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

{\begin{cases} \disp - (a + b \int_{R^{N}} | \nabla u |^{2} d x) Δ u + V (x) u = f (x, u) \qq 在 R^{N} 上, \\ u \in H^{1} (R^{N}) . \end{cases}

$\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

{\begin{cases} \disp - (a + b \int_{R^{N}} | \nabla u |^{2} d x) Δ u + V (x) u = f (x, u) \qq in \q R^{N}, \\ u \in H^{1} (R^{N}) \end{cases}

$\left\{ \begin{array}{ll} \disp -\left(a+b\int_{{\Bbb R}^N}|\nabla u|^2 {\rm d}x\right) \Delta {u}+{V(x)u}={f(x,u)}\qq \mbox{in}\q {\Bbb R}^N,\\ u\in H^1({\Bbb R}^N) \end{array} \right.$
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.

参考文献

[1] Kirchhoff G. Mechanik. Leipzig: Teubner, 1883

[2] Alves C O, Corr\^{e}a F, Ma T F. Positive solutions for a quasilinear elliptic equation of  Kirchhoff type. Computers and Mathematics with Applications, 2005, 49(1): 85--93
[3] Cavalcanti M M, Domingos Cavalcanti V N, Soriano J A. A global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear disspation.   Advanced Differential Equations, 2001, 6(1): 701--730
[4] D'Ancona P, Spagnolo S. Global solvability for the degenerate {K}irchhoff equation with real analytic data.  Inventiones Mathematicae, 1992, 108(1): 247--262
[5] Perera K, Zhang Z. Nontrivial solutions of {K}irchhoff-type problems via the {Y}ang index. Journal of Differential Equations, 2006, 221(1): 246--255
[6] Zhang Z, Perera K. Sign changing solutions of {K}irchhoff type problems via invariant sets of descent flow Journal of Mathematical Analysis and Applications, 2006, 317(2): 456--463
[7] Mao A, Zhang Z. Sign-changing and multiple solutions of {K}irchhoff type problems without the {PS} condition. Nonlinear Analysis: Theory, Methods and Applications, 2009, 70(3): 1275--1287
[8] He X, Zou W. Infinitely many positive solutions for {K}irchhoff-type problems. Nonlinear Analysis: Theory, Methods and Applications, 2009, 70(3): 1407--1414
[9] Cheng B, Wu X. Existence results of positive solutions of {K}irchhoff type problems. Nonlinear Analysis: Theory, Methods and Applications, 2009, 71(10): 4883--4892
[10] Jin J, Wu X. Infinitely many radial solutions for {K}irchhoff-type problems in $\100dpi \inline ${\Bbb R}^{N}$$ . Journal of Mathematical Analysis and Applications,
   2010, 369(2): 564--574
[11] Azzollini A. The elliptic Kirchhoff equation in $\100dpi \inline ${\Bbb R}^N$$ perturbed by a local nonlinearity. Diferential Integral Equations, 2012, 25: 543--554
[12] Pomponio A, Azzollini A, d'Avenia P. Multiple critical points for a class of nonlinear functionals. Ann Mat Pura Appl, 2011, 190(4): 507--523
[13] Wu X. Existence of nontrivial solutions and high energy solutions for {S}chr{\"o}dinger-{K}irchhoff-type equations in $\100dpi \inline ${\Bbb R}^{N}$$ . Nonlinear Analysis: Real World Applications, 2011, 12(2): 1278--1287
[14] Sun J J, Tang C L. Existence and multiplicity of solutions for {K}irchhoff type equations. Nonlinear Analysis: Theory, Methods and Applications, 2010, 74(4): 1212--1222
[15] Schechter M, Zou W M. Critical Point Theory and its Applications. New York: Springer, 2006

[16] Willem M. Minimax Theorems. Boston: Birkhauser, 1996

[17] Kajikiya R. A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations. Journal of Functional Analysis, 2005, 225(2): 352--370
[18] Ding Y H. Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Analysis, 1995, 25(11): 1095--1113

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