{"title":"涉及 p-Laplacian 的准线性薛定谔方程解的存在性和渐近行为","authors":"Jiaxin Cao, Youjun Wang","doi":"10.1007/s11784-024-01118-7","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we investigate the existence and asymptotic behavior of positive solutions for quasilinear Schrödinger equations involving <i>p</i>-Laplacian </p><span>$$\\begin{aligned} -\\Delta _{p}u + \\kappa \\Delta _{p}(u^2)u + (\\lambda A( x) + 1)|u|^{p-2}u = h(u), \\quad u\\in W^{1,p}(\\mathbb {R}^N), \\end{aligned}$$</span><p>where <span>\\(2<p<N\\)</span>, <span>\\(\\kappa ,\\)</span> <span>\\(\\lambda \\)</span> are parameters and <i>A</i>(<i>x</i>) is a potential. The problem is quite sensitive to the sign of <span>\\(\\kappa \\)</span> and there have been many results for <span>\\(\\kappa \\le 0.\\)</span> By means of minimization on the Nehari manifold together with perturbation type techniques, we establish the existence of positive solutions for small <span>\\(\\kappa >0\\)</span> and large <span>\\(\\lambda \\)</span>. Moreover, we show that the solutions <span>\\(u_{\\kappa ,\\lambda }\\)</span> converge in <span>\\(W^{1,p}\\)</span> to a positive solution of <i>p</i>-Laplacian in a bounded domain as <span>\\((\\kappa ,\\lambda )\\rightarrow (0^+,+\\infty )\\)</span>. Our results extend some known results of <span>\\(\\kappa \\le 0\\)</span>.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Existence and asymptotic behavior of solutions for quasilinear Schrödinger equations involving p-Laplacian\",\"authors\":\"Jiaxin Cao, Youjun Wang\",\"doi\":\"10.1007/s11784-024-01118-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper, we investigate the existence and asymptotic behavior of positive solutions for quasilinear Schrödinger equations involving <i>p</i>-Laplacian </p><span>$$\\\\begin{aligned} -\\\\Delta _{p}u + \\\\kappa \\\\Delta _{p}(u^2)u + (\\\\lambda A( x) + 1)|u|^{p-2}u = h(u), \\\\quad u\\\\in W^{1,p}(\\\\mathbb {R}^N), \\\\end{aligned}$$</span><p>where <span>\\\\(2<p<N\\\\)</span>, <span>\\\\(\\\\kappa ,\\\\)</span> <span>\\\\(\\\\lambda \\\\)</span> are parameters and <i>A</i>(<i>x</i>) is a potential. The problem is quite sensitive to the sign of <span>\\\\(\\\\kappa \\\\)</span> and there have been many results for <span>\\\\(\\\\kappa \\\\le 0.\\\\)</span> By means of minimization on the Nehari manifold together with perturbation type techniques, we establish the existence of positive solutions for small <span>\\\\(\\\\kappa >0\\\\)</span> and large <span>\\\\(\\\\lambda \\\\)</span>. Moreover, we show that the solutions <span>\\\\(u_{\\\\kappa ,\\\\lambda }\\\\)</span> converge in <span>\\\\(W^{1,p}\\\\)</span> to a positive solution of <i>p</i>-Laplacian in a bounded domain as <span>\\\\((\\\\kappa ,\\\\lambda )\\\\rightarrow (0^+,+\\\\infty )\\\\)</span>. Our results extend some known results of <span>\\\\(\\\\kappa \\\\le 0\\\\)</span>.</p>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-07-05\",\"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/s11784-024-01118-7\",\"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/s11784-024-01118-7","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
where \(2<p<N\), \(\kappa ,\)\(\lambda \) are parameters and A(x) is a potential. The problem is quite sensitive to the sign of \(\kappa \) and there have been many results for \(\kappa \le 0.\) By means of minimization on the Nehari manifold together with perturbation type techniques, we establish the existence of positive solutions for small \(\kappa >0\) and large \(\lambda \). Moreover, we show that the solutions \(u_{\kappa ,\lambda }\) converge in \(W^{1,p}\) to a positive solution of p-Laplacian in a bounded domain as \((\kappa ,\lambda )\rightarrow (0^+,+\infty )\). Our results extend some known results of \(\kappa \le 0\).
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.