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Concentrating standing waves for the fractional Schr?dinger equation with critical nonlinearities
Li,Suhong1,2; Ding,Yanheng2; Chen,Yu2
2015-12-24
Source PublicationBoundary Value Problems
ISSN1687-2770
Volume2015Issue:1
AbstractAbstractWe 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.
Keywordground state concentration standing waves nonlocal 35Q40 49J35
DOI10.1186/s13661-015-0507-1
Language英语
WOS IDBMC:10.1186/s13661-015-0507-1
PublisherSpringer International Publishing
Citation statistics
Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/303
Collection数学所
Corresponding AuthorLi,Suhong
Affiliation1.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
Recommended Citation
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).
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