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, Vol. 37 ›› Issue (3): 510-518.

一类约束变分问题极小元的存在性及其集中行为

谷龙江¹, 孙志禹¹, 曾小雨²

1. 中国科学院武汉物理与数学研究所武汉 430071;
2. 武汉理工大学理学院武汉 430070

收稿日期:2016-10-17 修回日期:2017-03-21 出版日期:2017-06-26 发布日期:2017-06-26
通讯作者: 曾小雨,E-mail:zengxy09@126.com E-mail:zengxy09@126.com
作者简介:谷龙江,E-mail:gulongjiang0@163.com;孙志禹,E-mail:onlyamoment@126.com
基金资助:
国家自然科学基金（11471331，11501555）和中央高校基本科研业务费专项基金（WUT：2017IVA076）

The Existence of Minimizers for a Class of Constrained Variational Problem with Its Concentration Behavior

Gu Longjiang¹, Sun Zhiyu¹, Zeng Xiaoyu²

1. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071;
2 School of Sciences, Wuhan University of Technology, Wuhan 430070

Received:2016-10-17 Revised:2017-03-21 Online:2017-06-26 Published:2017-06-26
Supported by:
Supported by the NSFC(11471331, 11501555) and the Fundamental Research Funds for the Central University (WUT:2017IVA076)

摘要/Abstract

摘要： 该文主要讨论了一类带有调和位势p-Laplacian方程特征值问题对应的变分泛函极小元的存在性与非存在性，并且使用能量估计的方法分析了当方程中相关参数逼近其临界值时极小元的集中行为.

关键词: 约束变分, p-Laplacian方程, 极小化问题, 集中行为

Abstract: In this paper, we mainly discuss the existence and non-existence of minimizers for the variational functional of a p-Laplacian eigenvalue problem involving harmonic potential. Moreover, the concentration behavior of the minimizers is also investigated by using the energy method when the related parameter closes to a critical value.

Key words: Constrained variation, p-Laplacian equation, Minimizing problem, Concentration behavior

中图分类号:

O175.25

谷龙江, 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[J]. 数学物理学报, 2017, 37(3): 510-518.

Gu Longjiang, Sun Zhiyu, Zeng Xiaoyu. The Existence of Minimizers for a Class of Constrained Variational Problem with Its Concentration Behavior[J]. Acta mathematica scientia,Series A, 2017, 37(3): 510-518.

参考文献

[1] Adams R A, Fournier J J. Sobolev Spaces. New York:Academic press, 2003
[2] Agueh M. Sharp Gagliardo-Nirenberg inequalities via p-Laplacian type equations. NoDEA Nonlinear Differential Equations Appl, 2008, 15:457-472
[3] Bao W Z, Cai Y Y. Mathematical theory and numerical methods for Bose-Einstein condensation. Kinet Relat Models, 2013, 6:1-135
[4] Bartsch T, Wang Z Q. Existence and multiplicity results for some superlinear elliptic problems on R^N. Comm Partial Differential Equations, 1995, 20:1725-1741
[5] Brezis H. Functional Analysis, Sobolev Spaces and Partial Differential Equations. New York:Springer Science & Business Media, 2010
[6] Damascelli L, Pacella F, Ramaswamy M. Symmetry of ground states of p-Laplace equations via the moving plane method. Arch Ration Mech Anal, 1999, 148:291-308
[7] Gidas B, Ni W M, Nirenberg L. Symmetry of positive solutions of nonlinear elliptic equations in Rn. Math Anal Appl Part A Adv Math Suppl Studies, 1981, 7:369-402
[8] Guo Y J, Seiringer R J. On the mass concentration for Bose-Einstein condensates with attractive interactions. Lett Math Phys, 2014, 104:141-156
[9] Kwong M K. Uniqueness of positive solutions of △u -u + u^p=0 in R^N. Arch Rational Mech Anal, 1989, 105:243-266
[10] Li G B, Yan S S. Eigenvalue problems for quasilinear elliptic-equations on Rn. Comm Partial Differential Equations, 1989, 14:1291-1314
[11] Li Y, Ni W M. Radial symmetry of positive solutions of nonlinear elliptic equations in Rn. Comm Partial Differential Equations, 1993, 18:1043-1054
[12] Liu C G, Zheng Y Q. Existence of nontrivial solutions for p-Laplacian equations in R^N. J Math Anal Appl, 2011, 380(2):669-679
[13] McLeod K, Serrin J. Uniqueness of positive radial solutions of △u + f(u)=0 in Rn. Arch Rational Mech Anal, 1987, 99:115-145
[14] Pucci P, Serrin J. The strong maximum principle revisited. J Differential Equations, 2004, 196:1-66
[15] Serrin J, Tang M X. Uniqueness of ground states for quasilinear elliptic equations. Indiana Univ Math J, 2000, 49:897-923
[16] Zhang J. Stability of attractive Bose-Einstein condensates. J Stat Phys, 2000, 101(3-4):731-746