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Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity

Vitaly Moroz Orcid Logo, Shiwang Ma

Mathematische Zeitschrift, Volume: 304, Issue: 13

Swansea University Author: Vitaly Moroz Orcid Logo

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DOI (Published version): 10.1007/s00209-023-03271-0

Abstract

We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation−Δu+u=u2∗−1+λuq−1inRN,(Pλ)where N≥3 is an integer, 2∗=2NN−2 is the Sobolev critical exponent, 2<q<2∗ and λ>0 is a parameter. It is known that as λ→0, after a rescaling the ground state solu...

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Published in: Mathematische Zeitschrift
ISSN: 0025-5874 1432-1823
Published: Springer Nature 2023
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URI: https://cronfa.swan.ac.uk/Record/cronfa63145
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spelling v2 63145 2023-04-14 Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity 160965ff7131686ab9263d39886c8c1a 0000-0003-3302-8782 Vitaly Moroz Vitaly Moroz true false 2023-04-14 SMA We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation−Δu+u=u2∗−1+λuq−1inRN,(Pλ)where N≥3 is an integer, 2∗=2NN−2 is the Sobolev critical exponent, 2<q<2∗ and λ>0 is a parameter. It is known that as λ→0, after a rescaling the ground state solutions of (Pλ) converge to a particular solution of the critical Emden-Fowler equation −Δu=u2∗−1. We establish a novel sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the space dimension N=3, N=4 or N≥5. We also discuss a connection of these results with a mass constrained problem associated to (Pλ). Unlike previous work of this type, our method is based on the Nehari-Pohožaev manifold minimization, which allows to control the L2 norm of the groundstates. Journal Article Mathematische Zeitschrift 304 13 Springer Nature 0025-5874 1432-1823 Nonlinear Schrödinger equation, critical Sobolev exponent, concentration compactness, asymptotic behaviour 19 4 2023 2023-04-19 10.1007/s00209-023-03271-0 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University SU Library paid the OA fee (TA Institutional Deal) S.M. was supported by National Natural Science Foundation of China (Grant Nos.11571187, 11771182). 2023-06-12T14:03:32.6828746 2023-04-14T09:20:11.1338246 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Vitaly Moroz 0000-0003-3302-8782 1 Shiwang Ma 2 63145__27172__f234a29b05794fa487b7cd0b82d5af0f.pdf MSVM-Combined_accepted.pdf 2023-04-23T17:34:05.8193029 Output 501590 application/pdf Accepted Manuscript true false 63145__27433__dc08c8d63ddc418fbf764fd69a5decb5.pdf 63145.pdf 2023-05-11T11:57:49.4035496 Output 452447 application/pdf Version of Record true Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. true eng http://creativecommons.org/licenses/by/4.0/
title Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity
spellingShingle Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity
Vitaly Moroz
title_short Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity
title_full Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity
title_fullStr Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity
title_full_unstemmed Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity
title_sort Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity
author_id_str_mv 160965ff7131686ab9263d39886c8c1a
author_id_fullname_str_mv 160965ff7131686ab9263d39886c8c1a_***_Vitaly Moroz
author Vitaly Moroz
author2 Vitaly Moroz
Shiwang Ma
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container_title Mathematische Zeitschrift
container_volume 304
container_issue 13
publishDate 2023
institution Swansea University
issn 0025-5874
1432-1823
doi_str_mv 10.1007/s00209-023-03271-0
publisher Springer Nature
college_str Faculty of Science and Engineering
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hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
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department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
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description We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation−Δu+u=u2∗−1+λuq−1inRN,(Pλ)where N≥3 is an integer, 2∗=2NN−2 is the Sobolev critical exponent, 2<q<2∗ and λ>0 is a parameter. It is known that as λ→0, after a rescaling the ground state solutions of (Pλ) converge to a particular solution of the critical Emden-Fowler equation −Δu=u2∗−1. We establish a novel sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the space dimension N=3, N=4 or N≥5. We also discuss a connection of these results with a mass constrained problem associated to (Pλ). Unlike previous work of this type, our method is based on the Nehari-Pohožaev manifold minimization, which allows to control the L2 norm of the groundstates.
published_date 2023-04-19T14:03:31Z
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