{"title":"具有一般非线性的基尔霍夫方程的基态归一化解:质量超临界情况","authors":"Qun Wang, Aixia Qian","doi":"10.1186/s13660-024-03086-5","DOIUrl":null,"url":null,"abstract":"We study the following nonlinear mass supercritical Kirchhoff equation: $$ - \\biggl(a+b \\int _{\\mathbb{R}^{N}} \\vert \\nabla u \\vert ^{2} \\biggr) \\triangle u+ \\mu u=f(u) \\quad \\text{in } {\\mathbb{R}^{N}}, $$ where $a ,b,m>0$ are prescribed, and the normalized constrain $\\int _{\\mathbb{R}^{N}}|u|^{2}\\,dx =m$ is satisfied in the case $1\\leq N\\leq 3$ . The nonlinearity f is more general and satisfies weak mass supercritical conditions. Under some mild assumptions, we establish the existence of ground state when $1\\leq N\\leq 3$ and obtain infinitely many radial solutions when $2\\leq N\\leq 3$ by constructing a particular bounded Palais–Smale sequence.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Ground state normalized solutions to the Kirchhoff equation with general nonlinearities: mass supercritical case\",\"authors\":\"Qun Wang, Aixia Qian\",\"doi\":\"10.1186/s13660-024-03086-5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the following nonlinear mass supercritical Kirchhoff equation: $$ - \\\\biggl(a+b \\\\int _{\\\\mathbb{R}^{N}} \\\\vert \\\\nabla u \\\\vert ^{2} \\\\biggr) \\\\triangle u+ \\\\mu u=f(u) \\\\quad \\\\text{in } {\\\\mathbb{R}^{N}}, $$ where $a ,b,m>0$ are prescribed, and the normalized constrain $\\\\int _{\\\\mathbb{R}^{N}}|u|^{2}\\\\,dx =m$ is satisfied in the case $1\\\\leq N\\\\leq 3$ . The nonlinearity f is more general and satisfies weak mass supercritical conditions. Under some mild assumptions, we establish the existence of ground state when $1\\\\leq N\\\\leq 3$ and obtain infinitely many radial solutions when $2\\\\leq N\\\\leq 3$ by constructing a particular bounded Palais–Smale sequence.\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-04-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1186/s13660-024-03086-5\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1186/s13660-024-03086-5","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
Ground state normalized solutions to the Kirchhoff equation with general nonlinearities: mass supercritical case
We study the following nonlinear mass supercritical Kirchhoff equation: $$ - \biggl(a+b \int _{\mathbb{R}^{N}} \vert \nabla u \vert ^{2} \biggr) \triangle u+ \mu u=f(u) \quad \text{in } {\mathbb{R}^{N}}, $$ where $a ,b,m>0$ are prescribed, and the normalized constrain $\int _{\mathbb{R}^{N}}|u|^{2}\,dx =m$ is satisfied in the case $1\leq N\leq 3$ . The nonlinearity f is more general and satisfies weak mass supercritical conditions. Under some mild assumptions, we establish the existence of ground state when $1\leq N\leq 3$ and obtain infinitely many radial solutions when $2\leq N\leq 3$ by constructing a particular bounded Palais–Smale sequence.
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.