MULTIPLE SOLUTIONS AND THEIR LIMITING BEHAVIOR OF COUPLED NONLINEAR SCHRÖDINGER SYSTEMS

doi:10.1016/S0252-9602(10)60117-8

数学物理学报(英文版) ›› 2010, Vol. 30 ›› Issue (4): 1199-1218.doi: 10.1016/S0252-9602(10)60117-8

MULTIPLE SOLUTIONS AND THEIR LIMITING BEHAVIOR OF COUPLED NONLINEAR SCHRÖDINGER SYSTEMS

万优艳

Institute of Physics and Mathematics, the Chinese Academy of Sciences, Wuhan |430071, China; The Department of Mathematics, Jianghan University, Wuhan 430056, China

收稿日期:2008-09-12 修回日期:2008-12-18 出版日期:2010-07-20 发布日期:2010-07-20
基金资助:
Supported by CAS-KJCX3-SYW-S03, Grant Fondecyt No. 1050613, and Scientific Research Fund for Youth of Hubei Provincial Education Department (Q20083401)

MULTIPLE SOLUTIONS AND THEIR LIMITING BEHAVIOR OF COUPLED NONLINEAR SCHRÖDINGER SYSTEMS

WAN You-Yan

Institute of Physics and Mathematics, the Chinese Academy of Sciences, Wuhan |430071, China; The Department of Mathematics, Jianghan University, Wuhan 430056, China

Received:2008-09-12 Revised:2008-12-18 Online:2010-07-20 Published:2010-07-20
Supported by:
Supported by CAS-KJCX3-SYW-S03, Grant Fondecyt No. 1050613, and Scientific Research Fund for Youth of Hubei Provincial Education Department (Q20083401)

摘要/Abstract

摘要：

We study the multiplicity of positive solutions and their limiting behavior as ε tends to zero for a class of coupled nonlinear Schr\"odinger system in R^N. We relate the number of positive solutions to the topology of the set of minimum points of the least energy function for ε sufficiently small. Also, we verify that these solutions concentrate at a global minimum point of the least energy function.

关键词: Elliptic system, Schr\"odinger equation, variational methods, Nehari manifold, relative category, concentration

Abstract:

Key words: Elliptic system, Schr\"odinger equation, variational methods, Nehari manifold, relative category, concentration

中图分类号:

35J50

万优艳. MULTIPLE SOLUTIONS AND THEIR LIMITING BEHAVIOR OF COUPLED NONLINEAR SCHRÖDINGER SYSTEMS[J]. 数学物理学报(英文版), 2010, 30(4): 1199-1218.

WAN You-Yan. MULTIPLE SOLUTIONS AND THEIR LIMITING BEHAVIOR OF COUPLED NONLINEAR SCHRÖDINGER SYSTEMS[J]. Acta mathematica scientia,Series B, 2010, 30(4): 1199-1218.

参考文献

[1] Alves C O, Soares S H M. Singularly perturbed elliptic systems. Nonlinear Analysis, 2006, 64: 109--129

[2] Alves C O, Soares S H M, Yang J. On existence and concentration of solutions for a class of Hamiltonian systems in R^N. Adv Nonlinear Stud, 2003, 3: 161--180

[3]  Ävila A, Yang J. On the existence and shape of least energy solutions for some elliptic systems. J Diff Eq, 2003, 191: 348--376

[4]  Ävila A, Yang J. Multiple solutions of nonlinear elliptic systems. NoDEA, 2006, 12: 459--479

[5] Cingolani S, Lazzo M. Multiple positive solutions to nonlinear Schr\"{o}dinger equations with competing potnetial functions. J Diff Eqns, 2000, 160: 118--138

[6] De Figueiredo D, Felmer P. On superquadratic elliptic systems. Trans Amer Math Soc, 1994, 102: 188--207

[7] De Figueiredo D, Yang J.Decay, symmetry and existence of solutions to semilinear elliptic systems. Nonlinear Analysis TMA, 1998, 33(3): 211--234

[8] Del Pino M, Felmer P. Semi-classical states of nonlinear Schr\"odinger equations: a variational reduction method. Math Ann, 2002, 324: 1--32

[9] Hulshof J, Van der Vorst R. Differential systems with strongly indefinite variational sturcture. J Funct Anal, 1993, 114: 32--58

[10] Lions P. The concentration-compactness principle in the calculus of variations. The locally compact case part 1 and 2. Ann Inst H Poincaré Anal Non Linéaire, 1984, 1: 109--145; 223--283

[11] Rabinowitz P. On a calss of nonlinear Schr\"{o}dinger equations. Z Angew Math Phys, 1992, 43: 270--291

[12] Sirakov B. On the existence of solutions of Hamiltonian elliptic systems in R^N. Adv in Diff Eqns, 2000, 5: 1445--1464

[13] Struwe M. Variational Methods. Berlin, New York: Springer-Verlag, 1990

[14] Wan Y. Existence and concentration of positive solutions for coupled nonlinear Schr\"{o}dinger systems in R^N. Adv Nonlinear Stud, 2007, 7: 77--95

[15] Wan Y, Ávila A. Multiple solutions of a coupled nonlinear Schr\"odinger system. J Math Anal Appl, 2007, 334: 1308--1325

[16]   Wang X. On concentration of positive bound states of nonlinear Schr\"{o}ding equations. Comm Math Phys, 1993, 153: 229--244

[17] Wang X, Zeng B. On concentration of positive bound states of nonlinear Schr\"{o}dinger equations with competing potential functions. SIAM J Math Anal, 1997, 28: 633--655

[18] Willem M. Minimax Theroems. Boston-Basel-Berlin: Birkhäuser, 1996

[19] Yang J. Nontrivial solutions of semilinear elliptic systems in R^N. Eletron J Diff Equs, 2001, 6: 343--357

MULTIPLE SOLUTIONS AND THEIR LIMITING BEHAVIOR OF COUPLED NONLINEAR SCHRÖDINGER SYSTEMS

MULTIPLE SOLUTIONS AND THEIR LIMITING BEHAVIOR OF COUPLED NONLINEAR SCHRÖDINGER SYSTEMS

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