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.

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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

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

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