On the logarithmic fractional Schrödinger–Poisson system with saddle-like potential
In this paper, we use variational methods to prove the existence of a positive solution for the following class of logarithmic fractional Schrödinger–Poisson system: \begin{equation*} \begin{cases} \epsilon^{2s}\left(-\Delta\right)^{s} u+V(x)u-\phi(x)u= u \log {u^{2}}&\quad\text{ in }\mathbb...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2024-07-01
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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=10967 |
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| Summary: | In this paper, we use variational methods to prove the existence of a positive solution for the following class of logarithmic fractional Schrödinger–Poisson system:
\begin{equation*}
\begin{cases}
\epsilon^{2s}\left(-\Delta\right)^{s} u+V(x)u-\phi(x)u= u \log {u^{2}}&\quad\text{ in }\mathbb{R}^{3}, \\
\epsilon^{2t}\left(-\Delta\right)^{t}\phi=|u|^{2}&\quad\text{ in }\mathbb{R}^{3},
\end{cases}
\end{equation*}
where $\epsilon>0$, $s,t\in(0,1)$, $\left(-\Delta\right)^{\alpha}$ is the fractional Laplacian and $V$ is a saddle-like potential. |
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| ISSN: | 1417-3875 |