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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 1417-3875 |