{"title":"Normalized ground state solutions of the biharmonic Schrödinger equation with general mass supercritical nonlinearities","authors":"Ziheng Zhang, Ying Wang","doi":"10.1016/j.aml.2024.109415","DOIUrl":null,"url":null,"abstract":"We are interested in the following problem <ce:display><ce:formula><mml:math altimg=\"si1.svg\" display=\"block\"><mml:mfenced close=\"\" open=\"{\"><mml:mrow><mml:mtable align=\"axis\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"><mml:mtr><mml:mtd columnalign=\"left\"><mml:msup><mml:mrow><mml:mi>Δ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:mi>λ</mml:mi><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>g</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mspace width=\"0.16667em\"></mml:mspace><mml:mspace width=\"0.16667em\"></mml:mspace><mml:mspace width=\"0.16667em\"></mml:mspace><mml:mspace width=\"0.16667em\"></mml:mspace><mml:mtext>in</mml:mtext><mml:mspace width=\"0.16667em\"></mml:mspace><mml:mspace width=\"0.16667em\"></mml:mspace><mml:mspace width=\"0.16667em\"></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:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:msub><mml:mrow><mml:mo linebreak=\"badbreak\">∫</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>d</mml:mi><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:math></ce:formula></ce:display>where <mml:math altimg=\"si4.svg\" display=\"inline\"><mml:mrow><mml:mi>N</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">≥</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:math>, <mml:math altimg=\"si5.svg\" display=\"inline\"><mml:mrow><mml:mi>c</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> and <mml:math altimg=\"si6.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> appears as a Lagrange multiplier. When <mml:math altimg=\"si7.svg\" display=\"inline\"><mml:mrow><mml:mi>g</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> satisfies a class of general mass supercritical conditions, we introduce one more constraint and consider the corresponding infimum. After showing that the new constraint is natural and verifying the compactness of the minimizing sequence, we obtain the existence of normalized ground state solutions. In this sense, the existing results are generalized and improved significantly.","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"17 1","pages":""},"PeriodicalIF":2.9000,"publicationDate":"2024-12-05","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.2024.109415","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We are interested in the following problem Δ2u+λu=g(u)inRN,∫RN|u|2dx=c,where N≥5, c>0 and λ∈R appears as a Lagrange multiplier. When g(u) satisfies a class of general mass supercritical conditions, we introduce one more constraint and consider the corresponding infimum. After showing that the new constraint is natural and verifying the compactness of the minimizing sequence, we obtain the existence of normalized ground state solutions. In this sense, the existing results are generalized and improved significantly.
期刊介绍：
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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