带有零谱点的渐近线性薛定谔方程

数学物理学报 ›› 2016, Vol. 36 ›› Issue (6): 1103-1116.

带有零谱点的渐近线性薛定谔方程

秦栋栋, 李赟杨, 唐先华

中南大学数学与统计学院长沙 410083

收稿日期:2016-03-30 修回日期:2016-08-18 出版日期:2016-12-26 发布日期:2016-12-26
通讯作者: 唐先华 E-mail:tangxh@mail.csu.edu.cn
作者简介:秦栋栋,qindd132@163.com;李赟杨,1256578581@qq.com
基金资助:
国家自然科学基金（11571370，11471278，11471137）、湖南省研究生科研创新项目（CX2015B037）、国家留学基金委（201506370092）和高等学校博士学科点专项科研基金（20120162110021）资助

Schrödinger Equation with Asymptotically Linear Nonlinearity and Spectrum Point Zero

Qin Dongdong, Li Yunyang, Tang Xianhua

School of Mathematics and Statistics, Central South University, Changsha 410083

Received:2016-03-30 Revised:2016-08-18 Online:2016-12-26 Published:2016-12-26
Supported by:
Supported by the NSFC (11571370,11471278,11471137),Hunan Provincial Innovation Foundation for Postgraduates (CX2015B037),China Scholarship Council (201506370092) and the Specialized Research Fund for the Doctroral Program of Higher Education (20120162110021)

摘要/Abstract

摘要：

该文考虑了如下薛定谔方程
-△u+V（x）u=f（x，u），对x∈R^N，
u（x）→0，当|x|→∞，
其中V与f关于x是周期的，0是谱σ（-△+V）的一个边界点.受最近的文献[35]的启发，进一步考虑了f（x，u）在|u|→∞时是渐近线性的情况，并利用非Nehari流形方法得到了该方程的基态解.与广义Nehari流形方法相比，该方法更加简便、直接.

关键词: 薛定谔方程, 渐近线性, 非Nehari流形方法, 谱点零, 基态解

Abstract:

This paper is concerned with the following Schrödinger equation
-△u+V(x)u=f(x, u), for x∈R^N,
u(x)→0, as |x|→∞,
where V and f are both periodic in x and 0 is a boundary point of the spectrum σ(-△+V). Inspired by recent work of Tang^[35], we consider further the case that f(x,u) is asymptotically linear as |u|→∞, and obtain the existence of ground state solutions using the non-Nehari manifold method which is more direct and simpler than the generalized manifold method.

Key words: Schrödinger equation, Asymptotically linear, Spectrum point zero, Non-Nehari manifold method, Ground states solutions

中图分类号:

O175.29

秦栋栋, 李赟杨, 唐先华. 带有零谱点的渐近线性薛定谔方程[J]. 数学物理学报, 2016, 36(6): 1103-1116.

Qin Dongdong, Li Yunyang, Tang Xianhua. Schrödinger Equation with Asymptotically Linear Nonlinearity and Spectrum Point Zero[J]. Acta mathematica scientia,Series A, 2016, 36(6): 1103-1116.

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