变号深阱位势分数阶Schrödinger方程非平凡解的存在性和集中性

数学物理学报 ›› 2018, Vol. 38 ›› Issue (5): 893-902.

变号深阱位势分数阶Schrödinger方程非平凡解的存在性和集中性

王文波¹(),李全清^2,*()

¹ 云南大学数学与统计学院昆明 650500
² 红河学院数学学院云南蒙自 661100

收稿日期:2017-07-28 出版日期:2018-11-09 发布日期:2018-11-09
通讯作者: 李全清 E-mail:wenbowangmath@163.com;shili06171987@126.com
作者简介:王文波, E-mail: wenbowangmath@163.com
基金资助:
云南省应用基础研究青年项目和红河学院科研基金博士专项项目(XJ17B11)

Existence and Concentration of Nontrivial Solutions for the Fractional Schrödinger Equations with Sign-Changing Steep Well Potential

Wenbo Wang¹(),Quanqing Li^2,*()

¹ School of Mathematics and Statistics, Yunnan University, Kunming 650500
² Department of Mathematics, Honghe University, Yunnan Mengzi 661100

Received:2017-07-28 Online:2018-11-09 Published:2018-11-09
Contact: Quanqing Li E-mail:wenbowangmath@163.com;shili06171987@126.com
Supported by:
the Yunnan Province Applied Basic Research for Youths and Honghe University Doctoral Research Program(XJ17B11)

摘要/Abstract

摘要：

考虑分数阶Schrödinger方程

$ \begin{equation}\renewcommand{\theequation}{$P_{\lambda}$} (-\Delta)^{s}u+\lambda V(x)u+V_{0}(x)u=P(x)|u|^{p-2}u+Q(x)|u|^{q-2}u, ~~~~~x\in {\Bbb R}^{N}\;\;\;\;\;\;\;\left( {{P_\lambda }} \right) \end{equation}$

非平凡解的存在性和集中性,其中$\lambda>0$, $s\in(0, 1)$, $N>2s$, $2<q<p<2_{s}^{\ast}$ ($2_{s}^{\ast}=\frac{2N}{N-2s}$), $P\in L^{\infty}$有正的下界, $Q\in L^{\infty}$可正可负或变号, $V$是深势阱位势, $V_{0}\in L^{\infty}$.当$\lambda$充分大时,此方程存在非平凡解.进一步,如果$V(x)\geq0$,其解序列拥有某种集中现象.特别地,对于解的存在性, $V$允许变号.

关键词: 分数阶Schrödinger方程, 势阱位势, 变号位势, 集中性

Abstract:

Consider the following fractional Schrödinger equation

$ \begin{equation}\renewcommand{\theequation}{$P_{\lambda}$} (-\Delta)^{s}u+\lambda V(x)u+V_{0}(x)u=P(x)|u|^{p-2}u+Q(x)|u|^{q-2}u, ~~~~~x\in {\Bbb R}^{N}, \end{equation}$

where $\lambda>0$, $s\in(0, 1)$, $N>2s$, $2<q<p<2_{s}^{\ast}$ ($2_{s}^{\ast}=\frac{2N}{N-2s}$), $P\in L^{\infty}$ is positive, $Q\in L^{\infty}$ may be positive, sign-changing or negative, $V$ is steep well potential, and $V_{0}\in L^{\infty}$. When $\lambda$ is large, the existence of nontrivial solutions is obtained via variational methods. Furthermore, if $V(x)\geq0$, concentration results are also obtained. In particular, the potential $V$ is allowed to be sign-changing for the existence.

Key words: Fractional Schrödinger equations, Steep well potential, Sign-changing potential, Concentration

中图分类号:

O175.29

王文波,李全清. 变号深阱位势分数阶Schrödinger方程非平凡解的存在性和集中性[J]. 数学物理学报, 2018, 38(5): 893-902.

Wenbo Wang,Quanqing Li. Existence and Concentration of Nontrivial Solutions for the Fractional Schrödinger Equations with Sign-Changing Steep Well Potential[J]. Acta mathematica scientia,Series A, 2018, 38(5): 893-902.

参考文献 34

1	Alves C O , Miyagaki O H . Existence and concentration of solution for a class of fractional elliptic equation in ${{\mathbb{R}}^N}$ via penalization mehtod. Calc Var Partial Differential Equations, 2016, 55: 1- 19 doi: 10.1007/s00526-015-0942-y
2	Brändle C , Colorado E , de Pablo A , Sánchez U . A concave-convex elliptic problem involving the fractional Laplacian. Proc Roy Soc Edinburgh Sect A, 2013, 143: 39- 71 doi: 10.1017/S0308210511000175
3	Barrios B , Colorado E , de Pablo A , Sánchez U . On some critical problems for the fractional Laplacian operator. J Differential Equations, 2012, 252: 6133- 6162 doi: 10.1016/j.jde.2012.02.023
4	Bartsch T , Pankov A , Wang Z Q . Nonlinear Schrödinger equations with steep potential well. Commun Contemp Math, 2001, 3: 549- 569 doi: 10.1142/S0219199701000494
5	Cheng M . Bound state for the fractional Schrödinger equations with unbounded potential. J Math Phys, 2012, 53: 043507 doi: 10.1063/1.3701574
6	Chen G . Multiple semiclassical standing waves for fractional nonlinear Schrödinger equations. Nonlinearity, 2015, 28: 927- 949 doi: 10.1088/0951-7715/28/4/927
7	Cabré X , Sire Y . Nonlinear equations for fractional Laplacians, Ⅰ:Regularity, maximum principles, and Hamiltonian estimates. Ann Inst H Poincaré Anal Non Linéeaire, 2014, 31: 23- 53 doi: 10.1016/j.anihpc.2013.02.001
8	Caffarelli L , Silvestre L . An extension problems related to the fractional Laplacian. Comm Partial Differential Equations, 2007, 32: 1245- 1260 doi: 10.1080/03605300600987306
9	Chang X , Wang Z Q . Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian. J Differential Equations, 2014, 256: 2965- 2992 doi: 10.1016/j.jde.2014.01.027
10	Chen G , Zheng Y . Concentration phenomena for fractional nonlinear Schrödinger equations. Commun Pure Appl Anal, 2014, 13: 2359- 2376 doi: 10.3934/cpaa
11	do ó J M , Miyagaki O H , Squassina M . Critical and subcritical fractional problems with vanishing potentials. Commun Contemp Math, 2016, 18: 1550063 doi: 10.1142/S0219199715500637
12	Ding Y , Szulkin A . Bound states for semilinear Schrödinger equations with sign-changing potential. Calc Var Partial Differential Equations, 2007, 29: 397- 419 doi: 10.1007/s00526-006-0071-8
13	Davila J , del Pino M , Wei J . Concentrating standing waves for the fractional nonlinear Schrödinger equation. J Differential Equations, 2014, 256: 858- 892 doi: 10.1016/j.jde.2013.10.006
14	Frank R , Lenzman E , Silvestre L . Uniqueness of radial solutions for the fractional Laplacian. Commun Pure Appl Anal, 2016, 69: 1671- 1726 doi: 10.1002/cpa.v69.9
15	Fall M , Mahmoudi F , Valdinoci E . Ground states and concentration phenomena for the fractional Schrödinger equation. Nonlinearity, 2015, 28: 1937- 1961 doi: 10.1088/0951-7715/28/6/1937
16	Figueiredo G , Sicilinao G . A multiplicity results via Ljusternik-schnirelmann category and Morse theory for a fractional Schrödinger equation in RN. Nonlinear Differ Equ Appl, 2016, 23: 1- 22 doi: 10.1007/s00030-016-0354-5
17	Guo Q , He X . Semiclassical states for fractional Schrödinger equations with critical growth. Nonlinear Analysis, 2017, 151: 164- 186 doi: 10.1016/j.na.2016.12.004
18	He X , Zou W . Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities. Calc Var Partial Differential Equations, 2016, 55: 1- 39 doi: 10.1007/s00526-015-0942-y
19	Jiang Y , Zhou H S . Schrödinger-Poisson system with steep potential well. J Differential Equations, 2011, 251: 582- 608 doi: 10.1016/j.jde.2011.05.006
20	Laskin N . Fractional Schrödinger equations. Phys Rev E, 2002, 66: 056108 doi: 10.1103/PhysRevE.66.056108
21	Ledesma C T . Existence and concentration of solutions for a nonlinear fractional Schrödinger equations with steep potential well. Commun Pure Appl Anal, 2016, 15: 535- 547 doi: 10.3934/cpaa
22	陆文端. 微分方程中的的变分法. 北京: 中国科学出版社, 2002
	Lu W . Variational Methods in Partial Differential Equations. Beijing: Scientific Publishing House in China, 2002
23	Nezza E D , Palatucci G , Valdinoci E . Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math, 2012, 136: 521- 573 doi: 10.1016/j.bulsci.2011.12.004
24	Rudin W . Functional Analysis. Beijin: China Machine Press, 2004
25	Rabinowitz P H . Minimax Methods in Critical Point Theory with Applications to Differential Equations. Providence, RI: Amer Math Soc, 1986
26	Servadei R . A critical fractional Laplace equation in resonant case. Topol Methods Nonlinear Anal, 2014, 43: 251- 267
27	Secchi S . Ground state solutions for nonlinear fractional Schrödinger equations in ${{\mathbb{R}}^N}$. J Math Phys, 2013, 54: 031501 doi: 10.1063/1.4793990
28	Shang X , Zhang J . Ground states for fractional Schrödinger equations with critical growth. Nonlinearity, 2014, 27: 187- 207 doi: 10.1088/0951-7715/27/2/187
29	Shang X , Zhang J , Yang Y . On fractional Schrödinger equation in ${{\mathbb{R}}^N}$ with critical growth. J Math Phys, 2013, 54: 877- 895
30	Teng K . Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent. J Differential Equations, 2016, 261: 3061- 3106 doi: 10.1016/j.jde.2016.05.022
31	Willem M . Minimax Theorems. Boston: Birkhäuser Boston Inc, 1996
32	Wang Z , Zhou H S . Ground state for nonlinear Schrödinger equation with sign-changing and vanishing potential. J Math Phys, 2011, 52: 113704 doi: 10.1063/1.3663434
33	Zhao L , Liu H , Zhao F . Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential. J Differential Equations, 2013, 255: 1- 23 doi: 10.1016/j.jde.2013.03.005
34	Zhang J, Jiang W. Existence and concentration of solutions for a fractional Schrödinger equations with sublinear nonlinearity. 2015, arXiv: 1502.02221v1

[1]	李琴, 杨作东. 含变号位势的p-Kirchhoff型方程组无穷多个高能量解的存在性[J]. 数学物理学报, 2017, 37(5): 869-876.
[2]	罗琳;徐国进. 四阶椭圆方程基态解的集中性态[J]. 数学物理学报, 2007, 27(4): 761-768.