{"title":"具有非自主非局部扰动的 HLS 上临界聚焦 Choquard 方程的归一化解","authors":"Ziheng Zhang, Jianlun Liu, Hong-Rui Sun","doi":"10.1007/s13324-024-00979-y","DOIUrl":null,"url":null,"abstract":"<div><p>This paper is concerned with the following HLS upper critical focusing Choquard equation with a non-autonomous nonlocal perturbation </p><div><div><span>$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} -{\\Delta }u-\\mu (I_\\alpha *[h|u|^p])h|u|^{p-2}u-(I_\\alpha *|u|^{2^*_\\alpha })|u|^{2^*_\\alpha -2}u=\\lambda u\\ \\ \\text{ in }\\ \\mathbb {R}^N, \\\\ \\int _{\\mathbb {R}^N} u^2 dx = c, \\end{array}\\right. } \\end{aligned}$$</span></div></div><p>where <span>\\(\\mu ,c>0\\)</span>, <span>\\(N \\ge 3\\)</span>, <span>\\(0<\\alpha <N\\)</span>, <span>\\(2_\\alpha :=\\frac{N+\\alpha }{N}<p<2^*_\\alpha :=\\frac{N+\\alpha }{N-2}\\)</span>, <span>\\(\\lambda \\in \\mathbb {R}\\)</span> is a Lagrange multiplier, <span>\\(I_\\alpha \\)</span> is the Riesz potential and <span>\\(h:\\mathbb {R}^N\\rightarrow (0,\\infty )\\)</span> is a continuous function. Under a class of reasonable assumptions on <i>h</i>, we prove the existence of normalized solutions to the above problem for the case <span>\\(\\frac{N+\\alpha +2}{N}\\le p<\\frac{N+\\alpha }{N-2}\\)</span> and discuss its asymptotical behaviors as <span>\\(\\mu \\rightarrow 0^+\\)</span> and <span>\\(c\\rightarrow 0^+\\)</span> respectively. When <span>\\(\\frac{N+\\alpha }{N}<p<\\frac{N+\\alpha +2}{N}\\)</span>, we obtain the existence of one local minimizer after considering a suitable minimization problem.</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"14 6","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Normalized solutions to HLS upper critical focusing Choquard equation with a non-autonomous nonlocal perturbation\",\"authors\":\"Ziheng Zhang, Jianlun Liu, Hong-Rui Sun\",\"doi\":\"10.1007/s13324-024-00979-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>This paper is concerned with the following HLS upper critical focusing Choquard equation with a non-autonomous nonlocal perturbation </p><div><div><span>$$\\\\begin{aligned} {\\\\left\\\\{ \\\\begin{array}{ll} -{\\\\Delta }u-\\\\mu (I_\\\\alpha *[h|u|^p])h|u|^{p-2}u-(I_\\\\alpha *|u|^{2^*_\\\\alpha })|u|^{2^*_\\\\alpha -2}u=\\\\lambda u\\\\ \\\\ \\\\text{ in }\\\\ \\\\mathbb {R}^N, \\\\\\\\ \\\\int _{\\\\mathbb {R}^N} u^2 dx = c, \\\\end{array}\\\\right. } \\\\end{aligned}$$</span></div></div><p>where <span>\\\\(\\\\mu ,c>0\\\\)</span>, <span>\\\\(N \\\\ge 3\\\\)</span>, <span>\\\\(0<\\\\alpha <N\\\\)</span>, <span>\\\\(2_\\\\alpha :=\\\\frac{N+\\\\alpha }{N}<p<2^*_\\\\alpha :=\\\\frac{N+\\\\alpha }{N-2}\\\\)</span>, <span>\\\\(\\\\lambda \\\\in \\\\mathbb {R}\\\\)</span> is a Lagrange multiplier, <span>\\\\(I_\\\\alpha \\\\)</span> is the Riesz potential and <span>\\\\(h:\\\\mathbb {R}^N\\\\rightarrow (0,\\\\infty )\\\\)</span> is a continuous function. Under a class of reasonable assumptions on <i>h</i>, we prove the existence of normalized solutions to the above problem for the case <span>\\\\(\\\\frac{N+\\\\alpha +2}{N}\\\\le p<\\\\frac{N+\\\\alpha }{N-2}\\\\)</span> and discuss its asymptotical behaviors as <span>\\\\(\\\\mu \\\\rightarrow 0^+\\\\)</span> and <span>\\\\(c\\\\rightarrow 0^+\\\\)</span> respectively. When <span>\\\\(\\\\frac{N+\\\\alpha }{N}<p<\\\\frac{N+\\\\alpha +2}{N}\\\\)</span>, we obtain the existence of one local minimizer after considering a suitable minimization problem.</p></div>\",\"PeriodicalId\":48860,\"journal\":{\"name\":\"Analysis and Mathematical Physics\",\"volume\":\"14 6\",\"pages\":\"\"},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2024-11-05\",\"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/s13324-024-00979-y\",\"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/s13324-024-00979-y","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
where \(\mu ,c>0\), \(N \ge 3\), \(0<\alpha <N\), \(2_\alpha :=\frac{N+\alpha }{N}<p<2^*_\alpha :=\frac{N+\alpha }{N-2}\), \(\lambda \in \mathbb {R}\) is a Lagrange multiplier, \(I_\alpha \) is the Riesz potential and \(h:\mathbb {R}^N\rightarrow (0,\infty )\) is a continuous function. Under a class of reasonable assumptions on h, we prove the existence of normalized solutions to the above problem for the case \(\frac{N+\alpha +2}{N}\le p<\frac{N+\alpha }{N-2}\) and discuss its asymptotical behaviors as \(\mu \rightarrow 0^+\) and \(c\rightarrow 0^+\) respectively. When \(\frac{N+\alpha }{N}<p<\frac{N+\alpha +2}{N}\), we obtain the existence of one local minimizer after considering a suitable minimization problem.
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.