Concentrating standing waves for the fractional Schr?dinger equation with critical nonlinearities
文献类型：期刊论文
| 作者 | Li,Suhong1,2; Ding,Yanheng2 ; Chen,Yu2
|
| 刊名 | Boundary Value Problems
 |
| 出版日期 | 2015-12-24 |
| 卷号 | 2015期号:1 |
| 关键词 | ground state concentration standing waves nonlocal 35Q40 49J35 |
| ISSN号 | 1687-2770 |
| DOI | 10.1186/s13661-015-0507-1 |
| 英文摘要 | AbstractWe study the following nonlocal Schr?dinger equations: Iε2s(?Δ)su+V(x)u=W(x)f(u),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned}& \varepsilon^{2s}(-\Delta)^{s}u+V(x)u=W(x)f(u), \end{aligned}$$ \end{document}IIε2s(?Δ)su+V(x)u=W(x)(f(u)+u2s??1),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned}& \varepsilon^{2s}(-\Delta)^{s}u+V(x)u=W(x) \bigl(f(u)+u^{2^{*}_{s}-1}\bigr), \end{aligned}$$ \end{document} for u∈Hs(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u\in H^{s}( \mathbb{R}^{N})$\end{document}, where f(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(u)$\end{document} is superlinear and subcritical, 2s?=2NN?2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2^{*}_{s}= \frac{2N}{N-2s}$\end{document} if N>2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N>2s$\end{document}. V(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V(x)$\end{document} and W(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$W(x)$\end{document} are sufficiently smooth potential with infV(x)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\inf V(x)>0$\end{document}, infW(x)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\inf W(x)>0$\end{document}, and ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon>0$\end{document} is a small number. Under proper assumptions, we explore the existence, concentration phenomenon, convergence, and decay estimate of semiclassical solutions of (I) and (II), respectively. Compared with some existing issues, the most interesting results obtained here are therefore: the concentration phenomenon depends on competing potential functions; the nonlocal critical problem (II) is considered; unlike the classical case s=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$s=1$\end{document}, the decay estimate of solution to (I) or (II) is of polynomial instead of exponential form, due to the nonlocal effect. |
| 语种 | 英语 |
| WOS记录号 | BMC:10.1186/S13661-015-0507-1 |
| 出版者 | Springer International Publishing |
| 源URL | [http://ir.amss.ac.cn/handle/2S8OKBNM/302]  |
| 专题 | 数学所 |
| 通讯作者 | Li,Suhong |
| 作者单位 | 1.Hebei Normal University of Science and Technology; Institute of Mathematics and Information Technology 2.Chinese Academy of Sciences; Institute of Mathematics, Academy of Mathematics and Systems Science |
| 推荐引用方式 GB/T 7714 | Li,Suhong,Ding,Yanheng,Chen,Yu. Concentrating standing waves for the fractional Schr?dinger equation with critical nonlinearities[J]. Boundary Value Problems,2015,2015(1). |
| APA | Li,Suhong,Ding,Yanheng,&Chen,Yu.(2015).Concentrating standing waves for the fractional Schr?dinger equation with critical nonlinearities.Boundary Value Problems,2015(1). |
| MLA | Li,Suhong,et al."Concentrating standing waves for the fractional Schr?dinger equation with critical nonlinearities".Boundary Value Problems 2015.1(2015). |
入库方式： OAI收割
来源：数学与系统科学研究院
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