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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 Publication Boundary Value Problems ISSN 1687-2770 Volume 2015Issue:1 Abstract 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. Keyword ground state concentration standing waves nonlocal 35Q40 49J35 DOI 10.1186/s13661-015-0507-1 Language 英语 WOS ID BMC:10.1186/s13661-015-0507-1 Publisher Springer International Publishing Citation statistics Document Type 期刊论文 Identifier http://ir.amss.ac.cn/handle/2S8OKBNM/303 Collection 数学所 Corresponding Author Li,Suhong Affiliation 1.Hebei Normal University of Science and Technology; Institute of Mathematics and Information Technology2.Chinese Academy of Sciences; Institute of Mathematics, Academy of Mathematics and Systems Science Recommended CitationGB/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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