{"title":"Normalized solutions for Kirchhoff equations with Sobolev critical exponent and mixed nonlinearities","authors":"Sitong Chen, Xianhua Tang","doi":"10.1007/s00208-024-02982-x","DOIUrl":null,"url":null,"abstract":"<p>This paper focuses on the existence of normalized solutions for the following Kirchhoff equation: </p><span>$$\\begin{aligned} \\left\\{ \\begin{array}{ll} -\\left( a+b\\int _{{\\mathbb {R}}^3}|\\nabla u|^2\\textrm{d}x\\right) \\Delta u+\\lambda u=u^5+\\mu |u|^{q-2}u, & x\\in {\\mathbb {R}}^3, \\\\ \\int _{{\\mathbb {R}}^3}u^2\\textrm{d}x=c, \\\\ \\end{array} \\right. \\end{aligned}$$</span><p>where <span>\\(a,b,c>0\\)</span>, <span>\\(\\mu \\in {\\mathbb {R}}\\)</span> and <span>\\(2<q<6\\)</span>, <span>\\(\\lambda \\in {\\mathbb {R}}\\)</span> will arise as a Lagrange multiplier that is not a priori given. By using new analytical techniques, the paper establishes several existence results for the case <span>\\(\\mu >0\\)</span>: </p><ol>\n<li>\n<span>(1)</span>\n<p>The existence of two solutions, one being a local minimizer and the other of mountain-pass type, under explicit conditions on <i>c</i> when <span>\\(2<q<\\frac{10}{3}\\)</span>.</p>\n</li>\n<li>\n<span>(2)</span>\n<p>The existence of a mountain-pass type solution under explicit conditions on <i>c</i> when <span>\\(\\frac{10}{3}\\le q<\\frac{14}{3}\\)</span>.</p>\n</li>\n<li>\n<span>(3)</span>\n<p>The existence of a ground state solution for all <span>\\(c>0\\)</span> when <span>\\(\\frac{14}{3}\\le q<6\\)</span>.</p>\n</li>\n</ol><p> Furthermore, the paper presents the first non-existence result for the case <span>\\(\\mu \\le 0\\)</span> and <span>\\(2<q<6\\)</span>. In particular, refined estimates of energy levels are proposed, suggesting a new threshold of compactness in the <span>\\(L^2\\)</span>-constraint. This study addresses an open problem for <span>\\(2<q<\\frac{10}{3}\\)</span> and fills a gap in the case <span>\\(\\frac{10}{3}\\le q<\\frac{14}{3}\\)</span>. We believe that our approach can be applied to a broader range of nonlinear terms with Sobolev critical growth, and the underlying ideas have potential for future development and applicability.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"158 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematische Annalen","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00208-024-02982-x","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
This paper focuses on the existence of normalized solutions for the following Kirchhoff equation:
where \(a,b,c>0\), \(\mu \in {\mathbb {R}}\) and \(2<q<6\), \(\lambda \in {\mathbb {R}}\) will arise as a Lagrange multiplier that is not a priori given. By using new analytical techniques, the paper establishes several existence results for the case \(\mu >0\):
(1)
The existence of two solutions, one being a local minimizer and the other of mountain-pass type, under explicit conditions on c when \(2<q<\frac{10}{3}\).
(2)
The existence of a mountain-pass type solution under explicit conditions on c when \(\frac{10}{3}\le q<\frac{14}{3}\).
(3)
The existence of a ground state solution for all \(c>0\) when \(\frac{14}{3}\le q<6\).
Furthermore, the paper presents the first non-existence result for the case \(\mu \le 0\) and \(2<q<6\). In particular, refined estimates of energy levels are proposed, suggesting a new threshold of compactness in the \(L^2\)-constraint. This study addresses an open problem for \(2<q<\frac{10}{3}\) and fills a gap in the case \(\frac{10}{3}\le q<\frac{14}{3}\). We believe that our approach can be applied to a broader range of nonlinear terms with Sobolev critical growth, and the underlying ideas have potential for future development and applicability.
期刊介绍:
Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin.
The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin.
Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.