无界域中的索波列夫紧凑嵌入及其在椭圆方程中的应用

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-10-26 DOI:10.1016/j.jmaa.2024.129001

Ryuji Kajikiya

{"title":"无界域中的索波列夫紧凑嵌入及其在椭圆方程中的应用","authors":"Ryuji Kajikiya","doi":"10.1016/j.jmaa.2024.129001","DOIUrl":null,"url":null,"abstract":"<div><div>We give a necessary and sufficient condition for the compact embedding of the Sobolev space <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mi>m</mi><mo>,</mo><mi>p</mi></mrow></msubsup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> for unbounded domains Ω. Applying this condition, we can decide whether the compact embedding holds or not. We give several examples of unbounded domains Ω satisfying the compact embedding. Using our condition, we study a semilinear elliptic equation in unbounded domains and prove the existence of a positive solution and infinitely many solutions.</div></div>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Sobolev compact embeddings in unbounded domains and its applications to elliptic equations\",\"authors\":\"Ryuji Kajikiya\",\"doi\":\"10.1016/j.jmaa.2024.129001\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We give a necessary and sufficient condition for the compact embedding of the Sobolev space <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mi>m</mi><mo>,</mo><mi>p</mi></mrow></msubsup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> for unbounded domains Ω. Applying this condition, we can decide whether the compact embedding holds or not. We give several examples of unbounded domains Ω satisfying the compact embedding. Using our condition, we study a semilinear elliptic equation in unbounded domains and prove the existence of a positive solution and infinitely many solutions.</div></div>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-10-26\",\"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://www.sciencedirect.com/science/article/pii/S0022247X24009235\",\"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://www.sciencedirect.com/science/article/pii/S0022247X24009235","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}

引用次数: 0

摘要

我们给出了无界域 Ω 的 Sobolev 空间 W0m,p(Ω) 的紧凑嵌入的必要条件和充分条件。利用这个条件，我们就能判断紧凑嵌入是否成立。我们举了几个满足紧凑嵌入的无界域 Ω 的例子。利用我们的条件，我们研究了无界域中的一个半线性椭圆方程，并证明了一个正解和无穷多个解的存在。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Sobolev compact embeddings in unbounded domains and its applications to elliptic equations

We give a necessary and sufficient condition for the compact embedding of the Sobolev space

W_{0}^{m, p} (Ω)

for unbounded domains Ω. Applying this condition, we can decide whether the compact embedding holds or not. We give several examples of unbounded domains Ω satisfying the compact embedding. Using our condition, we study a semilinear elliptic equation in unbounded domains and prove the existence of a positive solution and infinitely many solutions.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

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.