{"title":"具有规定质量的薛定谔-泊松方程的解的存在性和多重性","authors":"Xueqin Peng","doi":"10.1007/s13324-024-00963-6","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we study the existence and multiplicity for the following Schrödinger–Poisson equation </p><div><div><span>$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} -\\Delta u+\\lambda u-\\kappa (|x|^{-1}*|u|^2)u=f(u),&{}\\text {in}~~{\\mathbb {R}}^{3},\\\\ u>0,~\\displaystyle \\int _{{\\mathbb {R}}^{3}}u^2dx=a^2, \\end{array}\\right. } \\end{aligned}$$</span></div></div><p>where <span>\\(a>0\\)</span> is a prescribed mass, <span>\\(\\kappa \\in {\\mathbb {R}}\\setminus \\{0\\}\\)</span> and <span>\\(\\lambda \\in {\\mathbb {R}}\\)</span> is an undetermined parameter which appears as a Lagrange multiplier. Our results are threefold: (i) for the case <span>\\(\\kappa <0\\)</span>, we obtain the normalized ground state solution for <span>\\(a>0\\)</span> small by working on the Pohozaev manifold, where <i>f</i> satisfies the <span>\\(L^2\\)</span>-supercritical and Sobolev subcritical conditions, and the behavior of the normalized ground state energy <span>\\(c_a\\)</span> is also obtained; (ii) we prove that the above equation possesses infinitely many radial solutions whose energy converges to infinity; (iii) for <span>\\(\\kappa >0\\)</span> and <span>\\(f(u)=|u|^{4}u\\)</span>, we revisit the Brézis–Nirenberg problem with a nonlocal perturbation and obtain infinitely many radial solutions with negative energy. Our results implement some existing results about the Schrödinger–Poisson equation in the <span>\\(L^2\\)</span>-constraint setting.</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"14 5","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Existence and multiplicity of solutions for the Schrödinger–Poisson equation with prescribed mass\",\"authors\":\"Xueqin Peng\",\"doi\":\"10.1007/s13324-024-00963-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we study the existence and multiplicity for the following Schrödinger–Poisson equation </p><div><div><span>$$\\\\begin{aligned} {\\\\left\\\\{ \\\\begin{array}{ll} -\\\\Delta u+\\\\lambda u-\\\\kappa (|x|^{-1}*|u|^2)u=f(u),&{}\\\\text {in}~~{\\\\mathbb {R}}^{3},\\\\\\\\ u>0,~\\\\displaystyle \\\\int _{{\\\\mathbb {R}}^{3}}u^2dx=a^2, \\\\end{array}\\\\right. } \\\\end{aligned}$$</span></div></div><p>where <span>\\\\(a>0\\\\)</span> is a prescribed mass, <span>\\\\(\\\\kappa \\\\in {\\\\mathbb {R}}\\\\setminus \\\\{0\\\\}\\\\)</span> and <span>\\\\(\\\\lambda \\\\in {\\\\mathbb {R}}\\\\)</span> is an undetermined parameter which appears as a Lagrange multiplier. Our results are threefold: (i) for the case <span>\\\\(\\\\kappa <0\\\\)</span>, we obtain the normalized ground state solution for <span>\\\\(a>0\\\\)</span> small by working on the Pohozaev manifold, where <i>f</i> satisfies the <span>\\\\(L^2\\\\)</span>-supercritical and Sobolev subcritical conditions, and the behavior of the normalized ground state energy <span>\\\\(c_a\\\\)</span> is also obtained; (ii) we prove that the above equation possesses infinitely many radial solutions whose energy converges to infinity; (iii) for <span>\\\\(\\\\kappa >0\\\\)</span> and <span>\\\\(f(u)=|u|^{4}u\\\\)</span>, we revisit the Brézis–Nirenberg problem with a nonlocal perturbation and obtain infinitely many radial solutions with negative energy. Our results implement some existing results about the Schrödinger–Poisson equation in the <span>\\\\(L^2\\\\)</span>-constraint setting.</p></div>\",\"PeriodicalId\":48860,\"journal\":{\"name\":\"Analysis and Mathematical Physics\",\"volume\":\"14 5\",\"pages\":\"\"},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2024-08-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Analysis and Mathematical Physics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s13324-024-00963-6\",\"RegionNum\":3,\"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":"Analysis and Mathematical Physics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s13324-024-00963-6","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
where \(a>0\) is a prescribed mass, \(\kappa \in {\mathbb {R}}\setminus \{0\}\) and \(\lambda \in {\mathbb {R}}\) is an undetermined parameter which appears as a Lagrange multiplier. Our results are threefold: (i) for the case \(\kappa <0\), we obtain the normalized ground state solution for \(a>0\) small by working on the Pohozaev manifold, where f satisfies the \(L^2\)-supercritical and Sobolev subcritical conditions, and the behavior of the normalized ground state energy \(c_a\) is also obtained; (ii) we prove that the above equation possesses infinitely many radial solutions whose energy converges to infinity; (iii) for \(\kappa >0\) and \(f(u)=|u|^{4}u\), we revisit the Brézis–Nirenberg problem with a nonlocal perturbation and obtain infinitely many radial solutions with negative energy. Our results implement some existing results about the Schrödinger–Poisson equation in the \(L^2\)-constraint setting.
期刊介绍:
Analysis and Mathematical Physics (AMP) publishes current research results as well as selected high-quality survey articles in real, complex, harmonic; and geometric analysis originating and or having applications in mathematical physics. The journal promotes dialog among specialists in these areas.