Concentrating standing waves for the fractional Schr?dinger equation with critical nonlinearities

doi:10.1186/s13661-015-0507-1

CSpace > 数学所

	Concentrating standing waves for the fractional Schr?dinger equation with critical nonlinearities
	Li,Suhong 1,2; Ding,Yanheng2 ; Chen,Yu 2
	2015-12-24
发表期刊	Boundary Value Problems
ISSN	1687-2770
卷号	2015 期号:1
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.
关键词	ground state concentration standing waves nonlocal 35Q40 49J35
DOI	10.1186/s13661-015-0507-1
语种	英语
WOS记录号	BMC:10.1186/s13661-015-0507-1
出版者	Springer International Publishing
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/303
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).