Structures and evolution of bifurcation diagrams for a multiparameter $p$-Laplacian Dirichlet problem
We study the multiparameter $p$-Laplacian Dirichlet problem \begin{equation*} \begin{cases} \left( \varphi _{p}(u^{\prime }(x))\right) ^{\prime }+\lambda (ku^{p-1}+\sum _{i=1}^{m}a_{i}u^{q_{i}})-\mu \sum _{j=1}^{n}b_{j}u^{r_{j}}=0,% \text{ }-1<x<1, \\ u(-1)=u(1)=0,% \end{cases} \end{equation*}...
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| Language: | English |
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University of Szeged
2024-11-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
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| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=11205 |
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| author | Tsung-Yi Hsieh Shin-Hwa Wang |
| author_facet | Tsung-Yi Hsieh Shin-Hwa Wang |
| author_sort | Tsung-Yi Hsieh |
| collection | DOAJ |
| description | We study the multiparameter $p$-Laplacian Dirichlet problem
\begin{equation*}
\begin{cases}
\left( \varphi _{p}(u^{\prime }(x))\right) ^{\prime }+\lambda (ku^{p-1}+\sum
_{i=1}^{m}a_{i}u^{q_{i}})-\mu \sum _{j=1}^{n}b_{j}u^{r_{j}}=0,%
\text{ }-1<x<1, \\
u(-1)=u(1)=0,%
\end{cases}
\end{equation*}
where $p>1,$ $\varphi _{p}(y)=\left \vert y\right \vert ^{p-2}y$, $\left(
\varphi _{p}(u^{\prime })\right) ^{\prime }$ is the one-dimensional $p$-Laplacian, $\lambda >0$ and $\mu \geq 0$ are two bifurcation parameters. We assume that $k\geq 0,$ $0<p-1<q_{1}<q_{2}<\cdot \cdot \cdot <q_{m}<r_{1}<r_{2}<\cdot \cdot \cdot <r_{n},$ $m,n\geq 1,$ $a_{1}=1,$ $a_{i}>0$ for $i=1,2,\dots,m$ and $b_{1}=1,$ $b_{j}>0$ for $j=1,2,\dots,n$. We mainly prove that, on the $(\lambda ,\left \Vert u\right \Vert _{\infty })$-plane, the bifurcation diagram consists of a strictly decreasing curve for $\mu =0,$ and always consists of a $\subset $-shaped curve for fixed $\mu >0$. We then study the structures and evolution of the bifurcation diagrams with varying $\mu \geq 0$. |
| format | Article |
| id | doaj-art-1f4f979f5ba44f53b637deea4559a80d |
| institution | Kabale University |
| issn | 1417-3875 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj-art-1f4f979f5ba44f53b637deea4559a80d2025-01-15T21:24:59ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752024-11-0120246712610.14232/ejqtde.2024.1.6711205Structures and evolution of bifurcation diagrams for a multiparameter $p$-Laplacian Dirichlet problemTsung-Yi Hsieh0Shin-Hwa Wanghttps://orcid.org/0000-0003-2956-0290Department of Mathematics, National Tsing Hua University, Hsinchu, TaiwanWe study the multiparameter $p$-Laplacian Dirichlet problem \begin{equation*} \begin{cases} \left( \varphi _{p}(u^{\prime }(x))\right) ^{\prime }+\lambda (ku^{p-1}+\sum _{i=1}^{m}a_{i}u^{q_{i}})-\mu \sum _{j=1}^{n}b_{j}u^{r_{j}}=0,% \text{ }-1<x<1, \\ u(-1)=u(1)=0,% \end{cases} \end{equation*} where $p>1,$ $\varphi _{p}(y)=\left \vert y\right \vert ^{p-2}y$, $\left( \varphi _{p}(u^{\prime })\right) ^{\prime }$ is the one-dimensional $p$-Laplacian, $\lambda >0$ and $\mu \geq 0$ are two bifurcation parameters. We assume that $k\geq 0,$ $0<p-1<q_{1}<q_{2}<\cdot \cdot \cdot <q_{m}<r_{1}<r_{2}<\cdot \cdot \cdot <r_{n},$ $m,n\geq 1,$ $a_{1}=1,$ $a_{i}>0$ for $i=1,2,\dots,m$ and $b_{1}=1,$ $b_{j}>0$ for $j=1,2,\dots,n$. We mainly prove that, on the $(\lambda ,\left \Vert u\right \Vert _{\infty })$-plane, the bifurcation diagram consists of a strictly decreasing curve for $\mu =0,$ and always consists of a $\subset $-shaped curve for fixed $\mu >0$. We then study the structures and evolution of the bifurcation diagrams with varying $\mu \geq 0$.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=11205bifurcation diagramevolutionpositive solution$p$-laplacian$\subset$-shaped bifurcation curvetime map |
| spellingShingle | Tsung-Yi Hsieh Shin-Hwa Wang Structures and evolution of bifurcation diagrams for a multiparameter $p$-Laplacian Dirichlet problem Electronic Journal of Qualitative Theory of Differential Equations bifurcation diagram evolution positive solution $p$-laplacian $\subset$-shaped bifurcation curve time map |
| title | Structures and evolution of bifurcation diagrams for a multiparameter $p$-Laplacian Dirichlet problem |
| title_full | Structures and evolution of bifurcation diagrams for a multiparameter $p$-Laplacian Dirichlet problem |
| title_fullStr | Structures and evolution of bifurcation diagrams for a multiparameter $p$-Laplacian Dirichlet problem |
| title_full_unstemmed | Structures and evolution of bifurcation diagrams for a multiparameter $p$-Laplacian Dirichlet problem |
| title_short | Structures and evolution of bifurcation diagrams for a multiparameter $p$-Laplacian Dirichlet problem |
| title_sort | structures and evolution of bifurcation diagrams for a multiparameter p laplacian dirichlet problem |
| topic | bifurcation diagram evolution positive solution $p$-laplacian $\subset$-shaped bifurcation curve time map |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=11205 |
| work_keys_str_mv | AT tsungyihsieh structuresandevolutionofbifurcationdiagramsforamultiparameterplaplaciandirichletproblem AT shinhwawang structuresandevolutionofbifurcationdiagramsforamultiparameterplaplaciandirichletproblem |