{"title":"具有亚线性非线性的 p-Laplace 问题的半空间单调性","authors":"Phuong Le","doi":"10.1007/s11118-024-10157-1","DOIUrl":null,"url":null,"abstract":"<p>We prove the monotonicity of positive solutions to the equation <span>\\(-\\Delta _p u = f(u)\\)</span> in <span>\\(\\mathbb {R}^N_+\\)</span> with zero Dirichlet boundary condition, where <span>\\(1<p<2\\)</span> and <span>\\(f:[0,+\\infty )\\rightarrow \\mathbb {R}\\)</span> is a continuous function which is positive and locally Lipschitz continuous in <span>\\((0,+\\infty )\\)</span> and <span>\\(\\liminf _{t\\rightarrow 0^+}\\frac{f(t)}{t^{p-1}}>0\\)</span>. Furthermore, we allow <i>f</i> to be sign-changing in the case <span>\\(\\frac{2N+2}{N+2}<p<2\\)</span>. The celebrated moving plane method will be used in the proofs of our results.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Monotonicity in Half-spaces for p-Laplace Problems with a Sublinear Nonlinearity\",\"authors\":\"Phuong Le\",\"doi\":\"10.1007/s11118-024-10157-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We prove the monotonicity of positive solutions to the equation <span>\\\\(-\\\\Delta _p u = f(u)\\\\)</span> in <span>\\\\(\\\\mathbb {R}^N_+\\\\)</span> with zero Dirichlet boundary condition, where <span>\\\\(1<p<2\\\\)</span> and <span>\\\\(f:[0,+\\\\infty )\\\\rightarrow \\\\mathbb {R}\\\\)</span> is a continuous function which is positive and locally Lipschitz continuous in <span>\\\\((0,+\\\\infty )\\\\)</span> and <span>\\\\(\\\\liminf _{t\\\\rightarrow 0^+}\\\\frac{f(t)}{t^{p-1}}>0\\\\)</span>. Furthermore, we allow <i>f</i> to be sign-changing in the case <span>\\\\(\\\\frac{2N+2}{N+2}<p<2\\\\)</span>. The celebrated moving plane method will be used in the proofs of our results.</p>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-07-18\",\"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/10.1007/s11118-024-10157-1\",\"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/10.1007/s11118-024-10157-1","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
Monotonicity in Half-spaces for p-Laplace Problems with a Sublinear Nonlinearity
We prove the monotonicity of positive solutions to the equation \(-\Delta _p u = f(u)\) in \(\mathbb {R}^N_+\) with zero Dirichlet boundary condition, where \(1<p<2\) and \(f:[0,+\infty )\rightarrow \mathbb {R}\) is a continuous function which is positive and locally Lipschitz continuous in \((0,+\infty )\) and \(\liminf _{t\rightarrow 0^+}\frac{f(t)}{t^{p-1}}>0\). Furthermore, we allow f to be sign-changing in the case \(\frac{2N+2}{N+2}<p<2\). The celebrated moving plane method will be used in the proofs of our results.
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.