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{"title":"Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system","authors":"Jian Zhang, Huitao Zhou, Heilong Mi","doi":"10.1515/anona-2023-0139","DOIUrl":null,"url":null,"abstract":"\n <jats:p>This article is concerned with the following Hamiltonian elliptic system: <jats:disp-formula id=\"j_anona-2023-0139_eq_001\">\n <jats:alternatives>\n <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0139_eq_001.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\">\n <m:mfenced open=\"{\" close=\"\">\n <m:mrow>\n <m:mtable displaystyle=\"true\">\n <m:mtr>\n <m:mtd columnalign=\"left\">\n <m:mo>−</m:mo>\n <m:msup>\n <m:mrow>\n <m:mi>ε</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mn>2</m:mn>\n </m:mrow>\n </m:msup>\n <m:mi mathvariant=\"normal\">Δ</m:mi>\n <m:mi>u</m:mi>\n <m:mo>+</m:mo>\n <m:mi>ε</m:mi>\n <m:mover accent=\"true\">\n <m:mrow>\n <m:mi>b</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mo>→</m:mo>\n </m:mrow>\n </m:mover>\n <m:mo>⋅</m:mo>\n <m:mrow>\n <m:mo>∇</m:mo>\n </m:mrow>\n <m:mi>u</m:mi>\n <m:mo>+</m:mo>\n <m:mi>u</m:mi>\n <m:mo>+</m:mo>\n <m:mi>V</m:mi>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mi>x</m:mi>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n <m:mi>v</m:mi>\n <m:mo>=</m:mo>\n <m:msub>\n <m:mrow>\n <m:mi>H</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>v</m:mi>\n </m:mrow>\n </m:msub>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mi>u</m:mi>\n <m:mo>,</m:mo>\n <m:mi>v</m:mi>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n <m:mspace width=\"1em\" />\n <m:mspace width=\"0.1em\" />\n <m:mtext>in</m:mtext>\n <m:mspace width=\"0.1em\" />\n <m:mspace width=\"0.33em\" />\n <m:msup>\n <m:mrow>\n <m:mi mathvariant=\"double-struck\">R</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>N</m:mi>\n </m:mrow>\n </m:msup>\n <m:mo>,</m:mo>\n </m:mtd>\n </m:mtr>\n <m:mtr>\n <m:mtd columnalign=\"left\">\n <m:mo>−</m:mo>\n <m:msup>\n <m:mrow>\n <m:mi>ε</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mn>2</m:mn>\n </m:mrow>\n </m:msup>\n <m:mi mathvariant=\"normal\">Δ</m:mi>\n <m:mi>v</m:mi>\n <m:mo>−</m:mo>\n <m:mi>ε</m:mi>\n <m:mover accent=\"true\">\n <m:mrow>\n <m:mi>b</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mo>→</m:mo>\n </m:mrow>\n </m:mover>\n <m:mo>⋅</m:mo>\n <m:mrow>\n <m:mo>∇</m:mo>\n </m:mrow>\n <m:mi>v</m:mi>\n <m:mo>+</m:mo>\n <m:mi>v</m:mi>\n <m:mo>+</m:mo>\n <m:mi>V</m:mi>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mi>x</m:mi>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n <m:mi>u</m:mi>\n <m:mo>=</m:mo>\n <m:msub>\n <m:mrow>\n <m:mi>H</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>u</m:mi>\n </m:mrow>\n </m:msub>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mi>u</m:mi>\n <m:mo>,</m:mo>\n <m:mi>v</m:mi>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n <m:mspace width=\"1em\" />\n <m:mspace width=\"0.1em\" />\n <m:mtext>in</m:mtext>\n <m:mspace width=\"0.1em\" />\n <m:mspace width=\"0.33em\" />\n <m:msup>\n <m:mrow>\n <m:mi mathvariant=\"double-struck\">R</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>N</m:mi>\n </m:mrow>\n </m:msup>\n <m:mo>,</m:mo>\n </m:mtd>\n </m:mtr>\n </m:mtable>\n </m:mrow>\n </m:mfenced>\n </m:math>\n <jats:tex-math>\\left\\{\\begin{array}{l}-{\\varepsilon }^{2}\\Delta u+\\varepsilon \\overrightarrow{b}\\cdot \\nabla u+u+V\\left(x)v={H}_{v}\\left(u,v)\\hspace{1em}\\hspace{0.1em}\\text{in}\\hspace{0.1em}\\hspace{0.33em}{{\\mathbb{R}}}^{N},\\\\ -{\\varepsilon }^{2}\\Delta v-\\varepsilon \\overrightarrow{b}\\cdot \\nabla v+v+V\\left(x)u={H}_{u}\\left(u,v)\\hspace{1em}\\hspace{0.1em}\\text{in}\\hspace{0.1em}\\hspace{0.33em}{{\\mathbb{R}}}^{N},\\end{array}\\right.</jats:tex-math>\n </jats:alternatives>\n </jats:disp-formula> where <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":3.2000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Nonlinear Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/anona-2023-0139","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 3
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期刊介绍:
Advances in Nonlinear Analysis (ANONA) aims to publish selected research contributions devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. The Journal focuses on papers that address significant problems in pure and applied nonlinear analysis. ANONA seeks to present the most significant advances in this field to a wide readership, including researchers and graduate students in mathematics, physics, and engineering.