RN中Schrödinger-Poisson方程约束极小元的存在性

数学物理学报 ›› 2018, Vol. 38 ›› Issue (1): 61-70.

R^N中Schrödinger-Poisson方程约束极小元的存在性

朱新才

中国科学院武汉物理与数学研究所武汉 430071;中国科学院大学北京 100049

收稿日期:2016-11-15 修回日期:2017-04-07 出版日期:2018-02-26 发布日期:2018-02-26
作者简介:朱新才,zhuxc68@163.com
基金资助:
国家自然科学基金（11671394）

Existence of Constrained Minimizers for Schrödinger-Poisson Equations in R^N

Zhu Xincai

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071;University of Chinese Academy of Sciences, Beijing 100049

Received:2016-11-15 Revised:2017-04-07 Online:2018-02-26 Published:2018-02-26
Supported by:
Supported by the NSFC(11671394)

摘要/Abstract

摘要： 研究变分问题（1.2）约束极小元的存在性.该文对指标p进行了分类，而问题（1.2）极小元的存在性及非存在性依赖于指标p.对任意给定的系数a > 0，当p满足0 < p < （4/N）时，问题（1.2）至少存在一个极小元；而当p>（4/N）时，问题（1.2）不存在极小元.特别地，当p=（4/N）时，问题（1.2）存在极小元当且仅当0 < a ≤ a^*：=||φ||₂^（4/N），这里的φ（x）（在平移的意义下）是方程-Δu（x）+u（x）=u^1+（4/N）（x），x∈R^N唯一的径向对称正解.而当a > a^*时，问题（1.2）不存在极小元.

关键词: Schrödinger-Poisson方程, 约束极小元, 存在性

Abstract: In this paper, we concern with the existence of constrained minimizers for the variational problem (1.2). We give a classification of the exponent p determining the existence and nonexistence of minimizers. For any fixed a > 0, (1.2) admits minimizers if 0 < p < (4/N) and there is no minimizer of (1.2) if p > (4/N). Specially, if p=(4/N), the existence of minimizers is then proved if and only if a satisfies 0 < a ≤ a^*:=||φ||₂^(4/N), where φ(x) is the unique (up to translations) positive radial solution of -Δ u(x)+u(x)=u^1+(4/N)(x) in R^N. Moreover, there is no minimizer of (1.2) if a > a^*.

Key words: Schrödinger-Poisson Equations, Constrained minimizers, Existence

中图分类号:

O175.2

朱新才. R^N中Schrödinger-Poisson方程约束极小元的存在性[J]. 数学物理学报, 2018, 38(1): 61-70.

Zhu Xincai. Existence of Constrained Minimizers for Schrödinger-Poisson Equations in R^N[J]. Acta mathematica scientia,Series A, 2018, 38(1): 61-70.

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