an open interval with $0\leq \sigma _{1}<\sigma _{2}\leq \infty $. Under some suitable hypotheses on $f_{\varepsilon }$, we prove that there exists $\varepsilon _{0}\in \Theta $ such that, on the $(\lambda,\|u\|_{\infty })$-plane, the bifurcation curve is S-shaped for $\sigma_{1}<\varepsilon <\varepsilon _{0}$ and is monotone increasing for $\varepsilon _{0}\leq \varepsilon <\sigma _{2}$. We give an application to prove global bifurcation of bifurcation curves for the one-dimensional perturbed Gelfand problem.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=7633global bifurcationmultiparameter problems-shaped bifurcation curveexact multiplicitypositive solution |
| spellingShingle | Shao-Yuan Huang Kuo-Chih Hung Shin-Hwa Wang A global bifurcation theorem for a multiparameter positone problem and its application to the one-dimensional perturbed Gelfand problem Electronic Journal of Qualitative Theory of Differential Equations global bifurcation multiparameter problem s-shaped bifurcation curve exact multiplicity positive solution |
| title | A global bifurcation theorem for a multiparameter positone problem and its application to the one-dimensional perturbed Gelfand problem |
| title_full | A global bifurcation theorem for a multiparameter positone problem and its application to the one-dimensional perturbed Gelfand problem |
| title_fullStr | A global bifurcation theorem for a multiparameter positone problem and its application to the one-dimensional perturbed Gelfand problem |
| title_full_unstemmed | A global bifurcation theorem for a multiparameter positone problem and its application to the one-dimensional perturbed Gelfand problem |
| title_short | A global bifurcation theorem for a multiparameter positone problem and its application to the one-dimensional perturbed Gelfand problem |
| title_sort | global bifurcation theorem for a multiparameter positone problem and its application to the one dimensional perturbed gelfand problem |
| topic | global bifurcation multiparameter problem s-shaped bifurcation curve exact multiplicity positive solution |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=7633 |
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