一类修正Gross-Pitaevskii方程基态解的存在性

数学物理学报 ›› 2020, Vol. 40 ›› Issue (4): 850-856.

一类修正Gross-Pitaevskii方程基态解的存在性

黄小梦(),张贻民*()

^{武汉理工大学数学科学研究中心武汉 430070}

收稿日期:2020-02-17 出版日期:2020-08-26 发布日期:2020-08-20
通讯作者: 张贻民 E-mail:hhuangxiaomeng@126.com;zhangym802@126.com
作者简介:黄小梦, E-mail:hhuangxiaomeng@126.com
基金资助:
国家自然科学基金(11771127);中央高校基本科研业务费专项基金(2019IB009)

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

Xiaomeng Huang(),Yimin Zhang*()

^{Center for Mathematical Sciences, Wuhan University of Technology, Wuhan 430070}

Received:2020-02-17 Online:2020-08-26 Published:2020-08-20
Contact: Yimin Zhang E-mail:hhuangxiaomeng@126.com;zhangym802@126.com
Supported by:
the NSFC(11771127);the Fundamental Research Funds for the Central Universities(2019IB009)

摘要/Abstract

摘要：

该文利用伸缩变换结合重排不等式等技巧得到了修正Gross-Pitaevskii方程对应极小化问题极小元的存在性与非线性项指数 $p$ 的依赖关系.当 $0＜ p ＜ 2+\frac{4}{N}$ 时,对任意 $c>0$ ,极小化问题存在极小元.若 $p=2+\frac{4}{N}$ 且 $c\leq\|\phi\|_2$ 或者 $c>\left(\frac{3}{2}\right)^{\frac{N}{4}}\|\phi\|_2$ ( $\|\phi\|_2$ 的定义见第一节)或 $p>2+\frac{4}{N}$ ,问题不存在极小解.而对于 $p=2+\frac{4}{N}$ 且 $\|\phi\|_2 ＜ c ＜ \left(\frac{3}{2}\right)^{\frac{N}{4}}\|\phi\|_2$ ,不知道是否存在极小解.

关键词: 修正Gross-Pitaevskii方程, 基态解, 存在性

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

中图分类号:

O175.25

黄小梦,张贻民. 一类修正Gross-Pitaevskii方程基态解的存在性[J]. 数学物理学报, 2020, 40(4): 850-856.

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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