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{"title":"具有质量临界或超临界和Sobolev临界增长的HLS上临界Choquard方程的归一化基态解","authors":"Jianlun Liu, Ziheng Zhang, Hong-Rui Sun","doi":"10.1002/mma.11177","DOIUrl":null,"url":null,"abstract":"<div>\n \n <p>This paper is concerned with the following HLS upper critical Choquard equation with mass critical or supercritical and Sobolev critical growth \n<span></span><math>\n <semantics>\n <mrow>\n <mfenced>\n <mrow>\n <mtable>\n <mtr>\n <mtd>\n <mo>−</mo>\n <mspace></mspace>\n <mi>Δ</mi>\n <mi>u</mi>\n <mo>=</mo>\n <mi>λ</mi>\n <mi>u</mi>\n <mo>+</mo>\n <mi>μ</mi>\n <mo>|</mo>\n <mi>u</mi>\n <msup>\n <mrow>\n <mo>|</mo>\n </mrow>\n <mrow>\n <mi>p</mi>\n <mo>−</mo>\n <mn>2</mn>\n </mrow>\n </msup>\n <mi>u</mi>\n <mo>+</mo>\n <mo>|</mo>\n <mi>u</mi>\n <msup>\n <mrow>\n <mo>|</mo>\n </mrow>\n <mrow>\n <mn>4</mn>\n </mrow>\n </msup>\n <mi>u</mi>\n <mo>+</mo>\n <mo>(</mo>\n <msub>\n <mrow>\n <mi>I</mi>\n </mrow>\n <mrow>\n <mn>2</mn>\n </mrow>\n </msub>\n <mo>∗</mo>\n <mo>|</mo>\n <mi>u</mi>\n <msup>\n <mrow>\n <mo>|</mo>\n </mrow>\n <mrow>\n <mn>5</mn>\n </mrow>\n </msup>\n <mo>)</mo>\n <mo>|</mo>\n <mi>u</mi>\n <msup>\n <mrow>\n <mo>|</mo>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msup>\n <mi>u</mi>\n <mspace></mspace>\n <mspace></mspace>\n <mtext>in</mtext>\n <mspace></mspace>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msup>\n <mo>,</mo>\n </mtd>\n </mtr>\n <mtr>\n <mtd>\n <msub>\n <mrow>\n <mo>∫</mo>\n </mrow>\n <mrow>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msup>\n </mrow>\n </msub>\n <msup>\n <mrow>\n <mi>u</mi>\n </mrow>\n <mrow>\n <mn>2</mn>\n </mrow>\n </msup>\n <mi>d</mi>\n <mi>x</mi>\n <mo>=</mo>\n <mi>c</mi>\n </mtd>\n </mtr>\n </mtable>\n </mrow>\n </mfenced>\n <mo>,</mo>\n </mrow>\n <annotation>$$ \\left\\{\\begin{array}{l}-\\Delta u&amp;#x0003D;\\lambda u&amp;#x0002B;\\mu {\\left&amp;#x0007C;u\\right&amp;#x0007C;}&amp;#x0005E;{p-2}u&amp;#x0002B;{\\left&amp;#x0007C;u\\right&amp;#x0007C;}&amp;#x0005E;4u&amp;#x0002B;\\left({I}_2\\ast {\\left&amp;#x0007C;u\\right&amp;#x0007C;}&amp;#x0005E;5\\right){\\left&amp;#x0007C;u\\right&amp;#x0007C;}&amp;#x0005E;3u\\kern0.60em \\mathrm{in}\\kern0.3em {\\mathbb{R}}&amp;#x0005E;3,\\\\ {}{\\int}_{{\\mathbb{R}}&amp;#x0005E;3}{u}&amp;#x0005E;2 dx&amp;#x0003D;c\\end{array}\\right., $$</annotation>\n </semantics></math> where \n<span></span><math>\n <semantics>\n <mrow>\n <mi>μ</mi>\n <mo>,</mo>\n <mi>c</mi>\n <mo>></mo>\n <mn>0</mn>\n <mo>,</mo>\n <mspace></mspace>\n <mfrac>\n <mrow>\n <mn>10</mn>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </mfrac>\n <mo>≤</mo>\n <mi>p</mi>\n <mo><</mo>\n <mn>6</mn>\n <mo>,</mo>\n <mspace></mspace>\n <mi>λ</mi>\n <mo>∈</mo>\n <mi>ℝ</mi>\n </mrow>\n <annotation>$$ \\mu, c&gt;0,\\kern0.3em \\frac{10}{3}\\le p&lt;6,\\kern0.3em \\lambda \\in \\mathbb{R} $$</annotation>\n </semantics></math> is a Lagrange multiplier and \n<span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mrow>\n <mi>I</mi>\n </mrow>\n <mrow>\n <mn>2</mn>\n </mrow>\n </msub>\n </mrow>\n <annotation>$$ {I}_2 $$</annotation>\n </semantics></math> is the Riesz potential. The novelty of this paper is that, utilizing the classical strategy due to Jeanjean (Nonlinear Analysis: Theory, Methods & Applications, 28(1997), 1633-1659) and after some subtle energy estimate, we show the existence of mountain pass type normalized ground state solutions for any \n<span></span><math>\n <semantics>\n <mrow>\n <mi>μ</mi>\n <mo>></mo>\n <mn>0</mn>\n </mrow>\n <annotation>$$ \\mu &gt;0 $$</annotation>\n </semantics></math>. In this sense, the novelty of this paper includes three aspects: when \n<span></span><math>\n <semantics>\n <mrow>\n <mfrac>\n <mrow>\n <mn>10</mn>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </mfrac>\n <mo><</mo>\n <mi>p</mi>\n <mo><</mo>\n <mn>6</mn>\n </mrow>\n <annotation>$$ \\frac{10}{3}&lt;p&lt;6 $$</annotation>\n </semantics></math>, we do not need to add any constraints to the parameter \n<span></span><math>\n <semantics>\n <mrow>\n <mi>μ</mi>\n <mo>></mo>\n <mn>0</mn>\n </mrow>\n <annotation>$$ \\mu &gt;0 $$</annotation>\n </semantics></math>; meanwhile, the mountain pass type solution happens to be one ground states; in addition, the \n<span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mrow>\n <mi>L</mi>\n </mrow>\n <mrow>\n <mn>2</mn>\n </mrow>\n </msup>\n </mrow>\n <annotation>$$ {L}&amp;#x0005E;2 $$</annotation>\n </semantics></math>-critical perturbation, that is, \n<span></span><math>\n <semantics>\n <mrow>\n <mi>p</mi>\n <mo>=</mo>\n <mfrac>\n <mrow>\n <mn>10</mn>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </mfrac>\n </mrow>\n <annotation>$$ p&amp;#x0003D;\\frac{10}{3} $$</annotation>\n </semantics></math> is considered, which generalize and improve the recent results.</p>\n </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 15","pages":"14276-14289"},"PeriodicalIF":1.8000,"publicationDate":"2025-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Normalized Ground State Solutions to HLS Upper Critical Choquard Equation With Mass Critical or Supercritical and Sobolev Critical Growth\",\"authors\":\"Jianlun Liu, Ziheng Zhang, Hong-Rui Sun\",\"doi\":\"10.1002/mma.11177\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>\\n \\n <p>This paper is concerned with the following HLS upper critical Choquard equation with mass critical or supercritical and Sobolev critical growth \\n<span></span><math>\\n <semantics>\\n <mrow>\\n <mfenced>\\n <mrow>\\n <mtable>\\n <mtr>\\n <mtd>\\n <mo>−</mo>\\n <mspace></mspace>\\n <mi>Δ</mi>\\n <mi>u</mi>\\n <mo>=</mo>\\n <mi>λ</mi>\\n <mi>u</mi>\\n <mo>+</mo>\\n <mi>μ</mi>\\n <mo>|</mo>\\n <mi>u</mi>\\n <msup>\\n <mrow>\\n <mo>|</mo>\\n </mrow>\\n <mrow>\\n <mi>p</mi>\\n <mo>−</mo>\\n <mn>2</mn>\\n </mrow>\\n </msup>\\n <mi>u</mi>\\n <mo>+</mo>\\n <mo>|</mo>\\n <mi>u</mi>\\n <msup>\\n <mrow>\\n <mo>|</mo>\\n </mrow>\\n <mrow>\\n <mn>4</mn>\\n </mrow>\\n </msup>\\n <mi>u</mi>\\n <mo>+</mo>\\n <mo>(</mo>\\n <msub>\\n <mrow>\\n <mi>I</mi>\\n </mrow>\\n <mrow>\\n <mn>2</mn>\\n </mrow>\\n </msub>\\n <mo>∗</mo>\\n <mo>|</mo>\\n <mi>u</mi>\\n <msup>\\n <mrow>\\n <mo>|</mo>\\n </mrow>\\n <mrow>\\n <mn>5</mn>\\n </mrow>\\n </msup>\\n <mo>)</mo>\\n <mo>|</mo>\\n <mi>u</mi>\\n <msup>\\n <mrow>\\n <mo>|</mo>\\n </mrow>\\n <mrow>\\n <mn>3</mn>\\n </mrow>\\n </msup>\\n <mi>u</mi>\\n <mspace></mspace>\\n <mspace></mspace>\\n <mtext>in</mtext>\\n <mspace></mspace>\\n <msup>\\n <mrow>\\n <mi>ℝ</mi>\\n </mrow>\\n <mrow>\\n <mn>3</mn>\\n </mrow>\\n </msup>\\n <mo>,</mo>\\n </mtd>\\n </mtr>\\n <mtr>\\n <mtd>\\n <msub>\\n <mrow>\\n <mo>∫</mo>\\n </mrow>\\n <mrow>\\n <msup>\\n <mrow>\\n <mi>ℝ</mi>\\n </mrow>\\n <mrow>\\n <mn>3</mn>\\n </mrow>\\n </msup>\\n </mrow>\\n </msub>\\n <msup>\\n <mrow>\\n <mi>u</mi>\\n </mrow>\\n <mrow>\\n <mn>2</mn>\\n </mrow>\\n </msup>\\n <mi>d</mi>\\n <mi>x</mi>\\n <mo>=</mo>\\n <mi>c</mi>\\n </mtd>\\n </mtr>\\n </mtable>\\n </mrow>\\n </mfenced>\\n <mo>,</mo>\\n </mrow>\\n <annotation>$$ \\\\left\\\\{\\\\begin{array}{l}-\\\\Delta u&amp;#x0003D;\\\\lambda u&amp;#x0002B;\\\\mu {\\\\left&amp;#x0007C;u\\\\right&amp;#x0007C;}&amp;#x0005E;{p-2}u&amp;#x0002B;{\\\\left&amp;#x0007C;u\\\\right&amp;#x0007C;}&amp;#x0005E;4u&amp;#x0002B;\\\\left({I}_2\\\\ast {\\\\left&amp;#x0007C;u\\\\right&amp;#x0007C;}&amp;#x0005E;5\\\\right){\\\\left&amp;#x0007C;u\\\\right&amp;#x0007C;}&amp;#x0005E;3u\\\\kern0.60em \\\\mathrm{in}\\\\kern0.3em {\\\\mathbb{R}}&amp;#x0005E;3,\\\\\\\\ {}{\\\\int}_{{\\\\mathbb{R}}&amp;#x0005E;3}{u}&amp;#x0005E;2 dx&amp;#x0003D;c\\\\end{array}\\\\right., $$</annotation>\\n </semantics></math> where \\n<span></span><math>\\n <semantics>\\n <mrow>\\n <mi>μ</mi>\\n <mo>,</mo>\\n <mi>c</mi>\\n <mo>></mo>\\n <mn>0</mn>\\n <mo>,</mo>\\n <mspace></mspace>\\n <mfrac>\\n <mrow>\\n <mn>10</mn>\\n </mrow>\\n <mrow>\\n <mn>3</mn>\\n </mrow>\\n </mfrac>\\n <mo>≤</mo>\\n <mi>p</mi>\\n <mo><</mo>\\n <mn>6</mn>\\n <mo>,</mo>\\n <mspace></mspace>\\n <mi>λ</mi>\\n <mo>∈</mo>\\n <mi>ℝ</mi>\\n </mrow>\\n <annotation>$$ \\\\mu, c&gt;0,\\\\kern0.3em \\\\frac{10}{3}\\\\le p&lt;6,\\\\kern0.3em \\\\lambda \\\\in \\\\mathbb{R} $$</annotation>\\n </semantics></math> is a Lagrange multiplier and \\n<span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mrow>\\n <mi>I</mi>\\n </mrow>\\n <mrow>\\n <mn>2</mn>\\n </mrow>\\n </msub>\\n </mrow>\\n <annotation>$$ {I}_2 $$</annotation>\\n </semantics></math> is the Riesz potential. The novelty of this paper is that, utilizing the classical strategy due to Jeanjean (Nonlinear Analysis: Theory, Methods & Applications, 28(1997), 1633-1659) and after some subtle energy estimate, we show the existence of mountain pass type normalized ground state solutions for any \\n<span></span><math>\\n <semantics>\\n <mrow>\\n <mi>μ</mi>\\n <mo>></mo>\\n <mn>0</mn>\\n </mrow>\\n <annotation>$$ \\\\mu &gt;0 $$</annotation>\\n </semantics></math>. In this sense, the novelty of this paper includes three aspects: when \\n<span></span><math>\\n <semantics>\\n <mrow>\\n <mfrac>\\n <mrow>\\n <mn>10</mn>\\n </mrow>\\n <mrow>\\n <mn>3</mn>\\n </mrow>\\n </mfrac>\\n <mo><</mo>\\n <mi>p</mi>\\n <mo><</mo>\\n <mn>6</mn>\\n </mrow>\\n <annotation>$$ \\\\frac{10}{3}&lt;p&lt;6 $$</annotation>\\n </semantics></math>, we do not need to add any constraints to the parameter \\n<span></span><math>\\n <semantics>\\n <mrow>\\n <mi>μ</mi>\\n <mo>></mo>\\n <mn>0</mn>\\n </mrow>\\n <annotation>$$ \\\\mu &gt;0 $$</annotation>\\n </semantics></math>; meanwhile, the mountain pass type solution happens to be one ground states; in addition, the \\n<span></span><math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mrow>\\n <mi>L</mi>\\n </mrow>\\n <mrow>\\n <mn>2</mn>\\n </mrow>\\n </msup>\\n </mrow>\\n <annotation>$$ {L}&amp;#x0005E;2 $$</annotation>\\n </semantics></math>-critical perturbation, that is, \\n<span></span><math>\\n <semantics>\\n <mrow>\\n <mi>p</mi>\\n <mo>=</mo>\\n <mfrac>\\n <mrow>\\n <mn>10</mn>\\n </mrow>\\n <mrow>\\n <mn>3</mn>\\n </mrow>\\n </mfrac>\\n </mrow>\\n <annotation>$$ p&amp;#x0003D;\\\\frac{10}{3} $$</annotation>\\n </semantics></math> is considered, which generalize and improve the recent results.</p>\\n </div>\",\"PeriodicalId\":49865,\"journal\":{\"name\":\"Mathematical Methods in the Applied Sciences\",\"volume\":\"48 15\",\"pages\":\"14276-14289\"},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2025-06-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Methods in the Applied Sciences\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/mma.11177\",\"RegionNum\":3,\"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":"Mathematical Methods in the Applied Sciences","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mma.11177","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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