扰动方法研究R3中自由边值问题

Abstract

Abstract:

In this paper, the authors use the perturbation method to consider the following free boundary problem
{ -Δu=λ+εg(u))H(u-1), x ∈Ω,
ul ∂Ω=0.
where λ is a constant and Ω is the unit ball in R³, H(t)={ 1, t ≥ 0, 0, t<0.
Under some conditions, the authors prove that there is a positive constant ε₀ >0, if 0< ε < ε₀ there is at least one solution for above problem and some results for the solution as ε tend to zero.

Key words: Boundary problem, Linear operator, Implicit theorem, Eigenvalue

CLC Number:

35J25

Cite this article

LIU Wei-Ming, ZHANG Zheng-Jie. Perturbation in a Free Boundary Problem in R³[J].Acta mathematica scientia,Series A, 2010, 30(6): 1514-1522.

share this article

0
/ / Recommend

Add to citation manager EndNote|Reference Manager|ProCite|BibTeX|RefWorks

URL: http://121.43.60.238/sxwlxbA/EN/

http://121.43.60.238/sxwlxbA/EN/Y2010/V30/I6/1514

Trendmd

References

[1] Abdelkander Boucherif, Mohamed Bouguima. Perturbation in a free boundary problem. Journal of Mathematical Analysis and Applications, 1994, 185: 430--437

[2] Alexander R. A discontinuous nonlinear eigenvalue/free boundary problem. Math Methods Appl Sci, 1982, 4: 131--134

[3] Schwartz J T. Nonlinear Functional Analysis. New York: Springer-Verlag, 1969

[4] Gilbarg D, Trudinger N. Elliptic Partial Differential Equations of Second-order. New York: Springer-Verlag, 1983

[5] Lawrence C Evans. Partial Differential Equations. Providence, RI: American Mathematical Society, 1998

[6] Berestycki H, Brezis H. On a free boundary problem arising in plasma physics. Nonlinear Analysis, 1980, 4: 415--436

[7] Wolansky G. Critical behavior of semi-linear elliptic equations with sub-critical exponents. Nonliear Analysis, 1996, 5: 971--995

[8] Cao Daomin, Peng Shuangjie, Yan Shusen. Multiplicity of solutions for the plasma problem in two dimensions. to appear

Related Articles 15

[1]	Tan Shenyang, Huang Tiren, Liu Dajin. Estimates for Eigenvalues of the Operator in Divergence Form on H-Type Group [J]. Acta mathematica scientia,Series A, 2018, 38(3): 467-475.
[2]	Wang Shengjun, Han Yazhou. Generalized Picone's Identity and Its Applications for Heisenberg-Greiner Operators [J]. Acta mathematica scientia,Series A, 2017, 37(5): 895-901.
[3]	Luo Hua. Spectral Theory of Linear Weighted Sturm-Liouville Eigenvalue Problems [J]. Acta mathematica scientia,Series A, 2017, 37(3): 427-449.
[4]	Wu Hongxing, Wang Shenghua, Cheng Guofei. The Stability of a Transport Equations Solution with Abstract Boundary Condition in Slab Geometry [J]. Acta mathematica scientia,Series A, 2016, 36(4): 703-714.
[5]	He Yue. New Estimates of Lower Bound for the First Eigenvalue on Compact Manifolds with Positive Ricci Curvature [J]. Acta mathematica scientia,Series A, 2016, 36(2): 215-230.
[6]	Li Kun, Zheng Zhaowen. Spectral Properties for Sturm-Liouville Equations with Transmission Conditions [J]. Acta mathematica scientia,Series A, 2015, 35(5): 910-926.
[7]	Zheng Qingyu, Zhang Lei, Shi Shaoguang. On Boundedness of Sublinear Operators on Generalized Weighted Grand Morrey Spaces [J]. Acta mathematica scientia,Series A, 2015, 35(3): 503-514.
[8]	ZHANG Chun-Guo, ZHANG Hai-Yan, GU Shang-Wu. Energy Decay Estimates for the Weakly Coupled Systems with Local Memory Damping [J]. Acta mathematica scientia,Series A, 2015, 35(1): 194-209.
[9]	WANG Hua, Alatanchang, HUANG Jun-Jie. Multiplicity and Completeness of a Class of \|Hamiltonian \|Operators and Its Applications [J]. Acta mathematica scientia,Series A, 2014, 34(6): 1507-1517.
[10]	LI Ding-Wu, LIU Wei, SHEN Jin-Rong, XIONG Hui-Jun. Left and Right Inverse Eigenpairs Problem on Disk of Reflexive Matrices [J]. Acta mathematica scientia,Series A, 2014, 34(6): 1542-1553.
[11]	SUN He-Jun. Inequalities for Lower Order Eigenvalues of the Folland-Stein Operator and Quadratic Polynomial Operator of the Kohn Laplacian [J]. Acta mathematica scientia,Series A, 2014, 34(5): 1133-1143.
[12]	Leszek GASI′ NSKI, Nikolaos S. PAPAGEORGIOU. POSITIVE SOLUTIONS FOR PARAMETRIC EQUIDIFFUSIVE p-LAPLACIAN EQUATIONS [J]. Acta mathematica scientia,Series A, 2014, 34(3): 610-618.
[13]	Wu Shi-Liang, LI Cui-Xia. Spectral Analysis on Some Discrete Systems From Maxwell Equations [J]. Acta mathematica scientia,Series A, 2014, 34(1): 109-114.
[14]	ZHAO Wei-Xia. The Global Classical Solution to a Free Boundary Problem in the Peeling Phenomenon [J]. Acta mathematica scientia,Series A, 2013, 33(6): 1001-1012.
[15]	QU Feng-Long, LI Xi-Liang. The Interior Transmission Problem for Isotropic Maxwell Equations and Some Estimates on the Index of Refraction [J]. Acta mathematica scientia,Series A, 2013, 33(6): 1178-1188.

Metrics

Viewed

Full text

Abstract

Cited

Shared

Discussed

Comments

Recommended 10

[1]	Chen Fengde;Chen Xiaoxing ;Zhang Huiying. Positive Periodic Solutions of a Delayed Predator-Prey System with Holling Type II Functional Response[J]. Acta mathematica scientia,Series A, 2006, 26(1): 93 -103 .
[2]	Cai Jingjing\|Liu Yongming. Nonlinear Stability of Generalized Eady Problem in Finite Periodic Channel[J]. Acta mathematica scientia,Series A, 2008, 28(6): 1002 -1014 .
[3]	Zhu Bo; Han Baoyan. Backward Doubly Stochastic Differential Equations with Non-Lipschitz Coefficients[J]. Acta mathematica scientia,Series A, 2008, 28(5): 977 -984 .
[4]	Lv Shengli\|Li Jingbo\|Yue Dequan. The M/M/2 Repairable Queue System with Two Repairmen[J]. Acta mathematica scientia,Series A, 2008, 28(6): 1109 -1118 .
[5]	Shao Dan; Shao Liang; Shao Changgui; Xiao Ming. Recoupling Matrix of Action on Rescaled Basis in Loop Gravity[J]. Acta mathematica scientia,Series A, 2009, 29(1): 158 -164 .
[6]	Ma Dengju; Ren Han; Lu Junjie. The Euler Genera of Two Classes of Generalized Petersen Graphs[J]. Acta mathematica scientia,Series A, 2009, 29(1): 25 -31 .
[7]	Huang Longguang. The Solution Sets and Connectedness for Weak Vector Variational Inequalities[J]. Acta mathematica scientia,Series A, 2009, 29(1): 114 -120 .
[8]	Yang Shouzhi; Yang Xiaozhong. Orthogonal Multiscaling Functions and Multiwavelets with General Cardinal Property[J]. Acta mathematica scientia,Series A, 2007, 27(3): 470 -475 .
[9]	Li Yuan;Yang Yidang. Testing Heteroscedasticity by Wavelets in a Nonparametric Regression Model with Random Design[J]. Acta mathematica scientia,Series A, 2007, 27(4): 727 -740 .
[10]	Zhang Xingqiu;Zhong Qiuyan. The Existence of Multiple Positive Solutions for a Class of Singular Boundary Value Problems of Impulsive Differential Equations in a Banach Space[J]. Acta mathematica scientia,Series A, 2007, 27(4): 753 -760 .

Abstract
References
Related Articles
Metrics
Comments
Recommended

TOP

Perturbation in a Free Boundary Problem in R3

PDF (PC)