一类相对非线性薛定谔方程解的存在性

数学物理学报 ›› 2019, Vol. 39 ›› Issue (1): 95-104.

一类相对非线性薛定谔方程解的存在性

邱雯^1,²,张贻民^1,^2,*(),AbdelgadirAhmed Adam³

¹ 武汉理工大学数学科学研究中心武汉 430070
² 武汉理工大学理学院数学系武汉 430070
³ Department of Mathematics, Taibah University Madinah Munawwarah, Saudi Arabia

收稿日期:2017-12-27 出版日期:2019-02-26 发布日期:2019-03-12
通讯作者: 张贻民 E-mail:loiolo@qq.com; zhangym802@126.com; ahmedguangzhou17@gmail.com
基金资助:
国家自然科学基金(11471330);国家自然科学基金国家自然科学基金(11501555);国家自然科学基金(11771127);中央高校基本科研业务费专项基金(2017IVA-075);中央高校基本科研业务费专项基金(2017IVA076);中央高校基本科研业务费专项基金(2018-zy-138);中央高校基本科研业务费专项基金(2018IB014)

Existence of Nontrivial Solutions for a Class of Relativistic Nonlinear Schrödinger Equations

Wen Qiu^1,²,Yimin Zhang^1,^2,*(),Ahmed Adam Abdelgadir³

¹ Center of Mathematical Sciences, Wuhan University of Technology, Wuhan 430070
² Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan 430070
³ Department of Mathematics, Taibah University Madinah Munawwarah, Saudi Arabia

Received:2017-12-27 Online:2019-02-26 Published:2019-03-12
Contact: Yimin Zhang E-mail:loiolo@qq.com; zhangym802@126.com; ahmedguangzhou17@gmail.com
Supported by:
the NSFC(11471330);the NSFC(11501555);the NSFC(11771127);the Fundamental Research Funds for the Central Universities(2017IVA-075);the Fundamental Research Funds for the Central Universities(2017IVA076);the Fundamental Research Funds for the Central Universities(2018-zy-138);the Fundamental Research Funds for the Central Universities(2018IB014)

摘要/Abstract

摘要：

利用临界点理论考虑了一类相对非线性薛定谔方程，主要通过变量代换将相对非线性薛定谔方程转化成半线性椭圆型方程.首先考虑位势函数为零时，将经典的场方程结果推广到了相对非线性薛定谔方程；而后利用临界点理论得到了有界位势情形方程非平凡解的存在性，在此情形，改进了文献[12-13]中的超线性条件.

关键词: 山路引理, 相对非线性薛定谔方程, 集中紧引理

Abstract:

Using the critical point theory, we consider a class of relativistic nonlinear Schrödinger equations. By introducing a change, we transform the relativistic nonlinear Schrödinger equations into the semilinear elliptic equations. First, we extend results of the classical field equation to the relativistic nonlinear Schrödinger equation. Then, the variational methods was used to obtain the existence of nontrivial solutions of the relativistic nonlinear Schrödinger equations with bounded potential. Moreover, we improved the general superlinear conditions which was used in [12-13].

Key words: Mountain pass theorem, Relativistic nonlinear Schrödinger equation, Concentration compactness principle

中图分类号:

O175.25

邱雯,张贻民,AbdelgadirAhmed Adam. 一类相对非线性薛定谔方程解的存在性[J]. 数学物理学报, 2019, 39(1): 95-104.

Wen Qiu,Yimin Zhang,Ahmed Adam Abdelgadir. Existence of Nontrivial Solutions for a Class of Relativistic Nonlinear Schrödinger Equations[J]. Acta mathematica scientia,Series A, 2019, 39(1): 95-104.

参考文献 16

1	Bouard A , Hayashi N , Naumkin P I , Saut J C . Scattering problem and asymptotics for a relativistic nonlinear Schrödinger equation. Nonlinearity, 1999, 12: 1415- 1425 doi: 10.1088/0951-7715/12/5/313
2	Bouard A , Hayashi N , Saut J C . Global existence of small solutions to a relativistic nonliear Schrödinger equation. Commum Math Phys, 1997, 189: 73- 105 doi: 10.1007/s002200050191
3	Berestycki H , Lions P L . Nonlinear scalar field equations, I existence of a ground state. Arch Ration Mech Anal, 1983, 82 (4): 313- 345 doi: 10.1007/BF00250555
4	Chen X L , Sudan R N . Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse. Phys Rev Lett, 1993, 70: 2082- 2085 doi: 10.1103/PhysRevLett.70.2082
5	Cheng Y K , Yang J . The existence result for a relativistic nonlinear Schrödinger equation. J Math Phys, 2015, 56: 3262- 3267
6	Cheng Y K , Yang J . Positive solution to a class of relativistic nonlinear Schrödinger equation. J Math Anal Appl, 2014, 411: 665- 674 doi: 10.1016/j.jmaa.2013.10.006
7	Cheng Y K , Yao Y X . Soliton solutions to a class of relativistic nonlinear Schrödinger equations. Appl Math Comput, 2015, 260: 342- 350
8	Jeanjean L , Tanaka K . A remark in least energy solutions in ${{\mathbb{R}}^N}$. Proc Ams Math Soc, 2003, 131: 2399- 2408 doi: 10.1090/S0002-9939-02-06821-1
9	Lions P L . The concentration-compactness principle in the calculus of variations. The locally compact case, Part 1 and 2. Ann I H Anal Nonlinear, 1984, 1: 109- 283 109-145, 223-283
10	Mao A M , Luan S H . Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems. J Math Anal Appl, 2011, 383: 239- 243 doi: 10.1016/j.jmaa.2011.05.021
11	Mao A M , Jing R N , Luan S X , Chu J L , Kong Y . Some nonlocal elliptic problem involving positive parameter. Topological Meth Nonlinear Anal, 2013, 42 (1): 207- 220
12	Shen Y T , Wang Y J . Soliton solutions for generalized quasilinear Schrödinger equations. Nonlinear Anal, 2013, 80: 194- 201 doi: 10.1016/j.na.2012.10.005
13	Shen Y T , Wang Y J . Standing waves for a class of quasilinear Schrödinger equations. Complex Variables and Elliptic Equ, 2016, 61 (6): 817- 842 doi: 10.1080/17476933.2015.1119818
14	Shen Y T , Wang Y J . A class of generalized quasilinear Schrödinger equations. Commun Pure Appl Anal, 2016, 15 (3): 853- 870
15	Zeng X Y , Zhang Y M . Existence and uniqueness of normalized solutions for the Kirchhoff equation. Applied Math Lett, 2017, 74: 52- 59 doi: 10.1016/j.aml.2017.05.012
16	Zeng X Y , Zhang Y M , Zhou H S . Existence and stability of standing waves for a coupled nonlinear Schrödinger system. Acta Math Sci, 2015, 35B (1): 45- 70

[1]	唐文娟, 张正杰. R^N上带Hardy项的拟线性椭圆方程两个解的存在性[J]. 数学物理学报, 2018, 38(6): 1153-1161.
[2]	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.
[3]	李金菊, 张正杰. 广义Choquard-Pekar方程两个非负解的存在性[J]. 数学物理学报, 2017, 37(3): 491-498.
[4]	袁海华, 张正杰, 徐国进. 具有临界增长及Hardy项的半线性椭圆方程多解的存在性[J]. 数学物理学报, 2016, 36(6): 1137-1144.
[5]	刘健, 赵增勤. 基于变分方法的奇异脉冲方程弱解的存在性[J]. 数学物理学报, 2016, 36(3): 521-530.
[6]	魏公明, 李青. 分数阶耦合非线性Schrödinger方程组的山路解[J]. 数学物理学报, 2016, 36(1): 65-79.
[7]	宋洪雪, 闫庆伦. 一类带有Neumann边值条件的拟线性椭圆外部问题的多解性[J]. 数学物理学报, 2015, 35(5): 845-854.
[8]	张正杰, 张莹. R^N拟线性椭圆型方程两个非负解的存在性[J]. 数学物理学报, 2015, 35(2): 225-233.
[9]	魏美春, 唐春雷. R^N上的Kirchhoff 型问题非平凡解的存在性和多解性[J]. 数学物理学报, 2015, 35(1): 151-162.
[10]	张兴永. 一阶带线性部分Hamilton系统的周期解[J]. 数学物理学报, 2013, 33(5): 894-905.
[11]	魏公明. 耦合非线性Schrödinger方程组的Neumann问题[J]. 数学物理学报, 2009, 29(5): 1398-1414.
[12]	王吉安, 邓雪梅. 一类拟线性椭圆方程非平凡广义解的存在性[J]. 数学物理学报, 2009, 29(3): 727-736.
[13]	刘祥清; 黄毅生; 邓志颖. 双调和方程特征值问题[J]. 数学物理学报, 2009, 29(1): 57-69.