Global bifurcation of positive solutions for a superlinear $p$-Laplacian system
We are concerned with the principal eigenvalue of \begin{equation*} \begin{cases} -\Delta_p u= \lambda\theta_1\varphi_p(v), &x\in \Omega,\\ -\Delta_p v= \lambda\theta_2\varphi_p(u), &x\in \Omega,\\ u=0=v,\ &x\in\partial\Omega \end{cases} \tag{P} \end{equation*...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2024-08-01
|
| 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=10984 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1841533480229404672 |
|---|---|
| author | Lijuan Yang Ruyun Ma |
| author_facet | Lijuan Yang Ruyun Ma |
| author_sort | Lijuan Yang |
| collection | DOAJ |
| description | We are concerned with the principal eigenvalue of
\begin{equation*}
\begin{cases}
-\Delta_p u= \lambda\theta_1\varphi_p(v), &x\in \Omega,\\
-\Delta_p v= \lambda\theta_2\varphi_p(u), &x\in \Omega,\\
u=0=v,\ &x\in\partial\Omega
\end{cases}
\tag{P}
\end{equation*}
and the global structure of positive solutions for the system
\begin{equation*}
\begin{cases}
-\Delta_p u= \lambda f(v), &x\in \Omega,\\
-\Delta_p v= \lambda g(u), &x\in \Omega,\\
u=0=v,\ &x\in\partial\Omega,
\end{cases}
\tag{Q}
\end{equation*}
where $\varphi_p(s)=|s|^{p-2}s$, $\Delta_p s=\text{div}(|\nabla s|^{p-2}\nabla s)$, $\lambda>0$ is a parameter, $\Omega\subset\mathbb{R}^N$, $N> 2$, is a bounded domain with smooth boundary $\partial\Omega$, $f,g:\mathbb{R}\to(0,\infty)$ are continuous functions with $p$-superlinear growth at infinity. We obtain the principal eigenvalue of $(P)$ by using a nonlinear Krein–Rutman theorem and the unbounded branch of positive solutions for $(Q)$ via bifurcation technology. |
| format | Article |
| id | doaj-art-49e86df053dd4901942323972733d66f |
| institution | Kabale University |
| issn | 1417-3875 |
| language | English |
| publishDate | 2024-08-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj-art-49e86df053dd4901942323972733d66f2025-01-15T21:24:58ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752024-08-0120244511910.14232/ejqtde.2024.1.4510984Global bifurcation of positive solutions for a superlinear $p$-Laplacian systemLijuan Yang0Ruyun MaNorthwest Normal University, Lanzhou, P. R. ChinaWe are concerned with the principal eigenvalue of \begin{equation*} \begin{cases} -\Delta_p u= \lambda\theta_1\varphi_p(v), &x\in \Omega,\\ -\Delta_p v= \lambda\theta_2\varphi_p(u), &x\in \Omega,\\ u=0=v,\ &x\in\partial\Omega \end{cases} \tag{P} \end{equation*} and the global structure of positive solutions for the system \begin{equation*} \begin{cases} -\Delta_p u= \lambda f(v), &x\in \Omega,\\ -\Delta_p v= \lambda g(u), &x\in \Omega,\\ u=0=v,\ &x\in\partial\Omega, \end{cases} \tag{Q} \end{equation*} where $\varphi_p(s)=|s|^{p-2}s$, $\Delta_p s=\text{div}(|\nabla s|^{p-2}\nabla s)$, $\lambda>0$ is a parameter, $\Omega\subset\mathbb{R}^N$, $N> 2$, is a bounded domain with smooth boundary $\partial\Omega$, $f,g:\mathbb{R}\to(0,\infty)$ are continuous functions with $p$-superlinear growth at infinity. We obtain the principal eigenvalue of $(P)$ by using a nonlinear Krein–Rutman theorem and the unbounded branch of positive solutions for $(Q)$ via bifurcation technology.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10984$p$-laplacianprincipal eigenvaluepositive solutionsbifurcation |
| spellingShingle | Lijuan Yang Ruyun Ma Global bifurcation of positive solutions for a superlinear $p$-Laplacian system Electronic Journal of Qualitative Theory of Differential Equations $p$-laplacian principal eigenvalue positive solutions bifurcation |
| title | Global bifurcation of positive solutions for a superlinear $p$-Laplacian system |
| title_full | Global bifurcation of positive solutions for a superlinear $p$-Laplacian system |
| title_fullStr | Global bifurcation of positive solutions for a superlinear $p$-Laplacian system |
| title_full_unstemmed | Global bifurcation of positive solutions for a superlinear $p$-Laplacian system |
| title_short | Global bifurcation of positive solutions for a superlinear $p$-Laplacian system |
| title_sort | global bifurcation of positive solutions for a superlinear p laplacian system |
| topic | $p$-laplacian principal eigenvalue positive solutions bifurcation |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10984 |
| work_keys_str_mv | AT lijuanyang globalbifurcationofpositivesolutionsforasuperlinearplaplaciansystem AT ruyunma globalbifurcationofpositivesolutionsforasuperlinearplaplaciansystem |