American Institute of Mathematical Sciences

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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

Tai-Chia Lin 1, and  Tsung-Fang Wu 2,

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.
Keywords: Lusternik-Schnirelmann category, Two coupled nonlinear Schrödinger equations, multiple positive solutions..
Mathematics Subject Classification: Primary: 35J50, 35J57; Secondary: 35J4.
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
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A. Ambrosetti, "Critical Points and Nonlinear Variational Problems," Bulletin Soc. Math. France, Mémoire, 1992.  Google Scholar

[2]

A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067.  Google Scholar

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A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrodinger equations, Journal of the London Mathematical Society, 75 (2007), 67-82. doi: 10.1112/jlms/jdl020.  Google Scholar

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A. Ambrosetti, A. Malchiodi and W. M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres, part I, Comm. Math. Phys., 235 (2003), 427-466. doi: 10.1007/s00220-003-0811-y.  Google Scholar

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A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear singularly perturbed elliptic problems on $\mathbbR^N$, Arch. Ration. Mech. Anal., 159 (2001), 253-271. doi: 10.1007/s002050100152.  Google Scholar

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T. Bartsch, M. Clapp and T. Weth, Configuration spaces, transfer, and 2-nodal solutions of a semiclassical nonlinear Schrödinger equation, Mathematische Annalen, 388 (2007), 147-185. doi: 10.1007/s00208-006-0071-1.  Google Scholar

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H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.  Google Scholar

[11]

F. E. Browder, Lusternik-Schnirelman category and nonlinear elliptic eigenvalue problems, Bull. Amer. Math. Soc., 71 (1965), 644-648.  Google Scholar

[12]

J. Byeon and Z. Q.Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 165 (2002), 295-316. doi: 10.1007/s00205-002-0225-6.  Google Scholar

[13]

S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differential Equations, 160 (2000), 118-138. doi: 10.1006/jdeq.1999.3662.  Google Scholar

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G. Cerami and D. Passaseo, The effect of concentrating potentials in some singularly perturbed problems, Calc. Var. Partial Differential Equations, 17 (2003), 257-281.  Google Scholar

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D. Cao and E. S. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in $\mathbbR^N$, Ann. Inst. H. Poincaré Anal. Non Lineairé, 13 (1996), 567-588.  Google Scholar

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M. del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265. doi: 10.1006/jfan.1996.3085.  Google Scholar

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M. del Pino, M. Kowalczyk and J. Wei, Concentrations on curve for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146. doi: 10.1002/cpa.20135.  Google Scholar

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A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

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Y. Y. Li, On a singularly perturbed elliptic equation, Adv. Differential Equations, 2 (1997), 955-980.  Google Scholar

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N. Ikoma, Uniqueness of positive solutions for a nonlinear elliptic system, NoDEA: Nonlinear Differential Equations and Applications, 16 (2009), 555-567. doi: 10.1007/s00030-009-0017-x.  Google Scholar

[24]

S. Inouye, M. R. Andrews, J. Stenger, H. J. Miesner, D. M. Stamper-Kurn and W. Ketterle, Observation of Feshbach resonances in a Bose-Einstein condensate, Nature, 392 (1998), 151-154. Google Scholar

[25]

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner and P. Zoller, Cold bosonic atoms in optical lattices, Phys. Rev. Lett., 81 (1998), 3108-3111. Google Scholar

[26]

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin and C. Salomon, Formation of a matter-wave bright soliton, Science, 296 (2002), 1290-1293. Google Scholar

[27]

M. K. Kwong, Uniqueness of positive solution of $\Delta u-u+u^p=0$ in $\mathbbR^N$, Arch. Rat. Math. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.  Google Scholar

[28]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case I, Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 109-145.  Google Scholar

[29]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case II, Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 223-283.  Google Scholar

[30]

T. C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. I. H. Poincaré-AN, 22 (2005), 403-439. doi: 10.1016/j.anihpc.2004.03.004.  Google Scholar

[31]

T. C. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbbR^N,$ $n\leq 3$, Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.  Google Scholar

[32]

T. C. Lin and J. Wei, Erratum: Ground state of N coupled nonlinear Schrödinger equations in $\mathbbR^N,$ $n\leq 3$ [Comm. Math. Phys. 255 (2005) 629-653; MR2135447], Comm. Math. Phys., 277 (2008), 573-576.  Google Scholar

[33]

T. C. Lin and J. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differential Equations, 229 (2006), 538-569. doi: 10.1016/j.jde.2005.12.011.  Google Scholar

[34]

C. H. Liu, H. Y. Wang and T. F. Wu, Multiplicity of 2-nodal solutions for semilinear elliptic problems in $\mathbbR^N$, J. Math. Anal. Appl., 348 (2008), 169-179. doi: 10.1016/j.jmaa.2008.06.042.  Google Scholar

[35]

L.A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767. doi: 10.1016/j.jde.2006.07.002.  Google Scholar

[36]

L. Pitaevskii and S. Stringari, "Bose-Einstein Condensation," Oxford, 2003.  Google Scholar

[37]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equation, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.  Google Scholar

[38]

B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equation in $\mathbbR^N$, Ann. Mat. Pura Appl., 4 (2002), 73-83. doi: 10.1007/s102310200029.  Google Scholar

[39]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbbR^N$, Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.  Google Scholar

[40]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., (1993), 229-244.  Google Scholar

[41]

X. Wang and B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equation with competing potential functions, SIAM J. Math. Anal., 28 (1997), 633-655. doi: 10.1137/S0036141095290240.  Google Scholar

[42]

M. Willem, "Minimax Theorems," Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

show all references

References:
[1]

A. Ambrosetti, "Critical Points and Nonlinear Variational Problems," Bulletin Soc. Math. France, Mémoire, 1992.  Google Scholar

[2]

A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067.  Google Scholar

[3]

A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrodinger equations, Journal of the London Mathematical Society, 75 (2007), 67-82. doi: 10.1112/jlms/jdl020.  Google Scholar

[4]

A. Ambrosetti, A. Malchiodi and W. M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres, part I, Comm. Math. Phys., 235 (2003), 427-466. doi: 10.1007/s00220-003-0811-y.  Google Scholar

[5]

A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear singularly perturbed elliptic problems on $\mathbbR^N$, Arch. Ration. Mech. Anal., 159 (2001), 253-271. doi: 10.1007/s002050100152.  Google Scholar

[6]

S. Adachi and K. Tanaka, Four positive solutions for the semilinear elliptic equation: $-\Delta u+u=a(x)u^p+f(x)$ in $\mathbbR^N$, Calc. Var. Partial Diff. Eqns., 11 (2000), 63-95. doi: 10.1007/s005260050003.  Google Scholar

[7]

T. Bartsch, M. Clapp and T. Weth, Configuration spaces, transfer, and 2-nodal solutions of a semiclassical nonlinear Schrödinger equation, Mathematische Annalen, 388 (2007), 147-185. doi: 10.1007/s00208-006-0071-1.  Google Scholar

[8]

T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 259-281. doi: 10.1016/j.anihpc.2004.07.005.  Google Scholar

[9]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.  Google Scholar

[10]

H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.  Google Scholar

[11]

F. E. Browder, Lusternik-Schnirelman category and nonlinear elliptic eigenvalue problems, Bull. Amer. Math. Soc., 71 (1965), 644-648.  Google Scholar

[12]

J. Byeon and Z. Q.Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 165 (2002), 295-316. doi: 10.1007/s00205-002-0225-6.  Google Scholar

[13]

S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differential Equations, 160 (2000), 118-138. doi: 10.1006/jdeq.1999.3662.  Google Scholar

[14]

G. Cerami and D. Passaseo, The effect of concentrating potentials in some singularly perturbed problems, Calc. Var. Partial Differential Equations, 17 (2003), 257-281.  Google Scholar

[15]

D. Cao and E. S. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in $\mathbbR^N$, Ann. Inst. H. Poincaré Anal. Non Lineairé, 13 (1996), 567-588.  Google Scholar

[16]

M. del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265. doi: 10.1006/jfan.1996.3085.  Google Scholar

[17]

M. del Pino, M. Kowalczyk and J. Wei, Concentrations on curve for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146. doi: 10.1002/cpa.20135.  Google Scholar

[18]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 17 (1974), 324-353.  Google Scholar

[19]

D. G. de Figueiredo and O. Lopes, Solitary waves for some nonlinear Schrödinger systems, Ann. I. H. Poincaré-AN, 25 (2008), 149-161. doi: 10.1016/j.anihpc.2006.11.006.  Google Scholar

[20]

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

[21]

M. Grossi, On the number of single-peak solutions of the nonlinear Schrödinger equation, Ann. Inst. H. Poincare Anal. NonLineaire, 19 (2002), 261-280. doi: 10.1016/S0294-1449(01)00089-0.  Google Scholar

[22]

Y. Y. Li, On a singularly perturbed elliptic equation, Adv. Differential Equations, 2 (1997), 955-980.  Google Scholar

[23]

N. Ikoma, Uniqueness of positive solutions for a nonlinear elliptic system, NoDEA: Nonlinear Differential Equations and Applications, 16 (2009), 555-567. doi: 10.1007/s00030-009-0017-x.  Google Scholar

[24]

S. Inouye, M. R. Andrews, J. Stenger, H. J. Miesner, D. M. Stamper-Kurn and W. Ketterle, Observation of Feshbach resonances in a Bose-Einstein condensate, Nature, 392 (1998), 151-154. Google Scholar

[25]

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner and P. Zoller, Cold bosonic atoms in optical lattices, Phys. Rev. Lett., 81 (1998), 3108-3111. Google Scholar

[26]

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin and C. Salomon, Formation of a matter-wave bright soliton, Science, 296 (2002), 1290-1293. Google Scholar

[27]

M. K. Kwong, Uniqueness of positive solution of $\Delta u-u+u^p=0$ in $\mathbbR^N$, Arch. Rat. Math. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.  Google Scholar

[28]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case I, Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 109-145.  Google Scholar

[29]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case II, Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 223-283.  Google Scholar

[30]

T. C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. I. H. Poincaré-AN, 22 (2005), 403-439. doi: 10.1016/j.anihpc.2004.03.004.  Google Scholar

[31]

T. C. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbbR^N,$ $n\leq 3$, Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.  Google Scholar

[32]

T. C. Lin and J. Wei, Erratum: Ground state of N coupled nonlinear Schrödinger equations in $\mathbbR^N,$ $n\leq 3$ [Comm. Math. Phys. 255 (2005) 629-653; MR2135447], Comm. Math. Phys., 277 (2008), 573-576.  Google Scholar

[33]

T. C. Lin and J. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differential Equations, 229 (2006), 538-569. doi: 10.1016/j.jde.2005.12.011.  Google Scholar

[34]

C. H. Liu, H. Y. Wang and T. F. Wu, Multiplicity of 2-nodal solutions for semilinear elliptic problems in $\mathbbR^N$, J. Math. Anal. Appl., 348 (2008), 169-179. doi: 10.1016/j.jmaa.2008.06.042.  Google Scholar

[35]

L.A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767. doi: 10.1016/j.jde.2006.07.002.  Google Scholar

[36]

L. Pitaevskii and S. Stringari, "Bose-Einstein Condensation," Oxford, 2003.  Google Scholar

[37]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equation, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.  Google Scholar

[38]

B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equation in $\mathbbR^N$, Ann. Mat. Pura Appl., 4 (2002), 73-83. doi: 10.1007/s102310200029.  Google Scholar

[39]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbbR^N$, Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.  Google Scholar

[40]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., (1993), 229-244.  Google Scholar

[41]

X. Wang and B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equation with competing potential functions, SIAM J. Math. Anal., 28 (1997), 633-655. doi: 10.1137/S0036141095290240.  Google Scholar

[42]

M. Willem, "Minimax Theorems," Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

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