Existence of normalized solutions for a Sobolev supercritical Schrödinger equation
This paper studies the existence of normalized solutions for the following Schrödinger equation with Sobolev supercritical growth: \begin{document}$ \begin{equation*} \begin{cases} -\Delta u+V(x)u+\lambda u = f(u)+\mu |u|^{p-2}u, \quad &\hbox{in}\;\mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^2dx = a...
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2024-12-01
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2024316 |
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| author | Quanqing Li Zhipeng Yang |
| author_facet | Quanqing Li Zhipeng Yang |
| author_sort | Quanqing Li |
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| description | This paper studies the existence of normalized solutions for the following Schrödinger equation with Sobolev supercritical growth: \begin{document}$ \begin{equation*} \begin{cases} -\Delta u+V(x)u+\lambda u = f(u)+\mu |u|^{p-2}u, \quad &\hbox{in}\;\mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^2dx = a^2, \end{cases} \end{equation*} $\end{document} where $ p > 2^*: = \frac{2N}{N-2} $, $ N\geq 3 $, $ a > 0 $, $ \lambda \in \mathbb{R} $ is an unknown Lagrange multiplier, $ V \in C(\mathbb{R}^N, \mathbb{R}) $, $ f $ satisfies weak mass subcritical conditions. By employing the truncation technique, we establish the existence of normalized solutions to this Sobolev supercritical problem. Our primary contribution lies in our initial exploration of the case $ p > 2^* $, which represents an unfixed frequency problem. |
| format | Article |
| id | doaj-art-afd7ae6139d1485fbcf7ee9b757dc43c |
| institution | Kabale University |
| issn | 2688-1594 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj-art-afd7ae6139d1485fbcf7ee9b757dc43c2025-01-23T07:53:06ZengAIMS PressElectronic Research Archive2688-15942024-12-0132126761677110.3934/era.2024316Existence of normalized solutions for a Sobolev supercritical Schrödinger equationQuanqing Li0Zhipeng Yang1Department of Mathematics, Honghe University, Mengzi 661100, ChinaDepartment of Mathematics, Yunnan Normal University, Kunming 650500, ChinaThis paper studies the existence of normalized solutions for the following Schrödinger equation with Sobolev supercritical growth: \begin{document}$ \begin{equation*} \begin{cases} -\Delta u+V(x)u+\lambda u = f(u)+\mu |u|^{p-2}u, \quad &\hbox{in}\;\mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^2dx = a^2, \end{cases} \end{equation*} $\end{document} where $ p > 2^*: = \frac{2N}{N-2} $, $ N\geq 3 $, $ a > 0 $, $ \lambda \in \mathbb{R} $ is an unknown Lagrange multiplier, $ V \in C(\mathbb{R}^N, \mathbb{R}) $, $ f $ satisfies weak mass subcritical conditions. By employing the truncation technique, we establish the existence of normalized solutions to this Sobolev supercritical problem. Our primary contribution lies in our initial exploration of the case $ p > 2^* $, which represents an unfixed frequency problem.https://www.aimspress.com/article/doi/10.3934/era.2024316normalized solutiontruncation techniquesobolev supercritical growth |
| spellingShingle | Quanqing Li Zhipeng Yang Existence of normalized solutions for a Sobolev supercritical Schrödinger equation Electronic Research Archive normalized solution truncation technique sobolev supercritical growth |
| title | Existence of normalized solutions for a Sobolev supercritical Schrödinger equation |
| title_full | Existence of normalized solutions for a Sobolev supercritical Schrödinger equation |
| title_fullStr | Existence of normalized solutions for a Sobolev supercritical Schrödinger equation |
| title_full_unstemmed | Existence of normalized solutions for a Sobolev supercritical Schrödinger equation |
| title_short | Existence of normalized solutions for a Sobolev supercritical Schrödinger equation |
| title_sort | existence of normalized solutions for a sobolev supercritical schrodinger equation |
| topic | normalized solution truncation technique sobolev supercritical growth |
| url | https://www.aimspress.com/article/doi/10.3934/era.2024316 |
| work_keys_str_mv | AT quanqingli existenceofnormalizedsolutionsforasobolevsupercriticalschrodingerequation AT zhipengyang existenceofnormalizedsolutionsforasobolevsupercriticalschrodingerequation |