Multiple normalized solutions to the nonlinear Schrödinger–Poisson system with the $L^2$-subcritical growth
In this paper, we study the existence of multiple normalized solutions to the following Schrödinger–Poisson system with general nonlinearities: \begin{equation*} \begin{cases} -\varepsilon^2\Delta u+V(x)u+\phi u=f(u)+\lambda u & \hbox{in $\mathbb{R}^3$,} \\ -\varepsilon^2\Delta\phi=u^...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2024-10-01
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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=11101 |
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| Summary: | In this paper, we study the existence of multiple normalized solutions to the following Schrödinger–Poisson system with general nonlinearities:
\begin{equation*}
\begin{cases}
-\varepsilon^2\Delta u+V(x)u+\phi u=f(u)+\lambda u & \hbox{in $\mathbb{R}^3$,} \\
-\varepsilon^2\Delta\phi=u^2& \hbox{in $\mathbb{R}^3$,}\\
\int_{\mathbb{R}^3}|u|^2{\rm d}x=\varepsilon^3 a^2,\
\end{cases}
\end{equation*}
where $\varepsilon$, $a>0$, $\lambda\in\mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $V(x):\mathbb{R}^3 \rightarrow [0,\infty)$ is a continuous function, and $f$ is a differentiable function satisfying $L^2$-subcritical growth. Through using the minimization techniques and the Lusternik–Schnirelmann category, we prove that the numbers of normalized solutions are related to the topology of the set where the potential $V(x)$ attains its minimum value. |
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| ISSN: | 1417-3875 |