NLS equation with competing inhomogeneous nonlinearities: ground states, blow-up, and scattering

[2311.12693] NLS equation with competing inhomogeneous nonlinearities: ground states, blow-up, and scattering Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Analysis of PDEs arXiv:2311.12693 (math) [Submitted on 21 Nov 2023 ( v1 ), last revised 14 Apr 2026 (this version, v3)] Title: NLS equation with competing inhomogeneous nonlinearities: ground states, blow-up, and scattering Authors: Tianxiang Gou , Mohamed Majdoub , Tarek Saanouni View a PDF of the paper titled NLS equation with competing inhomogeneous nonlinearities: ground states, blow-up, and scattering, by Tianxiang Gou and 1 other authors View PDF HTML (experimental) Abstract: We investigate a class of nonlinear equations of Schrödinger type with competing inhomogeneous nonlinearities in the non-radial inter-critical regime, \begin{align*} i \partial_t u +\Delta u &=|x|^{-b_1} |u|^{p_1-2} u - |x|^{-b_2} |u|^{p_2-2}u \quad \mbox{in} \,\, \mathbb{R} \times \mathbb{R}^N, \end{align*} where $N \geq 1$, $b_1, b_2>0$ and $p_1,p_2>2$. First, we establish the existence/nonexistence, symmetry, decay, uniqueness, non-degeneracy and instability of ground states. Then, we prove the scattering versus blowup below the ground state energy threshold. Our approach relies on Tao's scattering criterion and Dodson-Murphy's Virial/Morawetz inequalities. We also obtain an upper bound of the blow-up rate. The novelty here is that the equation does not enjoy any scaling invariance due to the presence of competing nonlinearities and the singular weights prevent the invariance by translation in the space variable. To the best of authors knowledge, this is the first time when inhomegeneous NLS equation with a focusing leading order nonlinearity and a defocusing perturbation is investigated. Comments: 54 pages Subjects: Analysis of PDEs (math.AP) ; Mathematical Physics (math-ph) Cite as: arXiv:2311.12693 [math.AP] (or arXiv:2311.12693v3 [math.AP] for this version) https://doi.org/10.48550/arXiv.2311.12693 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Calc. Var. 65, 162 (2026) Related DOI : https://doi.org/10.1007/s00526-026-03340-2 Focus to learn more DOI(s) linking to related resources Submission history From: Mohamed Majdoub [ view email ] [v1] Tue, 21 Nov 2023 15:57:51 UTC (45 KB) [v2] Thu, 23 Nov 2023 08:45:41 UTC (45 KB) [v3] Tue, 14 Apr 2026 09:44:50 UTC (42 KB) Full-text links: Access Paper: View a PDF of the paper titled NLS equation with competing inhomogeneous nonlinearities: ground states, blow-up, and scattering, by Tianxiang Gou and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2023-11 Change to browse by: math math-ph math.MP 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

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