July  2013, 33(7): 2911-2938. doi: 10.3934/dcds.2013.33.2911

Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations

1. 

Department of Mathematics & National Center for Theoretical Sciences at Taipei, National Taiwan University, Taipei, 10617, Taiwan

2. 

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811

Received  April 2012 Revised  October 2012 Published  January 2013

It is well known that a single nonlinear Schrödinger (NLS) equation with a potential $V$ and a small parameter $\varepsilon $ may have a unique positive solution that is concentrated at the nondegenerate minimum point of $V$ . However, the uniqueness may fail for two-component systems of NLS equations with a small parameter $\varepsilon $ and potentials $V_{1}$ and $V_{2}$ having the same nondegenerate minimum point. In this paper, we will use energy estimates and category theory to prove the nonuniqueness theorem.
Citation: Tai-Chia Lin, Tsung-Fang Wu. Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2911-2938. doi: 10.3934/dcds.2013.33.2911
References:
[1]

Bulletin Soc. Math. France, Mémoire, 1992.  Google Scholar

[2]

Arch. Ration. Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067.  Google Scholar

[3]

Journal of the London Mathematical Society, 75 (2007), 67-82. doi: 10.1112/jlms/jdl020.  Google Scholar

[4]

Comm. Math. Phys., 235 (2003), 427-466. doi: 10.1007/s00220-003-0811-y.  Google Scholar

[5]

Arch. Ration. Mech. Anal., 159 (2001), 253-271. doi: 10.1007/s002050100152.  Google Scholar

[6]

Calc. Var. Partial Diff. Eqns., 11 (2000), 63-95. doi: 10.1007/s005260050003.  Google Scholar

[7]

Mathematische Annalen, 388 (2007), 147-185. doi: 10.1007/s00208-006-0071-1.  Google Scholar

[8]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 259-281. doi: 10.1016/j.anihpc.2004.07.005.  Google Scholar

[9]

Arch. Ration. Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.  Google Scholar

[10]

Proc. Amer Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.  Google Scholar

[11]

Bull. Amer. Math. Soc., 71 (1965), 644-648.  Google Scholar

[12]

Arch. Ration. Mech. Anal., 165 (2002), 295-316. doi: 10.1007/s00205-002-0225-6.  Google Scholar

[13]

J. Differential Equations, 160 (2000), 118-138. doi: 10.1006/jdeq.1999.3662.  Google Scholar

[14]

Calc. Var. Partial Differential Equations, 17 (2003), 257-281.  Google Scholar

[15]

Ann. Inst. H. Poincaré Anal. Non Lineairé, 13 (1996), 567-588.  Google Scholar

[16]

J. Funct. Anal., 149 (1997), 245-265. doi: 10.1006/jfan.1996.3085.  Google Scholar

[17]

Comm. Pure Appl. Math., 60 (2007), 113-146. doi: 10.1002/cpa.20135.  Google Scholar

[18]

J. Math. Anal. Appl., 17 (1974), 324-353.  Google Scholar

[19]

Ann. I. H. Poincaré-AN, 25 (2008), 149-161. doi: 10.1016/j.anihpc.2006.11.006.  Google Scholar

[20]

J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

[21]

Ann. Inst. H. Poincare Anal. NonLineaire, 19 (2002), 261-280. doi: 10.1016/S0294-1449(01)00089-0.  Google Scholar

[22]

Adv. Differential Equations, 2 (1997), 955-980.  Google Scholar

[23]

NoDEA: Nonlinear Differential Equations and Applications, 16 (2009), 555-567. doi: 10.1007/s00030-009-0017-x.  Google Scholar

[24]

Nature, 392 (1998), 151-154. Google Scholar

[25]

Phys. Rev. Lett., 81 (1998), 3108-3111. Google Scholar

[26]

Science, 296 (2002), 1290-1293. Google Scholar

[27]

Arch. Rat. Math. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.  Google Scholar

[28]

Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 109-145.  Google Scholar

[29]

Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 223-283.  Google Scholar

[30]

Ann. I. H. Poincaré-AN, 22 (2005), 403-439. doi: 10.1016/j.anihpc.2004.03.004.  Google Scholar

[31]

Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.  Google Scholar

[32]

Comm. Math. Phys., 277 (2008), 573-576.  Google Scholar

[33]

J. Differential Equations, 229 (2006), 538-569. doi: 10.1016/j.jde.2005.12.011.  Google Scholar

[34]

J. Math. Anal. Appl., 348 (2008), 169-179. doi: 10.1016/j.jmaa.2008.06.042.  Google Scholar

[35]

J. Differential Equations, 229 (2006), 743-767. doi: 10.1016/j.jde.2006.07.002.  Google Scholar

[36]

Oxford, 2003.  Google Scholar

[37]

Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.  Google Scholar

[38]

Ann. Mat. Pura Appl., 4 (2002), 73-83. doi: 10.1007/s102310200029.  Google Scholar

[39]

Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.  Google Scholar

[40]

Comm. Math. Phys., (1993), 229-244.  Google Scholar

[41]

SIAM J. Math. Anal., 28 (1997), 633-655. doi: 10.1137/S0036141095290240.  Google Scholar

[42]

Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

show all references

References:
[1]

Bulletin Soc. Math. France, Mémoire, 1992.  Google Scholar

[2]

Arch. Ration. Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067.  Google Scholar

[3]

Journal of the London Mathematical Society, 75 (2007), 67-82. doi: 10.1112/jlms/jdl020.  Google Scholar

[4]

Comm. Math. Phys., 235 (2003), 427-466. doi: 10.1007/s00220-003-0811-y.  Google Scholar

[5]

Arch. Ration. Mech. Anal., 159 (2001), 253-271. doi: 10.1007/s002050100152.  Google Scholar

[6]

Calc. Var. Partial Diff. Eqns., 11 (2000), 63-95. doi: 10.1007/s005260050003.  Google Scholar

[7]

Mathematische Annalen, 388 (2007), 147-185. doi: 10.1007/s00208-006-0071-1.  Google Scholar

[8]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 259-281. doi: 10.1016/j.anihpc.2004.07.005.  Google Scholar

[9]

Arch. Ration. Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.  Google Scholar

[10]

Proc. Amer Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.  Google Scholar

[11]

Bull. Amer. Math. Soc., 71 (1965), 644-648.  Google Scholar

[12]

Arch. Ration. Mech. Anal., 165 (2002), 295-316. doi: 10.1007/s00205-002-0225-6.  Google Scholar

[13]

J. Differential Equations, 160 (2000), 118-138. doi: 10.1006/jdeq.1999.3662.  Google Scholar

[14]

Calc. Var. Partial Differential Equations, 17 (2003), 257-281.  Google Scholar

[15]

Ann. Inst. H. Poincaré Anal. Non Lineairé, 13 (1996), 567-588.  Google Scholar

[16]

J. Funct. Anal., 149 (1997), 245-265. doi: 10.1006/jfan.1996.3085.  Google Scholar

[17]

Comm. Pure Appl. Math., 60 (2007), 113-146. doi: 10.1002/cpa.20135.  Google Scholar

[18]

J. Math. Anal. Appl., 17 (1974), 324-353.  Google Scholar

[19]

Ann. I. H. Poincaré-AN, 25 (2008), 149-161. doi: 10.1016/j.anihpc.2006.11.006.  Google Scholar

[20]

J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

[21]

Ann. Inst. H. Poincare Anal. NonLineaire, 19 (2002), 261-280. doi: 10.1016/S0294-1449(01)00089-0.  Google Scholar

[22]

Adv. Differential Equations, 2 (1997), 955-980.  Google Scholar

[23]

NoDEA: Nonlinear Differential Equations and Applications, 16 (2009), 555-567. doi: 10.1007/s00030-009-0017-x.  Google Scholar

[24]

Nature, 392 (1998), 151-154. Google Scholar

[25]

Phys. Rev. Lett., 81 (1998), 3108-3111. Google Scholar

[26]

Science, 296 (2002), 1290-1293. Google Scholar

[27]

Arch. Rat. Math. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.  Google Scholar

[28]

Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 109-145.  Google Scholar

[29]

Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 223-283.  Google Scholar

[30]

Ann. I. H. Poincaré-AN, 22 (2005), 403-439. doi: 10.1016/j.anihpc.2004.03.004.  Google Scholar

[31]

Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.  Google Scholar

[32]

Comm. Math. Phys., 277 (2008), 573-576.  Google Scholar

[33]

J. Differential Equations, 229 (2006), 538-569. doi: 10.1016/j.jde.2005.12.011.  Google Scholar

[34]

J. Math. Anal. Appl., 348 (2008), 169-179. doi: 10.1016/j.jmaa.2008.06.042.  Google Scholar

[35]

J. Differential Equations, 229 (2006), 743-767. doi: 10.1016/j.jde.2006.07.002.  Google Scholar

[36]

Oxford, 2003.  Google Scholar

[37]

Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.  Google Scholar

[38]

Ann. Mat. Pura Appl., 4 (2002), 73-83. doi: 10.1007/s102310200029.  Google Scholar

[39]

Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.  Google Scholar

[40]

Comm. Math. Phys., (1993), 229-244.  Google Scholar

[41]

SIAM J. Math. Anal., 28 (1997), 633-655. doi: 10.1137/S0036141095290240.  Google Scholar

[42]

Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

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