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 and 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.

[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.

[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.

[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.

[5]

A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear singularly perturbed elliptic problems on $\mathbb{R}^N2$, Arch. Ration. Mech. Anal., 159 (2001), 253-271. doi: 10.1007/s002050100152.

[6]

S. Adachi and K. Tanaka, Four positive solutions for the semilinear elliptic equation: $-\Delta u+u=a(x)u^p+f(x)$ in $\mathbb{R}^N2$, Calc. Var. Partial Diff. Eqns., 11 (2000), 63-95. doi: 10.1007/s005260050003.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[15]

D. Cao and E. S. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in $\mathbb{R}^N2$, Ann. Inst. H. Poincaré Anal. Non Lineairé, 13 (1996), 567-588.

[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.

[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.

[18]

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

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[27]

M. K. Kwong, Uniqueness of positive solution of $\Delta u-u+u^p=0$ in $\mathbb{R}^N2$, Arch. Rat. Math. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.

[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.

[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.

[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.

[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.

[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.

[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.

[34]

C. H. Liu, H. Y. Wang and T. F. Wu, Multiplicity of 2-nodal solutions for semilinear elliptic problems in $\mathbb{R}^N2$, J. Math. Anal. Appl., 348 (2008), 169-179. doi: 10.1016/j.jmaa.2008.06.042.

[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.

[36]

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

[37]

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

[38]

B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equation in $\mathbb{R}^N2$, Ann. Mat. Pura Appl., 4 (2002), 73-83. doi: 10.1007/s102310200029.

[39]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^N2$, Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.

[40]

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

[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.

[42]

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

show all references

References:
[1]

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

[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.

[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.

[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.

[5]

A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear singularly perturbed elliptic problems on $\mathbb{R}^N2$, Arch. Ration. Mech. Anal., 159 (2001), 253-271. doi: 10.1007/s002050100152.

[6]

S. Adachi and K. Tanaka, Four positive solutions for the semilinear elliptic equation: $-\Delta u+u=a(x)u^p+f(x)$ in $\mathbb{R}^N2$, Calc. Var. Partial Diff. Eqns., 11 (2000), 63-95. doi: 10.1007/s005260050003.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[15]

D. Cao and E. S. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in $\mathbb{R}^N2$, Ann. Inst. H. Poincaré Anal. Non Lineairé, 13 (1996), 567-588.

[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.

[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.

[18]

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

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[27]

M. K. Kwong, Uniqueness of positive solution of $\Delta u-u+u^p=0$ in $\mathbb{R}^N2$, Arch. Rat. Math. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.

[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.

[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.

[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.

[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.

[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.

[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.

[34]

C. H. Liu, H. Y. Wang and T. F. Wu, Multiplicity of 2-nodal solutions for semilinear elliptic problems in $\mathbb{R}^N2$, J. Math. Anal. Appl., 348 (2008), 169-179. doi: 10.1016/j.jmaa.2008.06.042.

[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.

[36]

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

[37]

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

[38]

B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equation in $\mathbb{R}^N2$, Ann. Mat. Pura Appl., 4 (2002), 73-83. doi: 10.1007/s102310200029.

[39]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^N2$, Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.

[40]

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

[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.

[42]

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

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