[2404.13908] Normalized grounded states for a coupled nonlinear schrödinger system on $\mathbb{R}^3$ Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Analysis of PDEs arXiv:2404.13908 (math) This paper has been withdrawn by Chengcheng Wu [Submitted on 22 Apr 2024 ( v1 ), last revised 2 Aug 2026 (this version, v6)] Title: Normalized grounded states for a coupled nonlinear schrödinger system on $\mathbb{R}^3$ Authors: Chengcheng Wu View a PDF of the paper titled Normalized grounded states for a coupled nonlinear schr\"{o}dinger system on $\mathbb{R}^3$, by Chengcheng Wu No PDF available, click to view other formats Abstract: We investigate the existence of normalized ground states to the system of coupled Schrödinger equations: \begin{equation}\label{eq:0.1} \begin{cases} -\Delta u_1 + \lambda_1 u_1 = \mu_1 |u_1|^{p_1-2}u_1 + \beta r_1|u_1|^{r_1-2}u_1|u_2|^{r_2} & \text{ in } \mathbb{R}^{3}, -\Delta u_2 + \lambda_2 u_2 = \mu_2|u_2|^{p_2-2}u_2 + \beta r_2|u_1|^{r_1}|u_2|^{r_2-2}u_2 & \text{ in } \mathbb{R}^3, \end{cases} \end{equation} subject to the constraints $\mathcal{S}_{a_1} \times \mathcal{S}_{a_2} = \{(u_1 \in H^1(\mathbb{R}^3))|\int_{\mathbb{R}^3} u_1^2 dx = a_1^2\} \times \{(u_2 \in H^1(\mathbb{R}^3))|\int_{\mathbb{R}^3} u_2^2 dx = a_2^2\}$, where $\mu_1, \mu_2 > 0$, $r_1, r_2 > 1$, and $\beta \geq 0$. Our focus is on the coupled mass super-critical case, specifically, $$\frac{10}{3} < p_1, p_2, r_1 + r_2 < 2^* = 6.$$ We demonstrate that there exists a $\tilde{\beta} \geq 0$ such that equation (\ref{eq:0.1}) admits positive, radially symmetric, normalized ground state solutions when $\beta > \tilde{\beta}$. Furthermore, this result can be generalized to systems with an arbitrary number of components, and the corresponding standing wave is orbitally unstable. Comments: There exists some errors Subjects: Analysis of PDEs (math.AP) Cite as: arXiv:2404.13908 [math.AP] (or arXiv:2404.13908v6 [math.AP] for this version) https://doi.org/10.48550/arXiv.2404.13908 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Chengcheng Wu [ view email ] [v1] Mon, 22 Apr 2024 06:37:03 UTC (18 KB) [v2] Tue, 23 Apr 2024 02:10:54 UTC (18 KB) [v3] Mon, 29 Apr 2024 14:06:48 UTC (18 KB) [v4] Thu, 4 Sep 2025 08:24:04 UTC (1 KB) (withdrawn) [v5] Fri, 24 Apr 2026 00:43:31 UTC (1 KB) (withdrawn) [v6] Sun, 2 Aug 2026 01:09:02 UTC (1 KB) (withdrawn) Full-text links: Access Paper: View a PDF of the paper titled Normalized grounded states for a coupled nonlinear schr\"{o}dinger system on $\mathbb{R}^3$, by Chengcheng Wu Withdrawn No license for this version due to withdrawn Current browse context: math.AP < prev | next > new | recent | 2024-04 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from