Acta mathematica scientia, Series B >
MULTIPLE SOLUTIONS AND THEIR LIMITING BEHAVIOR OF COUPLED NONLINEAR SCHRÖDINGER SYSTEMS
Received date: 2008-09-12
Revised date: 2008-12-18
Online 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)
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 RN. 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.
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 . DOI: 10.1016/S0252-9602(10)60117-8
[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 RN. 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 RN. 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 RN. 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 RN. Eletron J Diff Equs, 2001, 6: 343--357
/
| 〈 |
|
〉 |