{"title":"Single Peak Solutions for a Schrödinger Equation with Variable Exponent","authors":"Zhong Yuan Liu, Peng Luo, Hua Fei Xie","doi":"10.1007/s10114-023-2616-6","DOIUrl":null,"url":null,"abstract":"<div><p>We study the following Schrödinger equation with variable exponent </p><div><div><span>$$ - \\Delta u + u = {u^{p + \\epsilon a(x)}},\\,\\,\\,u > 0\\,\\,{\\rm{in}}\\,\\,{\\mathbb{R}^N},$$</span></div></div><p> where <span>\\(\\epsilon > 0,\\,\\,1 < p < {{N + 2} \\over {N - 2}},\\,\\,a(x) \\in {C^1}({\\mathbb{R}^N}) \\cap {L^\\infty }({\\mathbb{R}^N}),\\,\\,N \\ge 3\\)</span> Under certain assumptions on a vector field related to <i>a</i>(<i>x</i>), we use the Lyapunov–Schmidt reduction to show the existence of single peak solutions to the above problem. We also obtain local uniqueness and exact multiplicity results for this problem by the Pohozaev type identity.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"39 11","pages":"2207 - 2218"},"PeriodicalIF":0.8000,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Sinica-English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10114-023-2616-6","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We study the following Schrödinger equation with variable exponent
$$ - \Delta u + u = {u^{p + \epsilon a(x)}},\,\,\,u > 0\,\,{\rm{in}}\,\,{\mathbb{R}^N},$$
where \(\epsilon > 0,\,\,1 < p < {{N + 2} \over {N - 2}},\,\,a(x) \in {C^1}({\mathbb{R}^N}) \cap {L^\infty }({\mathbb{R}^N}),\,\,N \ge 3\) Under certain assumptions on a vector field related to a(x), we use the Lyapunov–Schmidt reduction to show the existence of single peak solutions to the above problem. We also obtain local uniqueness and exact multiplicity results for this problem by the Pohozaev type identity.
期刊介绍:
Acta Mathematica Sinica, established by the Chinese Mathematical Society in 1936, is the first and the best mathematical journal in China. In 1985, Acta Mathematica Sinica is divided into English Series and Chinese Series. The English Series is a monthly journal, publishing significant research papers from all branches of pure and applied mathematics. It provides authoritative reviews of current developments in mathematical research. Contributions are invited from researchers from all over the world.