RN上的Kirchhoff 型问题非平凡解的存在性和多解性

数学物理学报 ›› 2015, Vol. 35 ›› Issue (1): 151-162.

R^N上的Kirchhoff 型问题非平凡解的存在性和多解性

魏美春, 唐春雷**

西南大学数学与统计学院重庆 400715

收稿日期:2013-12-13 修回日期:2014-05-06 出版日期:2015-02-25 发布日期:2015-02-25
通讯作者: 唐春雷 E-mail:tangcl@swu.edu.cn
基金资助:
国家自然科学基金(11471267)

Existence and Multiplicity of Nontrivial Solutions for Kirchhoff-Type Problem in R^N

WEI Mei-Chun, TANG Chun-Lei**

School of Mathematics and Statistics, Southwest |University, Chongqing 400715

Received:2013-12-13 Revised:2014-05-06 Online:2015-02-25 Published:2015-02-25
Contact: TANG Chun-Lei E-mail:tangcl@swu.edu.cn

摘要/Abstract

摘要：

考虑如下 Kirchhoff 型问题
$$
\left\{
   \begin{array}{ll}
\disp -\left(a+b\int_{{\Bbb R}^N}|\nabla u|^2 {\rm d}x\right)
\Delta {u}+{V(x)u}={f(x,u)}\qq \mbox{在 ${\Bbb R}^N$ 上,}\\
u\in H^1({\Bbb R}^N).
   \end{array}
   \right.
$$
通过山路引理, 喷泉定理和对称山路引理得到问题非平凡解的存在性和多解性.

关键词: Kirchhoff 型问题 , 山路引理 , 喷泉定理 , 对称山路引理

Abstract:

The existence and multiplicity of nontrivial solutions are
obtained for the Kirchhoff-type problem
$$
\left\{
   \begin{array}{ll}
\disp -\left(a+b\int_{{\Bbb R}^N}|\nabla u|^2 {\rm d}x\right)
\Delta {u}+{V(x)u}={f(x,u)}\qq \mbox{in}\q {\Bbb R}^N,\\
u\in H^1({\Bbb R}^N)
   \end{array}
   \right.
$$
by using the Mountain Pass Lemma, the Fountain Theorem and the
Symmetric Mountain Pass Lemma.

Key words: Kirchhoff-type problem , Mountain Pass Lemma , Fountain Theorem , Symmetric Mountain Pass Lemma

中图分类号:

35J35

魏美春, 唐春雷. R^N上的Kirchhoff 型问题非平凡解的存在性和多解性[J]. 数学物理学报, 2015, 35(1): 151-162.

WEI Mei-Chun, TANG Chun-Lei. Existence and Multiplicity of Nontrivial Solutions for Kirchhoff-Type Problem in R^N[J]. Acta mathematica scientia,Series A, 2015, 35(1): 151-162.

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