Multiple normalized solutions for $(2,q)$-Laplacian equation problems in whole $\mathbb{R}^{N}$
This paper considers the existence of multiple normalized solutions of the following $(2,q)$-Laplacian equation: \begin{equation*} \begin{cases} -\Delta u-\Delta_q u=\lambda u+h(\epsilon x)f(u), &\mathrm{in}\ \mathbb{R}^{N},\\ \int_{\mathbb{R}^{N}}|u|^2\mathrm{d}x=a^2, \end{cases} \end{equa...
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| Format: | Article |
| Language: | English |
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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=10957 |
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| _version_ | 1846091039465537536 |
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| author | Renhua Chen Li Wang Xin Song |
| author_facet | Renhua Chen Li Wang Xin Song |
| author_sort | Renhua Chen |
| collection | DOAJ |
| description | This paper considers the existence of multiple normalized solutions of the following $(2,q)$-Laplacian equation:
\begin{equation*}
\begin{cases}
-\Delta u-\Delta_q u=\lambda u+h(\epsilon x)f(u), &\mathrm{in}\ \mathbb{R}^{N},\\
\int_{\mathbb{R}^{N}}|u|^2\mathrm{d}x=a^2,
\end{cases}
\end{equation*}
where $2<q<N$, $\epsilon>0, a>0$ and $\lambda \in \mathbb{R}$ is a Lagrange multiplier which is unknown, $h$ is a continuous positive function and $f$ is also continuous satisfying $L^2$-subcritical growth. When $\epsilon$ is small enough, we show that the number of normalized solutions is at least the number of global maximum points of $h$ by Ekeland's variational principle. |
| format | Article |
| id | doaj-art-bef207812b0246c3b1f421305cb15370 |
| institution | Kabale University |
| issn | 1417-3875 |
| language | English |
| publishDate | 2024-08-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj-art-bef207812b0246c3b1f421305cb153702025-01-15T21:24:58ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752024-08-0120244811910.14232/ejqtde.2024.1.4810957Multiple normalized solutions for $(2,q)$-Laplacian equation problems in whole $\mathbb{R}^{N}$Renhua Chen0Li WangXin Song1College of Science, East China Jiaotong University, Nanchang, ChinaCollege of Science, East China Jiaotong University, Nanchang, ChinaThis paper considers the existence of multiple normalized solutions of the following $(2,q)$-Laplacian equation: \begin{equation*} \begin{cases} -\Delta u-\Delta_q u=\lambda u+h(\epsilon x)f(u), &\mathrm{in}\ \mathbb{R}^{N},\\ \int_{\mathbb{R}^{N}}|u|^2\mathrm{d}x=a^2, \end{cases} \end{equation*} where $2<q<N$, $\epsilon>0, a>0$ and $\lambda \in \mathbb{R}$ is a Lagrange multiplier which is unknown, $h$ is a continuous positive function and $f$ is also continuous satisfying $L^2$-subcritical growth. When $\epsilon$ is small enough, we show that the number of normalized solutions is at least the number of global maximum points of $h$ by Ekeland's variational principle.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10957normalized solutionmultiplicity$(2q)$-laplacianvariational methods |
| spellingShingle | Renhua Chen Li Wang Xin Song Multiple normalized solutions for $(2,q)$-Laplacian equation problems in whole $\mathbb{R}^{N}$ Electronic Journal of Qualitative Theory of Differential Equations normalized solution multiplicity $(2 q)$-laplacian variational methods |
| title | Multiple normalized solutions for $(2,q)$-Laplacian equation problems in whole $\mathbb{R}^{N}$ |
| title_full | Multiple normalized solutions for $(2,q)$-Laplacian equation problems in whole $\mathbb{R}^{N}$ |
| title_fullStr | Multiple normalized solutions for $(2,q)$-Laplacian equation problems in whole $\mathbb{R}^{N}$ |
| title_full_unstemmed | Multiple normalized solutions for $(2,q)$-Laplacian equation problems in whole $\mathbb{R}^{N}$ |
| title_short | Multiple normalized solutions for $(2,q)$-Laplacian equation problems in whole $\mathbb{R}^{N}$ |
| title_sort | multiple normalized solutions for 2 q laplacian equation problems in whole mathbb r n |
| topic | normalized solution multiplicity $(2 q)$-laplacian variational methods |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10957 |
| work_keys_str_mv | AT renhuachen multiplenormalizedsolutionsfor2qlaplacianequationproblemsinwholemathbbrn AT liwang multiplenormalizedsolutionsfor2qlaplacianequationproblemsinwholemathbbrn AT xinsong multiplenormalizedsolutionsfor2qlaplacianequationproblemsinwholemathbbrn |