{"title":"Competing Potentials Effect on Positive Solutions to a Schrödinger-Poisson System","authors":"Haining Fan, Xiaochun Liu","doi":"10.1007/s10114-024-3124-z","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we study the multiplicity and concentration of positive solutions for a Schrödinger-Poisson system involving sign-changing potential and the nonlinearity <i>K</i>(<i>x</i>)∣<i>u</i>∣<sup><i>p</i>−2</sup><i>u</i> (2 < <i>p</i> < 4) in ℝ<sup>3</sup>. Such a problem cannot be studied by variational methods in a standard way, even by restricting its corresponding energy functional on the Nehari manifold since its (PS) sequence may not be bounded. By some new analytic techniques and the Ljusternik-Schnirelmann category theory, we relate the concentration and the number of positive solutions to the category of the global minima set of a suitable ground energy function. Furthermore, we investigate the asymptotic behavior of the solutions. In particular, we do not use Pohozaev equality in this work.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"41 4","pages":"1055 - 1090"},"PeriodicalIF":0.8000,"publicationDate":"2024-12-26","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/s10114-024-3124-z","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
In this paper, we study the multiplicity and concentration of positive solutions for a Schrödinger-Poisson system involving sign-changing potential and the nonlinearity K(x)∣u∣p−2u (2 < p < 4) in ℝ3. Such a problem cannot be studied by variational methods in a standard way, even by restricting its corresponding energy functional on the Nehari manifold since its (PS) sequence may not be bounded. By some new analytic techniques and the Ljusternik-Schnirelmann category theory, we relate the concentration and the number of positive solutions to the category of the global minima set of a suitable ground energy function. Furthermore, we investigate the asymptotic behavior of the solutions. In particular, we do not use Pohozaev equality in this work.
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.