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*...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2024-08-01
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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=10984 |
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| Summary: | 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. |
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| ISSN: | 1417-3875 |