American Institute of Mathematical Sciences

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 - A, 2013, 33 (7) : 2911-2938. doi: 10.3934/dcds.2013.33.2911
References:
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Amer Math. Soc., 88 (1983), 486. 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. 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. 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. 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. 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. Google Scholar [16] M. del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations,, J. Funct. Anal., 149 (1997), 245. 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. doi: 10.1002/cpa.20135. Google Scholar [18] I. Ekeland, On the variational principle,, J. Math. Anal. Appl., 17 (1974), 324. 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. 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. 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. 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. Google Scholar [23] N. Ikoma, Uniqueness of positive solutions for a nonlinear elliptic system,, NoDEA: Nonlinear Differential Equations and Applications, 16 (2009), 555. 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. 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. 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. 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. 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. 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. Google Scholar [30] T. C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations,, Ann. I. H. Poincaré-AN, 22 (2005), 403. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1137/S0036141095290240. Google Scholar [42] M. Willem, "Minimax Theorems,", Birkhäuser, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

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References:
 [1] A. Ambrosetti, "Critical Points and Nonlinear Variational Problems,", Bulletin Soc. Math. France, (1992). Google Scholar [2] A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations,, Arch. Ration. Mech. Anal., 140 (1997), 285. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar [16] M. del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations,, J. Funct. Anal., 149 (1997), 245. 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. doi: 10.1002/cpa.20135. Google Scholar [18] I. Ekeland, On the variational principle,, J. Math. Anal. Appl., 17 (1974), 324. 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. 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. 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. 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. Google Scholar [23] N. Ikoma, Uniqueness of positive solutions for a nonlinear elliptic system,, NoDEA: Nonlinear Differential Equations and Applications, 16 (2009), 555. 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. 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. 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. 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. 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. 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. Google Scholar [30] T. C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations,, Ann. I. H. Poincaré-AN, 22 (2005), 403. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1137/S0036141095290240. Google Scholar [42] M. Willem, "Minimax Theorems,", Birkhäuser, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar
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