带Choquard项的拟线性薛定谔方程的基态解

数学物理学报 ›› 2022, Vol. 42 ›› Issue (3): 730-748.

带Choquard项的拟线性薛定谔方程的基态解

王亚男(),滕凯民*()

^{太原理工大学数学学院山西晋中 030600}

收稿日期:2021-04-14 出版日期:2022-06-26 发布日期:2022-05-09
通讯作者: 滕凯民 E-mail:862240639@qq.com;tengkaimin2013@163.com
作者简介:王亚男, E-mail: 862240639@qq.com
基金资助:
国家自然科学基金(11501403);山西省留学回国择优项目(2018);山西省自然科学基金面上项目(201901D111085)

Ground State Solutions for Quasilinear Schrödinger Equation of Choquard Type

Yanan Wang(),Kaimin Teng*()

^{School of Mathematical, Taiyuan University of Technology, Shanxi Jinzhong 030600}

Received:2021-04-14 Online:2022-06-26 Published:2022-05-09
Contact: Kaimin Teng E-mail:862240639@qq.com;tengkaimin2013@163.com
Supported by:
the NSFC(11501403);the Scientific Activities of Selected Returned Overseas Professionals in Shanxi Province(2018);the NSF of Shanxi Province(201901D111085)

摘要/Abstract

摘要：

该文研究如下Choquard型拟线性薛定谔方程 $ \begin{eqnarray*} -\triangle u+\frac{k}{2}u\triangle u^2+V(x)u=(I_{\alpha}\ast|u|^{p})|u|^{p-2}u, \, \, x\in\mathbb{R}^N, \end{eqnarray*} $ 其中$N\geq3$, 0 < $\alpha$ < $N$, $< p<\frac{N+\alpha}{N-2}$, $I_{\alpha}$是Riesz位势, $V(x)$是一个正连续位势, $k$是一个非负参数. 采用Pohožaev流形方法, 证明了基态解的存在性.

关键词: Choquard型方程, Pohožaev流形, 基态解

Abstract:

In this paper, we consider the following quasilinear Schrödinger equations of Choquard type $ \begin{eqnarray*} -\triangle u+\frac{k}{2}u\triangle u^2+V(x)u=(I_{\alpha}\ast|u|^{p})|u|^{p-2}u, \, \, x\in\mathbb{R}^N, \end{eqnarray*} $ where $N\geq3$, 0 < $\alpha$ < $N$, $<p<\frac{N+\alpha}{N-2}$, $I_{\alpha}$ is the Riesz potential, $V(x)$ is a positive continuous potential and $k$ is a non-negative parameter. The existence of ground state solutions is established via Pohožaev manifold approach.

Key words: Choquard type problems, Pohožaev manifold, Ground state solutions

中图分类号:

O175.25

王亚男,滕凯民. 带Choquard项的拟线性薛定谔方程的基态解[J]. 数学物理学报, 2022, 42(3): 730-748.

Yanan Wang,Kaimin Teng. Ground State Solutions for Quasilinear Schrödinger Equation of Choquard Type[J]. Acta mathematica scientia,Series A, 2022, 42(3): 730-748.

参考文献 30

1	Alves C O , Wang Y J , Shen Y T . Soliton solutions for a class of quasilinear Schrödinger equations with a parameter. J Differ Equ, 2015, 259, 318- 343 doi: 10.1016/j.jde.2015.02.030
2	Chen J H , Cheng B T , Huang X J . Ground state solutions for a class of quasilinear Schrödinger equations with Choquard type nonlinearity. Appl Math Lett, 2020, 102, 106141 doi: 10.1016/j.aml.2019.106141
3	Chen J H , Huang X J , Cheng B T , Zhu C X . Some results on standing wave solutions for a class of quasilinear Schrödinger equations. J Math Phys, 2019, 60, 091506 doi: 10.1063/1.5093720
4	Chen J H , Huang X J , Cheng B T . Positive solutions for a class of quasilinear Schrödinger equations with superlinear condition. Appl Math Lett, 2019, 87, 165- 171 doi: 10.1016/j.aml.2018.07.035
5	Chen J H , Wu Q F , Huang X J , Zhu C X . Positive solutions for a class of quasilinear Schrödinger equations with two parameters. Bull Malays Math Sci Soc, 2020, 43, 2321- 2341 doi: 10.1007/s40840-019-00803-y
6	Chen S X , Wu X . Existence of positive solutions for a class of quasilinear Schrödinger equations of Choquard type. J Math Anal Appl, 2019, 475, 1754- 1777 doi: 10.1016/j.jmaa.2019.03.051
7	Colin M , Jeanjean L . Solutions for a quasilinear Schrödinge equation: a dual approach. Nonlinear Anal, 2004, 56, 213- 226 doi: 10.1016/j.na.2003.09.008
8	Jeanjean L . On the existence of bounded Palais-Smale sequence and application to a Landesman-Lazer type problem set on $ {{\Bbb R}}^3 $. Proc Roy Soc Edinburgh Sect, 1999, 129A, 787- 809
9	Kurihara S . Large-amplitude quasi-solitons in superfluid films. J Phys Soc Japan, 1981, 50, 3262- 3267 doi: 10.1143/JPSJ.50.3262
10	Li G D , Tang C L . Existence of a ground state solution for Choquard equation with the upper critical exponent. Comput Math Appl, 2018, 76, 2635- 2647 doi: 10.1016/j.camwa.2018.08.052
11	Liu X Q , Liu J Q , Wang Z Q . Quasilinear elliptic equations via perturbation method. Proc Amer Math Soc, 2013, 141, 253- 263
12	Liu X Q , Liu J Q , Wang Z Q . Quasilinear elliptic equations with critical growth via perturbation method. J Differ Equ, 2013, 254, 102- 124 doi: 10.1016/j.jde.2012.09.006
13	Liu X Q , Liu J Q , Wang Z Q . Ground states for quasilinear Schrödinger equations with critical growth. Calc Var Partial Differential Equations, 2013, 46, 641- 669 doi: 10.1007/s00526-012-0497-0
14	Luo H X . Ground state solutions of Pohožaev type and Nehari type for a class of nonlinear Choquard equations. J Math Anal Appl, 2018, 467, 842- 862 doi: 10.1016/j.jmaa.2018.07.055
15	Laedke E W , Spatschek K H , Stenflo L . Evolution theorem for a class of perturbed envelope soliton solutions. J Math Phys, 1983, 24, 2764- 2769 doi: 10.1063/1.525675
16	Liu J Q , Wang Z Q . Soliton solutions for quasilinear Schrödinger equations. Ⅰ. Proc Amer Math Soc, 2003, 131, 441- 448
17	Liu J Q , Wang Y Q , Wang Z Q . Soliton solutions for quasilinear Schrödinger equations. Ⅱ. J Differ Equ, 2003, 187, 473- 493 doi: 10.1016/S0022-0396(02)00064-5
18	Liu J Q , Wang Y Q , Wang Z Q . Solutions for quasilinear Schrödinger equations via the Nehari method. Comm Partial Differential Equations, 2004, 29, 879- 901 doi: 10.1081/PDE-120037335
19	Makhankov V G , Fedanin V K . Nonlinear effects in quasi-one-dimensional models of condensed matter theory. Phys Rep, 1984, 104, 1- 86 doi: 10.1016/0370-1573(84)90106-6
20	Moroz V , Van Schaftingen J . Groundstates of nonlinear Choquard equations: Existence, qualitative prop erties and decay asymptotics. J Func Anal, 2013, 265, 153- 184 doi: 10.1016/j.jfa.2013.04.007
21	Poppenberg M , Schmitt K , Wang Z Q . On the existence of soliton solutions to quasilinear Schrödinger equations. Calc Var, 2002, 14, 329- 344 doi: 10.1007/s005260100105
22	Ritchie B . Relativistic self-focusing and channel formation in laser-plasma interactions. Phys Rev E, 1994, 50, 687- 689 doi: 10.1103/PhysRevE.50.R687
23	Silva E A B , Vieira G F . Quasilinear asymptotically periodic Schrödinger equations with critical growth. Calc Var, 2010, 39, 1- 33 doi: 10.1007/s00526-009-0299-1
24	Struwe M . A global compactness result for elliptic boundary value problems involving limiting nonlineari ties. Math Z, 1984, 187 (4): 511- 517 doi: 10.1007/BF01174186
25	Severo U B , Gloss E , Silva E D . On a class of quasilinear Schrödinger equations with superlinear or asymptotically linear terms. J Differ Equ, 2017, 263, 3550- 3580 doi: 10.1016/j.jde.2017.04.040
26	Wang Y J , Li Z X . Existence of solutions to quasilinear Schrödinger equations involving critical Sobolev exponent. Taiwan J Math, 2018, 22, 401- 420
27	Xu L X , Chen H B . Ground state solutions for quasilinear Schrödinger equations via Pohožaev manifold in Orlicz space. J Differ Equ, 2018, 265, 4417- 4441 doi: 10.1016/j.jde.2018.06.009
28	Xue Y F , Tang C L . Existence of a bound state solutions for quasilinear Schrödinger equations. Adv Nonlinear Anal, 2019, 8, 323- 338
29	Yang M B , Santos C A , Zhou J Z . Least action nodal solutions for a quasilinear defocusing Schrödinger equation with supercritical nonlinearity. Commun Contemp Math, 2019, 21, 1850026 doi: 10.1142/S0219199718500268
30	Yang X , Zhang W , Zhao F . Existence and multiplicity of solutions for a quasilinear Choquard equation via perturbation method. J Math Phys, 2018, 59, 081503 doi: 10.1063/1.5038762

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