一类二阶不定权特征值问题结点解的全局结构

数学物理学报 ›› 2011, Vol. 31 ›› Issue (4): 910-917.

一类二阶不定权特征值问题结点解的全局结构

马如云, |韩晓玲

西北师范大学数学与信息科学学院兰州 730070

收稿日期:2009-08-17 修回日期:2010-11-21 出版日期:2011-08-25 发布日期:2011-08-25
基金资助:
国家自然科学基金(11061030, 11026060)、甘肃省高校科研业务费和西北师范大学科技创新工程项目(03-69, 0303-23)资助

Nodal Solutions of a Nonlinear Eigenvalue Problem with Indefinite Weight Function

MA Ru-Yun, |Han-Xiao-Ling

College of Mathematics and Information Science, Northwest Normal University, Lanzhou 730070

Received:2009-08-17 Revised:2010-11-21 Online:2011-08-25 Published:2011-08-25
Supported by:
国家自然科学基金(11061030, 11026060)、甘肃省高校科研业务费和西北师范大学科技创新工程项目(03-69, 0303-23)资助

摘要/Abstract

摘要：

该文运用分歧方法研究二阶非线性不定权特征值问题
u''+rh(t)f(u)=0, 0u(0)=u(1)=0
结点解的全局结构, 这里r ∈R 是参数, 权函数h ∈C[0,1] 可变号.

关键词: 不定权, 分歧, 结点解

Abstract:

The authors are concerned with the existence of nodal solutions of the indefinite weight boundary value problem
u''+rh(t)f(u)=0, 0u(0)=u(1)=0,
where C[0,1] changes sign. The proof of the main result is based upon bifurcation techniques.

Key words: Indefinite weight, Bifurcation, Nodal solutions

中图分类号:

34B15

马如云, 韩晓玲. 一类二阶不定权特征值问题结点解的全局结构[J]. 数学物理学报, 2011, 31(4): 910-917.

MA Ru-Yun, Han-Xiao-Ling. Nodal Solutions of a Nonlinear Eigenvalue Problem with Indefinite Weight Function[J]. Acta mathematica scientia,Series A, 2011, 31(4): 910-917.

参考文献

[1] Henderson J, Wang H. Positive solutions for nonlinear eigenvalue problems. J Math Anal Appl, 1997, 208: 252--259

[2] Ma R, Thompson B. Nodal solutions for nonlinear eigenvalue problems. Nonlinear Analysis, 2004, 59: 707--718

[3]} Rabinowitz P, Paul H. Some global results for nonlinear eigenvalue problems. J Functional Analysis, 1971, 7: 487--513

[4] Rabinowitz P. On bifurcation from infinity. J Differential Equations, 1973, 14: 462--475

[5] Ma R. Global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems. Appl Math Lett, 2008, 21(7): 754--760

[6] Ince E. Ordinary Differential Equations. New York: Dover, 1926

[7] Binding P. Variational principles for indefinite eigenvalue problems. Linear Algebra and its Applications, 1995, 218: 251--262

[8] Brown K, Lin S. On the existence of positive eigenfunction for an eigenvalue problem with indefinite weight function. J Math Anal Appl, 1980, 75: 112--120

[9] Ruf B, Srikanth P. Multiplicity results for ODEs with nonlinearities crossing all but a finite number of eigenvalues. Nonlinear Analysis TMA, 1986, 10(2): 157--163

[10] Rynne B. Infinitely many solutions of superlinear fourth order boundary value problems. Nonlinear Analysis TMA, 2002, 19(2): 303--312

[11] Rynne B. Global bifurcation for 2mth-order boundary value problems and infinitely many solutions of superlinear problems. J
Differential Equations, 2003, 188(2): 461--472

[12] Naito Y, Tanaka S. On the existence of multiple solutions of the boundary value problem for nonlinear second-order differential equations. Nonlinear Analysis TMA, 2004, 56(6): 919--935

[13] Bong Hak Im, Eun Kyoung Lee, Yong-Hoon Lee. A global bifurcation phenomenon for second order singular boundary value
problems. J Math Anal Appl, 2005, 308: 61--78

[14] Sun J, Xu X, O'Regan D. Nodal solutions for $m$-point boundary value problems using bifurcation methods. Nonlinear Analysis TMA, 2008, 68: 3034--3046

[15] Protter M, Weinberger H. Maximum Principles in Differential Equations. New York: Springer-Verlag, 1984