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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University of Szeged
2024-10-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
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| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=11101 |
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| author | Siwei Wei Kaimin Teng |
| author_facet | Siwei Wei Kaimin Teng |
| author_sort | Siwei Wei |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-14bd9a871750475eb24fc054ee353885 |
| institution | Kabale University |
| issn | 1417-3875 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj-art-14bd9a871750475eb24fc054ee3538852025-01-15T21:24:59ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752024-10-0120246112110.14232/ejqtde.2024.1.6111101Multiple normalized solutions to the nonlinear Schrödinger–Poisson system with the $L^2$-subcritical growthSiwei Wei0Kaimin TengDepartment of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi, People's Republic of ChinaIn 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.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=11101schrödinger–poisson systemnormalized solutionslusternik–schnirelmann categoryvariational methods |
| spellingShingle | Siwei Wei Kaimin Teng Multiple normalized solutions to the nonlinear Schrödinger–Poisson system with the $L^2$-subcritical growth Electronic Journal of Qualitative Theory of Differential Equations schrödinger–poisson system normalized solutions lusternik–schnirelmann category variational methods |
| title | Multiple normalized solutions to the nonlinear Schrödinger–Poisson system with the $L^2$-subcritical growth |
| title_full | Multiple normalized solutions to the nonlinear Schrödinger–Poisson system with the $L^2$-subcritical growth |
| title_fullStr | Multiple normalized solutions to the nonlinear Schrödinger–Poisson system with the $L^2$-subcritical growth |
| title_full_unstemmed | Multiple normalized solutions to the nonlinear Schrödinger–Poisson system with the $L^2$-subcritical growth |
| title_short | Multiple normalized solutions to the nonlinear Schrödinger–Poisson system with the $L^2$-subcritical growth |
| title_sort | multiple normalized solutions to the nonlinear schrodinger poisson system with the l 2 subcritical growth |
| topic | schrödinger–poisson system normalized solutions lusternik–schnirelmann category variational methods |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=11101 |
| work_keys_str_mv | AT siweiwei multiplenormalizedsolutionstothenonlinearschrodingerpoissonsystemwiththel2subcriticalgrowth AT kaiminteng multiplenormalizedsolutionstothenonlinearschrodingerpoissonsystemwiththel2subcriticalgrowth |