Existence of normalized solutions for the Schrödinger equation
<p>In this paper, we devote to studying the existence of normalized solutions for the following Schrödinger equation with Sobolev critical nonlinearities.</p> <p class="disp_formula">$ \begin{align*} &\left\{\begin{array}{ll} -\Delta u = \lambda u+\mu\lvert...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2023-09-01
|
| Series: | Communications in Analysis and Mechanics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/cam.2023028?viewType=HTML |
| _version_ | 1797361196691619840 |
|---|---|
| author | Shengbing Deng Qiaoran Wu |
| author_facet | Shengbing Deng Qiaoran Wu |
| author_sort | Shengbing Deng |
| collection | DOAJ |
| description | <p>In this paper, we devote to studying the existence of normalized solutions for the following Schrödinger equation with Sobolev critical nonlinearities.</p>
<p class="disp_formula">$ \begin{align*} &\left\{\begin{array}{ll} -\Delta u = \lambda u+\mu\lvert u \rvert^{q-2}u+\lvert u \rvert^{p-2}u&{\mbox{in}}\ \mathbb{R}^N,\\ \int_{\mathbb{R}^N}\lvert u\rvert^2dx = a^2, \end{array}\right. \end{align*} $</p>
<p>where $ N\geqslant 3 $, $ 2 < q < 2+\frac{4}{N} $, $ p = 2^* = \frac{2N}{N-2} $, $ a, \mu > 0 $ and $ \lambda\in\mathbb{R} $ is a Lagrange multiplier. Since the existence result for $ 2+\frac{4}{N} < p < 2^* $ has been proved, using an approximation method, that is let $ p\rightarrow 2^* $, we obtain that there exists a mountain-pass type solution for $ p = 2^* $.</p> |
| first_indexed | 2024-03-08T15:50:28Z |
| format | Article |
| id | doaj.art-2fea7bb8ea3e4373b8642e4bc507b932 |
| institution | Directory Open Access Journal |
| issn | 2836-3310 |
| language | English |
| last_indexed | 2024-03-08T15:50:28Z |
| publishDate | 2023-09-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Communications in Analysis and Mechanics |
| spelling | doaj.art-2fea7bb8ea3e4373b8642e4bc507b9322024-01-09T05:58:22ZengAIMS PressCommunications in Analysis and Mechanics2836-33102023-09-0115357558510.3934/cam.2023028Existence of normalized solutions for the Schrödinger equationShengbing Deng0Qiaoran Wu1School of Mathematics and Statistics, Southwest University, Chongqing, 400715, P.R. ChinaSchool of Mathematics and Statistics, Southwest University, Chongqing, 400715, P.R. China<p>In this paper, we devote to studying the existence of normalized solutions for the following Schrödinger equation with Sobolev critical nonlinearities.</p> <p class="disp_formula">$ \begin{align*} &\left\{\begin{array}{ll} -\Delta u = \lambda u+\mu\lvert u \rvert^{q-2}u+\lvert u \rvert^{p-2}u&{\mbox{in}}\ \mathbb{R}^N,\\ \int_{\mathbb{R}^N}\lvert u\rvert^2dx = a^2, \end{array}\right. \end{align*} $</p> <p>where $ N\geqslant 3 $, $ 2 < q < 2+\frac{4}{N} $, $ p = 2^* = \frac{2N}{N-2} $, $ a, \mu > 0 $ and $ \lambda\in\mathbb{R} $ is a Lagrange multiplier. Since the existence result for $ 2+\frac{4}{N} < p < 2^* $ has been proved, using an approximation method, that is let $ p\rightarrow 2^* $, we obtain that there exists a mountain-pass type solution for $ p = 2^* $.</p>https://www.aimspress.com/article/doi/10.3934/cam.2023028?viewType=HTMLnormalized solutionsschrödinger equationsobolev critical nonlinearitiesapproximation methodmountain-pass type solution |
| spellingShingle | Shengbing Deng Qiaoran Wu Existence of normalized solutions for the Schrödinger equation Communications in Analysis and Mechanics normalized solutions schrödinger equation sobolev critical nonlinearities approximation method mountain-pass type solution |
| title | Existence of normalized solutions for the Schrödinger equation |
| title_full | Existence of normalized solutions for the Schrödinger equation |
| title_fullStr | Existence of normalized solutions for the Schrödinger equation |
| title_full_unstemmed | Existence of normalized solutions for the Schrödinger equation |
| title_short | Existence of normalized solutions for the Schrödinger equation |
| title_sort | existence of normalized solutions for the schrodinger equation |
| topic | normalized solutions schrödinger equation sobolev critical nonlinearities approximation method mountain-pass type solution |
| url | https://www.aimspress.com/article/doi/10.3934/cam.2023028?viewType=HTML |
| work_keys_str_mv | AT shengbingdeng existenceofnormalizedsolutionsfortheschrodingerequation AT qiaoranwu existenceofnormalizedsolutionsfortheschrodingerequation |