{"title":"具有Hardy-Littlewood-Sobolev临界指数的Hartree-Fock系统的定性分析","authors":"Meng Li , Haoyuan Xu , Meihua Yang , Maoding Zhen","doi":"10.1016/j.jde.2025.113565","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper, we develop an exhaustive analysis on standing waves with prescribed mass for the coupled Hartree-Fock system as following, which is introduced by Hartree in the 1920's and developed by Fock for describing large systems of identical fermions,<span><span><span><math><mrow><mrow><mo>{</mo><mtable><mtr><mtd></mtd><mtd><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>u</mi><mo>=</mo><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup><mo>⋆</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>u</mi><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>γ</mi></mrow></msup><mo>⋆</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>u</mi><mo>+</mo><mi>β</mi><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>γ</mi></mrow></msup><mo>⋆</mo><mo>|</mo><mi>v</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>u</mi></mtd></mtr><mtr><mtd></mtd><mtd><mo>−</mo><mi>Δ</mi><mi>v</mi><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>v</mi><mo>=</mo><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup><mo>⋆</mo><mo>|</mo><mi>v</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>v</mi><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>γ</mi></mrow></msup><mo>⋆</mo><mo>|</mo><mi>v</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>v</mi><mo>+</mo><mi>β</mi><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>γ</mi></mrow></msup><mo>⋆</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>v</mi></mtd></mtr></mtable></mrow><mspace></mspace><mtext>in </mtext><msup><mrow><mi>R</mi></mrow><mrow><mn>5</mn></mrow></msup></mrow></math></span></span></span> under mass constraint conditions<span><span><span><math><mrow><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mn>5</mn></mrow></msup></mrow></munder><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mn>5</mn></mrow></msup></mrow></munder><mo>|</mo><mi>v</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo></mrow></math></span></span></span> where <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>></mo><mn>0</mn><mo>(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo><mo>,</mo><mspace></mspace><mi>β</mi><mo>></mo><mn>0</mn></math></span>, <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mi>γ</mi><mo>≤</mo><mn>4</mn></math></span>, and the frequency <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>R</mi><mo>(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span> are a part of unknown and appear as Lagrange multipliers. Firstly, when <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mi>γ</mi><mo>=</mo><mn>2</mn></math></span> or <span><math><mi>α</mi><mo>=</mo><mn>2</mn><mo><</mo><mi>γ</mi><mo><</mo><mn>4</mn></math></span>, we establish an existence result by using the constrained minimization method. Then turning to the case of <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>2</mn><mo><</mo><mi>γ</mi><mo><</mo><mn>4</mn></math></span> or <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>2</mn><mo>,</mo><mi>γ</mi><mo>=</mo><mn>4</mn></math></span>, we show that the problem admits a ground state and an excited state, which are characterized respectively by a local minimizer and a mountain-pass critical point of the corresponding energy functional. Moreover, we give a precise mass collapse behaviors of the ground state solutions for <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>2</mn><mo><</mo><mi>γ</mi><mo><</mo><mn>4</mn></math></span> and <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>2</mn><mo>,</mo><mi>γ</mi><mo>=</mo><mn>4</mn></math></span> when the masses of two components vanish at the same rate. We also give a precise limit profiles of the mountain pass solutions when <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>2</mn><mo>,</mo><mi>γ</mi><mo>=</mo><mn>4</mn></math></span> as <span><math><mo>(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>)</mo><mo>→</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></span>. This seems to be the first contribution regarding the multiplicity as well as the synchronized mass collapse behaviors of the normalized solutions to Hartree-Fock system with Hardy-Littlewood-Sobolev critical exponent. Finally, when <span><math><mn>2</mn><mo>≤</mo><mi>α</mi><mo><</mo><mn>4</mn></math></span> and <span><math><mi>γ</mi><mo>=</mo><mn>4</mn></math></span>, we prove an existence result of the ground states, which are characterized by constrained mountain-pass critical points of the corresponding energy functional. Furthermore, precise limit profiles of the ground states are obtained when the masses of two components vanish. These results are a continuation of the works <span><span>[5]</span></span> and <span><span>[14]</span></span> concerning normalized solutions for Hartree equations to Hartree-Fock system with Hardy-Littlewood-Sobolev critical exponent.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"444 ","pages":"Article 113565"},"PeriodicalIF":2.4000,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Qualitative analysis for Hartree-Fock system with Hardy-Littlewood-Sobolev critical exponent\",\"authors\":\"Meng Li , Haoyuan Xu , Meihua Yang , Maoding Zhen\",\"doi\":\"10.1016/j.jde.2025.113565\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>In this paper, we develop an exhaustive analysis on standing waves with prescribed mass for the coupled Hartree-Fock system as following, which is introduced by Hartree in the 1920's and developed by Fock for describing large systems of identical fermions,<span><span><span><math><mrow><mrow><mo>{</mo><mtable><mtr><mtd></mtd><mtd><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>u</mi><mo>=</mo><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup><mo>⋆</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>u</mi><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>γ</mi></mrow></msup><mo>⋆</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>u</mi><mo>+</mo><mi>β</mi><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>γ</mi></mrow></msup><mo>⋆</mo><mo>|</mo><mi>v</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>u</mi></mtd></mtr><mtr><mtd></mtd><mtd><mo>−</mo><mi>Δ</mi><mi>v</mi><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>v</mi><mo>=</mo><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup><mo>⋆</mo><mo>|</mo><mi>v</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>v</mi><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>γ</mi></mrow></msup><mo>⋆</mo><mo>|</mo><mi>v</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>v</mi><mo>+</mo><mi>β</mi><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>γ</mi></mrow></msup><mo>⋆</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>v</mi></mtd></mtr></mtable></mrow><mspace></mspace><mtext>in </mtext><msup><mrow><mi>R</mi></mrow><mrow><mn>5</mn></mrow></msup></mrow></math></span></span></span> under mass constraint conditions<span><span><span><math><mrow><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mn>5</mn></mrow></msup></mrow></munder><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mn>5</mn></mrow></msup></mrow></munder><mo>|</mo><mi>v</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo></mrow></math></span></span></span> where <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>></mo><mn>0</mn><mo>(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo><mo>,</mo><mspace></mspace><mi>β</mi><mo>></mo><mn>0</mn></math></span>, <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mi>γ</mi><mo>≤</mo><mn>4</mn></math></span>, and the frequency <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>R</mi><mo>(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span> are a part of unknown and appear as Lagrange multipliers. Firstly, when <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mi>γ</mi><mo>=</mo><mn>2</mn></math></span> or <span><math><mi>α</mi><mo>=</mo><mn>2</mn><mo><</mo><mi>γ</mi><mo><</mo><mn>4</mn></math></span>, we establish an existence result by using the constrained minimization method. Then turning to the case of <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>2</mn><mo><</mo><mi>γ</mi><mo><</mo><mn>4</mn></math></span> or <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>2</mn><mo>,</mo><mi>γ</mi><mo>=</mo><mn>4</mn></math></span>, we show that the problem admits a ground state and an excited state, which are characterized respectively by a local minimizer and a mountain-pass critical point of the corresponding energy functional. Moreover, we give a precise mass collapse behaviors of the ground state solutions for <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>2</mn><mo><</mo><mi>γ</mi><mo><</mo><mn>4</mn></math></span> and <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>2</mn><mo>,</mo><mi>γ</mi><mo>=</mo><mn>4</mn></math></span> when the masses of two components vanish at the same rate. We also give a precise limit profiles of the mountain pass solutions when <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>2</mn><mo>,</mo><mi>γ</mi><mo>=</mo><mn>4</mn></math></span> as <span><math><mo>(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>)</mo><mo>→</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></span>. This seems to be the first contribution regarding the multiplicity as well as the synchronized mass collapse behaviors of the normalized solutions to Hartree-Fock system with Hardy-Littlewood-Sobolev critical exponent. Finally, when <span><math><mn>2</mn><mo>≤</mo><mi>α</mi><mo><</mo><mn>4</mn></math></span> and <span><math><mi>γ</mi><mo>=</mo><mn>4</mn></math></span>, we prove an existence result of the ground states, which are characterized by constrained mountain-pass critical points of the corresponding energy functional. Furthermore, precise limit profiles of the ground states are obtained when the masses of two components vanish. These results are a continuation of the works <span><span>[5]</span></span> and <span><span>[14]</span></span> concerning normalized solutions for Hartree equations to Hartree-Fock system with Hardy-Littlewood-Sobolev critical exponent.</div></div>\",\"PeriodicalId\":15623,\"journal\":{\"name\":\"Journal of Differential Equations\",\"volume\":\"444 \",\"pages\":\"Article 113565\"},\"PeriodicalIF\":2.4000,\"publicationDate\":\"2025-06-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Differential Equations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022039625005923\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022039625005923","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Qualitative analysis for Hartree-Fock system with Hardy-Littlewood-Sobolev critical exponent
In this paper, we develop an exhaustive analysis on standing waves with prescribed mass for the coupled Hartree-Fock system as following, which is introduced by Hartree in the 1920's and developed by Fock for describing large systems of identical fermions, under mass constraint conditions where , , and the frequency are a part of unknown and appear as Lagrange multipliers. Firstly, when or , we establish an existence result by using the constrained minimization method. Then turning to the case of or , we show that the problem admits a ground state and an excited state, which are characterized respectively by a local minimizer and a mountain-pass critical point of the corresponding energy functional. Moreover, we give a precise mass collapse behaviors of the ground state solutions for and when the masses of two components vanish at the same rate. We also give a precise limit profiles of the mountain pass solutions when as . This seems to be the first contribution regarding the multiplicity as well as the synchronized mass collapse behaviors of the normalized solutions to Hartree-Fock system with Hardy-Littlewood-Sobolev critical exponent. Finally, when and , we prove an existence result of the ground states, which are characterized by constrained mountain-pass critical points of the corresponding energy functional. Furthermore, precise limit profiles of the ground states are obtained when the masses of two components vanish. These results are a continuation of the works [5] and [14] concerning normalized solutions for Hartree equations to Hartree-Fock system with Hardy-Littlewood-Sobolev critical exponent.
期刊介绍:
The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools.
Research Areas Include:
• Mathematical control theory
• Ordinary differential equations
• Partial differential equations
• Stochastic differential equations
• Topological dynamics
• Related topics