A uniqueness result for a Schrödinger–Poisson system with strong singularity
In this paper, we consider the following Schrödinger–Poisson system with strong singularity $$\begin{cases} -\Delta{u}+\phi u=f(x)u^{-\gamma}, & x\in \Omega,\\ -\Delta{\phi}=u^2, & x\in\Omega,\\ u>0, & x\in\Omega,\\ u=\phi=0, & x\in\partial\Omega, \end{cases}$$ where $\Omega\subs...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2019-11-01
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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=7605 |
| Summary: | In this paper, we consider the following Schrödinger–Poisson system with strong singularity
$$\begin{cases}
-\Delta{u}+\phi u=f(x)u^{-\gamma}, & x\in \Omega,\\
-\Delta{\phi}=u^2, & x\in\Omega,\\
u>0, & x\in\Omega,\\
u=\phi=0, & x\in\partial\Omega,
\end{cases}$$
where $\Omega\subset \mathbb{R}^3$ is a smooth bounded domain, $\gamma>1$, $f\in L^1(\Omega)$ is a positive function (i.e. $f(x)>0$ a.e. in $\Omega$). A necessary and sufficient condition on the existence and uniqueness of positive weak solution of the system is obtained. The results supplement the main conclusions in recent literature. |
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| ISSN: | 1417-3875 1417-3875 |