一类修正Gross-Pitaevskii方程基态解的存在性

Abstract

Abstract:

In this paper, using scaling technique and some rearrangement inequalities, existence and classification of ground states for a class of Modified Gross-Pitaevskii equations with respect to the nonlinear exponent p. If $2＜ p ＜ 2+\frac{4}{N}$ , for any $c>0$ , there is at least a minimizer for this problem. If $p=2+\frac{4}{N}$ and $c\leq\|\phi\|_2$ or $c>\left(\frac{3}{2}\right)^{\frac{N}{4}}\|\phi\|_2$ (the definition of $\|\phi\|_2$ see section 1) or $p>2+\frac{4}{N}$ , there is no minimizer for this problem. But it is unclear if $p=2+\frac{4}{N}$ and $\|\phi\|_2＜c＜\left(\frac{3}{2}\right)^{\frac{N}{4}}\|\phi\|_2$.

Key words: Modified Gross-Pitaevskii equation, Ground state, Existence

CLC Number:

O175.25

Xiaomeng Huang,Yimin Zhang. Existence of Ground States for a Class of Modified Gross-Pitaevskii Equations[J].Acta mathematica scientia,Series A, 2020, 40(4): 850-856.

Trendmd

References 19

1	Agueh M . Sharp Gagliardo-Nirenberg inequalities via p-Laplacian type equations. Nonlinear Differ Equ Appl, 2008, 15, 457- 472
2	De Bouard A , Hayashi N , NaumkinSaut P I , Saut J C . Scattering problem and asymptotics for a relativistic nonlinear Schrödinger equation. Nonlinearity, 1999, 12, 1415- 1425
3	De Bouard A , Hayashi N , Saut J C . Global existence of small solutions to a relativisti nonlinear Schrödinger equation. Comm Math Phys, 1997, 189, 73- 105
4	Cheng Y K , Yao Y X . Soliton solutions to a class of relativistic nonlinear Schrödinger equations. Applied Math Compu, 2015, 260, 342- 350
5	Cheng Y K , Yang J . Positive solution to a class of relativistic nonlinear Schrödinger equation. J Math Anal Appl, 2014, 441, 665- 674
6	Colin M , Jeanjean L , Squassina M . Stability and instability results for standing waves of quasi-linear Schrödinger equations. Nonlinearity, 2010, 23, 1353- 1385
7	古龙江, 孙志禹, 曾小雨. 一类约束变分问题极小元的存在性及其集中行为. 数学物理学报, 2017, 37 (3): 510- 518
	Gu L J , Sun Z Y , Zeng X Y . The existence of minimizers for a class of constrained variational problem with its concentration behavior. Acta Math Sci, 2017, 37A (3): 510- 518
8	Hajaiej H, Squassina M. Generalized Polya-Szego inequality and applications to some quasi-linear elliptic problems. 2009, arXiv: 0903.3975
9	Ioffe A . On lower semicontinuity of integral functionals I. Siam J control optmization, 1977, 15, 521- 538
10	Ioffe A . On lower semicontinuity of integral functionals II. Siam J control optmization, 1977, 15, 991- 1000
11	Li G B , Yan S S . Eigenvalue problems for quasilinear elliptic equations on RN. Commu Partial Differ Equ, 1989, 14 (8/9): 1291- 1314
12	邱雯, 张贻民, AbdelgadirAhmed Ada. 一类相对非线性薛定谔方程解的存在性. 数学物理学报, 2019, 39A (1): 95- 104
	Qiu W , Zhang Y M , Abdelgadir A A . Existence of nontrivial solutions for a class of relativistic nonlinear Schrödinger equations. Acta Math Sci, 2019, 39A (1): 95- 104
13	Shen Y T , Wang Y J . Soliton solutions for generalized quasilinear Schröinger equations. Nonlinear Anal, 2013, 80, 194- 201
14	Zeng X Y , Zhang Y M . Existence and uniqueness of normalized solutions for the Kirchhoff equation. Applied Math Lett, 2017, 74, 52- 59
15	Zeng X Y , Zhang Y M . Existence and asymptotic behavior for the ground state of quasi-linear elliptic equation. Adv Nonlinear Stud, 2018, 18 (4): 725- 744
16	Zeng X Y , Zhang Y M . Asymptotic behaviors of ground stats for a modified Gross-Pitaevskii equation. Discrete Cont Dyn Sys, 2019, 39 (9): 5263- 5273
17	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
18	朱新才. . 数学物理学报, 2018, 38 (1): 61- 70
	Zhu X C . . Acta Math Sci, 2018, 38A (1): 61- 70
19	Zhu X C . Existence and blow-up behavior of constrained minimizers for Schrödinger-Poisson-Slater system. Acta Math Sci, 2018, 38B (2): 733- 744

[1]	Wang Hanyi, Huang Shiyu, Xiang Jianlin. Mountain-pass Solution and Ground State Solution for a Kirchhoff Type Elliptic Equation [J]. Acta mathematica scientia,Series A, 2025, 45(4): 1041-1057.
[2]	Zhu Weipeng, Li Jinlu, Wu Xing. Global Smooth Solutions of the Damped Boussinesq Equations with a Class of Large Initial Data [J]. Acta mathematica scientia,Series A, 2025, 45(4): 1077-1085.
[3]	Liang Qing. Existence, Uniqueness and Stability of the Global Solutions to Two Classes of Pantogragh Stochastic Functional Differential Equations with Markovian Switching [J]. Acta mathematica scientia,Series A, 2025, 45(4): 1184-1205.
[4]	Meng Xiaoying,Lu Lu. Existence and Asymptotic Behavior of Solutions for Kirchhoff Equations Involving the Fractional $ p$-Laplacian [J]. Acta mathematica scientia,Series A, 2025, 45(2): 434-449.
[5]	Wang Guozheng,Shi Zhenxia. Existence of Periodic Forced Waves for a Lotka-Volterra Cooperative System with Shifting Habitat [J]. Acta mathematica scientia,Series A, 2025, 45(2): 479-492.
[6]	Chen Zhengyan,Zhang Jiafeng. Ground State Solutions for a Class of Critical Kirchhoff Type Equation in $ \mathbb{R}^4$ with Steep Potential Well [J]. Acta mathematica scientia,Series A, 2025, 45(2): 450-464.
[7]	Li Jianjun,Li Yangchen. The Existence and Blow-Up of Solutions for a Class of Fractional $ p$-Laplace Diffusion Equation with Logarithmic Nonlinearity [J]. Acta mathematica scientia,Series A, 2025, 45(2): 465-478.
[8]	Li Bin,Xie Li. The Effects of Police Deployment in a Chemotaxis System with Singular Sensitivity for Criminal Activities [J]. Acta mathematica scientia,Series A, 2025, 45(2): 512-533.
[9]	Cao Lei,Chen Xiao. The Existence of Domain Wall Solution Arising in Abelian Higgs Model Subject to Born-Infeld Theory [J]. Acta mathematica scientia,Series A, 2025, 45(2): 567-575.
[10]	Shi Jincheng, Liu Yan. Global Existence and Blow-Up for Semilinear Third Order Evolution Equation with Different Power Nonlinearities [J]. Acta mathematica scientia,Series A, 2024, 44(6): 1550-1562.
[11]	Xiao Suping, Zhao Yuanzhang. Existence and Nonexistence of Solutions for Nonhomogeneous Dirichlet Exterior Problem to a Semilinear Hyperbolic Differential Inequality [J]. Acta mathematica scientia,Series A, 2024, 44(5): 1167-1182.
[12]	Zhang Guangxin, Yang Yunrui, Song Xue. Periodic Traveling Wave Solutions of Delayed SEIR Systems with Nonlocal Effects [J]. Acta mathematica scientia,Series A, 2024, 44(1): 140-159.
[13]	Yuan Yongjun. A Normalized Gradient Flow with Lagrange Multipliers for Computing Ground States of Spin-Orbit Coupled Spin-2 Bose-Einstein Condensates [J]. Acta mathematica scientia,Series A, 2023, 43(5): 1607-1619.
[14]	He Xuyang,Mao Mingzhi,Zhang Tengfei. Existence and Stability of a Class of Impulsive Neutral Stochastic Functional Differential Equations with Poisson Jump [J]. Acta mathematica scientia,Series A, 2023, 43(4): 1221-1243.
[15]	Li Yixian,Zhang Zhengjie. The Existence of Ground State Solutions for a Class of Equations Related to Klein-Gordon-Maxwell Systems [J]. Acta mathematica scientia,Series A, 2023, 43(3): 680-690.

Existence of Ground States for a Class of Modified Gross-Pitaevskii Equations

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 19

Related Articles 15

Metrics

Comments

Recommended 10