A BIHARMONIC EIGENVALUE PROBLEM AND ITS APPLICATION

doi:10.1016/S0252-9602(12)60093-9

数学物理学报(英文版) ›› 2012, Vol. 32 ›› Issue (3): 1213-1225.doi: 10.1016/S0252-9602(12)60093-9

A BIHARMONIC EIGENVALUE PROBLEM AND ITS APPLICATION

王江潮, 张贻民*

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences,Wuhan 430071, China

收稿日期:2011-01-13 出版日期:2012-05-20 发布日期:2012-05-20
通讯作者: 张贻民,zhangym802@126.com E-mail:rightatyou@gmail.com; zhangym802@126.com
基金资助:
Project supported by the National Science Foundation of China (11071245). †Corresponding author.

A BIHARMONIC EIGENVALUE PROBLEM AND ITS APPLICATION

WANG Jiang-Chao, ZHANG Yi-Min*

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences,Wuhan 430071, China

Received:2011-01-13 Online:2012-05-20 Published:2012-05-20
Contact: ZHANG Yi-Min,zhangym802@126.com E-mail:rightatyou@gmail.com; zhangym802@126.com
Supported by:
Project supported by the National Science Foundation of China (11071245). †Corresponding author.

摘要/Abstract

摘要：

In this article, we focus on the eigenvalue problem of the following linear biharmonic equation in RN:
△²u−αu+λg(x)u = 0 with u ∈ H²(R^N), u ≠0, N ≥ 5. (*)
Note that there are two parameters and α in it, which is different from the usual eigen-value problems. Here, we consider λ as an eigenvalue and seek for a suitable range of parameter α, which ensures that problem (*) has a maximal eigenvalue. As the loss of
strong maximum principle for our problem, we can only get the existence of non-trivial so-lutions, not positive solutions, in this article. As an application, by using these results, we studied also the existence of non-trivial solutions for an asymptotically linear biharmonic
equation in R^N.

关键词: Biharmonic equation, potential well, eigenvalue problem, asymptotically lin-ear

Abstract:

Key words: Biharmonic equation, potential well, eigenvalue problem, asymptotically lin-ear

中图分类号:

35J20

王江潮, 张贻民. A BIHARMONIC EIGENVALUE PROBLEM AND ITS APPLICATION[J]. 数学物理学报(英文版), 2012, 32(3): 1213-1225.

WANG Jiang-Chao, ZHANG Yi-Min. A BIHARMONIC EIGENVALUE PROBLEM AND ITS APPLICATION[J]. Acta mathematica scientia,Series B, 2012, 32(3): 1213-1225.

参考文献

[1] Bernis F, Grunau H C. Critical Exponent and Multiple Critical Dimensions for Polyharmoninc Operators. Journal of Differential Equations, 1995, 117: 469–486

[2] Chabrowski J, Jo˜ao Marcos do ´O. On Some Fourth-order Semilinear Elliptic Problems in RN. Nonlinear Analysis, 2002, 46: 861–884

[3] Chen Z H, Shen Y T. Some Existence Results of Solution for p-Laplacian. Acta Mathematica Scientia, 2003, 23B: 487–496

[4] Deng Y B, Gao Q, Jin L Y. On the Existence of Nontrivial solutions for p-harmonic Equations on Unbounded Domains. Nonlinear Analysis, 2008, 69: 4713–4731

[5] Deng Y B, Li Y. Branches of Solutions to Semilinear Biharmoninc Equations on RN. Proceedings of the Royal Society of Edinburgh, 2006, 136A: 733–758

[6] Deng Y B, Li Y. Exponential Decay of the Solutions for Nonlinear Biharmonic Equations. Communications in Contemporary Mathematics, 2007, 9(5): 753–768

[7] Deng Y B, Li Y. Regularity of the Solutions for Nonlinear Biharmonic Equations in RN. Acta Mathematica Scientia, 2009, 29B: 1469–1480

[8] Francois A van Heerden, Wang Z Q. Schr¨odinger Type Equations with Asymptotically Linear Nonlinearities. Differential Integral Equations, 2003, 16(3): 257–280

[9] Garabedian P R. A Partial Differential Equation Arising in Conformal Mapping. Pacific J Math, 1951, 1: 485–524

[10] Gazzola F, Grunau H G, Squassina M. Existence and Nonexistence Results for Critical Growth Biharmonic Ellipitc Equations. Calculus of Variatious, 2003, 18: 117–143

[11] Grunau H C, Sweers G. The Maximum Principle and Positive Principal Eigenfunctions for Polyharmonic Equations//Reaction Diffusion Systems (Trieste, 1995). Lecture Notes in Pure and Appl Math 194. New York: Dekker, 1998: 163–182

[12] Liu Y, Wang Z P. Biharmonic Equations with Asymptotically Linear Nonlinearities. Acta Mathematics Scientia, 2007, 27B(3): 549–560

[13] Pucci P, Serrin J. A General Variational Identity. Indiana University Mathematics Journal, 1986, 35(3): 681–703

[14] Stuart C A. An Introduction to Elliptic Equations on RN. Ambrosetti A, Chang K C, Ekeland E. Nonlinear Functional Analysis and its Applications. World Scientific, 1998

[15] Stuart C A, Zhou H S. Global Branch of Solutions for Non-linear Schr¨odinger Equations with Deepening Potential Well. Proc London Math Soc, 2006, 92(3): 655–681

[16] Stuart C A, Zhou H S. Positive Eigenfunctions of a Schr¨odinger Operator. J London Math Soc, 2005, 72(2): 429–441

[17] Vorst V D. Variational Identities and Applications to Differential Systems. Arch Rational Mech Anal, 1991, 116: 375–398

[18] Wang Y J, Shen Y T. Infinitely Many Sign-changing Solutions for a Class of Biharmonic Equation Without Symmetry. Nonlinear Analysis, 2009, 71: 967–977

[19] Wang Z P, Zhou H S. Positive Solution for Nonlinear Schr¨odinger Equation with Deepening Potential Well. J Eur Math Soc, 2009, 11: 545–573

[20] Xiong H, Shen Y T. Nonlinear Biharmonic Equations with Critical Potential. Acta Mathematica Sinica, English Series, Dec, 2005, 21(6): 1285–1294

A BIHARMONIC EIGENVALUE PROBLEM AND ITS APPLICATION

A BIHARMONIC EIGENVALUE PROBLEM AND ITS APPLICATION

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 8

Metrics

本文评价

推荐阅读 10

[1]	杨瑞瑞, 张薇, 刘祥清. SIGN-CHANGING SOLUTIONS FOR p-BIHARMONIC EQUATIONS WITH HARDY POTENTIAL IN R^N[J]. 数学物理学报(英文版), 2017, 37(3): 593-606.
[2]	蒋毅, 罗川. OPTIMAL CONDITIONS OF GLOBAL EXISTENCE AND BLOW-UP FOR A NONLINEAR PARABOLIC EQUATION[J]. 数学物理学报(英文版), 2016, 36(3): 872-880.
[3]	A.S. BERDYSHEV, A. CABADA, B.Kh. TURMETOV. ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR A NONHOMOGENEOUS BIHARMONIC EQUATION WITH A BOUNDARY#br# OPERATOR OF A FRACTIONAL ORDER[J]. 数学物理学报(英文版), 2014, 34(6): 1695-1706.
[4]	刘春根, 张清业. NONTRIVIAL SOLUTIONS FOR ASYMPTOTICALLY LINEAR HAMILTONIAN SYSTEMS WITH LAGRANGIAN BOUNDARY CONDITIONS[J]. 数学物理学报(英文版), 2012, 32(4): 1545-1558.
[5]	邓引斌, 李亦. REGULARITY OF THE SOLUTIONS FOR NONLINEAR BIHARMONIC EQUATIONS IN \|R^N[J]. 数学物理学报(英文版), 2009, 29(5): 1469-1480.
[6]	黄飞敏. TWO PROBLEMS OF HERMITE ELLIPTIC EQUATIONSTWO PROBLEMS OF HERMITE ELLIPTIC EQUATIONS[J]. 数学物理学报(英文版), 2009, 29(2): 409-414.
[7]	姚仰新; 王荣鑫; 沈尧天. NONTRIVIAL SOLUTION FOR A CLASS OF SEMILINEAR BIHARMONIC EQUATION INVOLVING CRITICAL EXPONENTS[J]. 数学物理学报(英文版), 2007, 27(3): 509-514.
[8]	邵志强; 洪家兴. THE EIGENVALUE PROBLEM FOR THE LAPLACIAN EQUATIONS[J]. 数学物理学报(英文版), 2007, 27(2): 329-337.