{"title":"非线性标量场方程的无穷多变号归一化解","authors":"Jiaxin Zhan, Jianjun Zhang, Xuexiu Zhong, Jinfang Zhou","doi":"10.1016/j.aml.2024.109426","DOIUrl":null,"url":null,"abstract":"We study the existence of infinitely many sign-changing solutions to the following nonlinear scalar Schrödinger equation <ce:display><ce:formula><mml:math altimg=\"si1.svg\" display=\"block\"><mml:mrow><mml:mo>−</mml:mo><mml:mi>Δ</mml:mi><mml:mi>u</mml:mi><mml:mo linebreak=\"goodbreak\">+</mml:mo><mml:mi>λ</mml:mi><mml:mi>u</mml:mi><mml:mo linebreak=\"goodbreak\">=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mspace width=\"1em\"></mml:mspace><mml:mtext>in</mml:mtext><mml:mspace width=\"1em\"></mml:mspace><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></ce:formula></ce:display>with a prescribed mass <mml:math altimg=\"si2.svg\" display=\"inline\"><mml:mrow><mml:msub><mml:mrow><mml:mo>∫</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi>u</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant=\"normal\">d</mml:mi><mml:mi>x</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">=</mml:mo><mml:mi>a</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math> Here <mml:math altimg=\"si3.svg\" display=\"inline\"><mml:mrow><mml:mi>f</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">∈</mml:mo><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi mathvariant=\"double-struck\">R</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"double-struck\">R</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, <mml:math altimg=\"si4.svg\" display=\"inline\"><mml:mrow><mml:mi>a</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> is a given constant and <mml:math altimg=\"si5.svg\" display=\"inline\"><mml:mrow><mml:mi>λ</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">∈</mml:mo><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow></mml:math> is an unknown parameter appearing as a Lagrange multiplier. Jeanjean and Lu have established the existence of infinitely many sign-changing normalized solutions in [Nonlinearity 32 (2019), no. 12, 4942–4966] and [Calc. Var. Partial Differential Equations 59 (2020), no. 5, Paper No. 174, 43 pp.] for <mml:math altimg=\"si6.svg\" display=\"inline\"><mml:mrow><mml:mi>N</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">=</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math> or <mml:math altimg=\"si7.svg\" display=\"inline\"><mml:mrow><mml:mi>N</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">≥</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:math>. After fully utilizing the properties of positive solutions given by Jeanjean,Zhang and Zhong[J. Math. Pures Appl. (9) 183 (2024), 44–75], we give an alternative approach and extend the existence of infinitely many sign-changing normalized solutions to all <mml:math altimg=\"si8.svg\" display=\"inline\"><mml:mrow><mml:mi>N</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">≥</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>.","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"22 1","pages":""},"PeriodicalIF":2.9000,"publicationDate":"2024-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Infinitely many sign-changing normalized solutions for nonlinear scalar field equations\",\"authors\":\"Jiaxin Zhan, Jianjun Zhang, Xuexiu Zhong, Jinfang Zhou\",\"doi\":\"10.1016/j.aml.2024.109426\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the existence of infinitely many sign-changing solutions to the following nonlinear scalar Schrödinger equation <ce:display><ce:formula><mml:math altimg=\\\"si1.svg\\\" display=\\\"block\\\"><mml:mrow><mml:mo>−</mml:mo><mml:mi>Δ</mml:mi><mml:mi>u</mml:mi><mml:mo linebreak=\\\"goodbreak\\\">+</mml:mo><mml:mi>λ</mml:mi><mml:mi>u</mml:mi><mml:mo linebreak=\\\"goodbreak\\\">=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mspace width=\\\"1em\\\"></mml:mspace><mml:mtext>in</mml:mtext><mml:mspace width=\\\"1em\\\"></mml:mspace><mml:msup><mml:mrow><mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></ce:formula></ce:display>with a prescribed mass <mml:math altimg=\\\"si2.svg\\\" display=\\\"inline\\\"><mml:mrow><mml:msub><mml:mrow><mml:mo>∫</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi>u</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant=\\\"normal\\\">d</mml:mi><mml:mi>x</mml:mi><mml:mo linebreak=\\\"goodbreak\\\" linebreakstyle=\\\"after\\\">=</mml:mo><mml:mi>a</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math> Here <mml:math altimg=\\\"si3.svg\\\" display=\\\"inline\\\"><mml:mrow><mml:mi>f</mml:mi><mml:mo linebreak=\\\"goodbreak\\\" linebreakstyle=\\\"after\\\">∈</mml:mo><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, <mml:math altimg=\\\"si4.svg\\\" display=\\\"inline\\\"><mml:mrow><mml:mi>a</mml:mi><mml:mo linebreak=\\\"goodbreak\\\" linebreakstyle=\\\"after\\\">></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> is a given constant and <mml:math altimg=\\\"si5.svg\\\" display=\\\"inline\\\"><mml:mrow><mml:mi>λ</mml:mi><mml:mo linebreak=\\\"goodbreak\\\" linebreakstyle=\\\"after\\\">∈</mml:mo><mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi></mml:mrow></mml:math> is an unknown parameter appearing as a Lagrange multiplier. Jeanjean and Lu have established the existence of infinitely many sign-changing normalized solutions in [Nonlinearity 32 (2019), no. 12, 4942–4966] and [Calc. Var. Partial Differential Equations 59 (2020), no. 5, Paper No. 174, 43 pp.] for <mml:math altimg=\\\"si6.svg\\\" display=\\\"inline\\\"><mml:mrow><mml:mi>N</mml:mi><mml:mo linebreak=\\\"goodbreak\\\" linebreakstyle=\\\"after\\\">=</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math> or <mml:math altimg=\\\"si7.svg\\\" display=\\\"inline\\\"><mml:mrow><mml:mi>N</mml:mi><mml:mo linebreak=\\\"goodbreak\\\" linebreakstyle=\\\"after\\\">≥</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:math>. After fully utilizing the properties of positive solutions given by Jeanjean,Zhang and Zhong[J. Math. Pures Appl. (9) 183 (2024), 44–75], we give an alternative approach and extend the existence of infinitely many sign-changing normalized solutions to all <mml:math altimg=\\\"si8.svg\\\" display=\\\"inline\\\"><mml:mrow><mml:mi>N</mml:mi><mml:mo linebreak=\\\"goodbreak\\\" linebreakstyle=\\\"after\\\">≥</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>.\",\"PeriodicalId\":55497,\"journal\":{\"name\":\"Applied Mathematics Letters\",\"volume\":\"22 1\",\"pages\":\"\"},\"PeriodicalIF\":2.9000,\"publicationDate\":\"2024-12-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Applied Mathematics Letters\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1016/j.aml.2024.109426\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics Letters","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1016/j.aml.2024.109426","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Infinitely many sign-changing normalized solutions for nonlinear scalar field equations
We study the existence of infinitely many sign-changing solutions to the following nonlinear scalar Schrödinger equation −Δu+λu=f(u)inRNwith a prescribed mass ∫RN|u|2dx=a. Here f∈C1(R,R), a>0 is a given constant and λ∈R is an unknown parameter appearing as a Lagrange multiplier. Jeanjean and Lu have established the existence of infinitely many sign-changing normalized solutions in [Nonlinearity 32 (2019), no. 12, 4942–4966] and [Calc. Var. Partial Differential Equations 59 (2020), no. 5, Paper No. 174, 43 pp.] for N=4 or N≥6. After fully utilizing the properties of positive solutions given by Jeanjean,Zhang and Zhong[J. Math. Pures Appl. (9) 183 (2024), 44–75], we give an alternative approach and extend the existence of infinitely many sign-changing normalized solutions to all N≥2.
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The purpose of Applied Mathematics Letters is to provide a means of rapid publication for important but brief applied mathematical papers. The brief descriptions of any work involving a novel application or utilization of mathematics, or a development in the methodology of applied mathematics is a potential contribution for this journal. This journal''s focus is on applied mathematics topics based on differential equations and linear algebra. Priority will be given to submissions that are likely to appeal to a wide audience.
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