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Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger Systems
1. | School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China |
2. | School of Mathematical Sciences, Capital Normal University, Beijing 100048, China |
3. | HLM, CEMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China |
4. | School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China |
$ \begin{equation} \left\{ \begin{array}{ll} -\Delta u+\lambda_1 u = \mu_1 u^3+\beta uv^2-\gamma v &\text{in } \mathbb{R}^N, \\ -\Delta v+\lambda_2 v = \mu_2 v^3+\beta vu^2-\gamma u &\text{in } \mathbb{R}^N, \\ u(x), v(x)\rightarrow 0 \text{ as } \vert x\vert\rightarrow+\infty. \end{array} \right.\nonumber \end{equation} $ |
$ \lambda_1 = \lambda_2, \mu_1 = \mu_2 $ |
$ \gamma $ |
$ \beta\in(-1, +\infty) $ |
$ \gamma $ |
$ [-1, 0]\times H_r^1\left( \mathbb{R} ^N\right)\times H_r^1\left( \mathbb{R} ^N\right) $ |
$ \gamma = 0 $ |
References:
[1] |
N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 1-4. Google Scholar |
[2] |
J. C. Alexander and S. S. Antman,
Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems, Arch. Ration. Mech. Anal., 76 (1981), 339-354.
doi: 10.1007/BF00249970. |
[3] |
A. Ambrosetti and E. Colorado,
Bound and ground states of coupled nonlinear Schrödinger equations, C.R. Math. Acad. Sci. Paris, 342 (2006), 453-458.
doi: 10.1016/j.crma.2006.01.024. |
[4] |
A. Ambrosetti and E. Colorado,
Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.
doi: 10.1112/jlms/jdl020. |
[5] |
A. Ambrosetti, E. Colorado and D. Ruiz,
Standing waves multi-bump solitions to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var., 30 (2007), 85-112.
doi: 10.1007/s00526-006-0079-0. |
[6] |
T. Bartsch, E. N. Dancer and Z. Wang,
A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var., 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[7] |
T. Bartsch and Z.-Q. Wang,
Note on ground states of nonlinear Schrödinger systems, J. Partial Differ. Equ., 19 (2006), 200-207.
|
[8] |
T. Bartsch, Z.-Q. Wang and J. Wei,
Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.
doi: 10.1007/s11784-007-0033-6. |
[9] |
J. Belmonte-Beitia, V. M. Prez-Garca and P. J. Torres,
Solitary waves for linearly coupled nonlinear Schrödinger equations with inhomogeneous coefficients, J. Nonlinear Sci., 19 (2009), 437-451.
doi: 10.1007/s00332-008-9037-7. |
[10] |
J. Busca and B. Sirakov,
Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56.
doi: 10.1006/jdeq.1999.3701. |
[11] |
Z. Chen and W. Zou,
An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var., 48 (2013), 695-711.
doi: 10.1007/s00526-012-0568-2. |
[12] |
E. N. Dancer,
Boundary-value problems for ordinary differential equations on infinite intervals, Proc. Lond. Math. Soc., 30 (1975), 76-94.
doi: 10.1112/plms/s3-30.1.76. |
[13] |
E. N. Dancer, K. Wang and Z. Zhang,
The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture, J. Funct. Anal., 262 (2012), 1087-1131.
doi: 10.1016/j.jfa.2011.10.013. |
[14] |
E. N. Dancer, K. Wang and Z. Zhang,
Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species, J. Differential Equations, 251 (2011), 2737-2769.
doi: 10.1016/j.jde.2011.06.015. |
[15] |
E. N. Dancer and J. Wei,
Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction, Trans. Amer. Math. Soc., 361 (2009), 1189-1208.
doi: 10.1090/S0002-9947-08-04735-1. |
[16] |
E. N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non. Linéaire, 27 (2010), 953-969.
doi: 10.1016/j.anihpc.2010.01.009. |
[17] |
B. Deconinck, P. G. Kevrekidis, H. E. Nistazakis and D. J. Frantzeskakis, Linearly coupled Bose-Einstein condensates: from Rabi oscillations and quasiperiodic solutions to oscillating domain walls and spiral waves, Phys. Rev. A, 70 (2004), 705-706. Google Scholar |
[18] |
N. Dunford and J. T. Schwartz, Linear Operators. Part II. Spectral Theory. Selfadjoint Operators in Hilbert Space, Wiley, New York, 1988. |
[19] |
B. D. Esry, C. H. Greene, J. P. Burke Jr and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597. Google Scholar |
[20] |
P. M. Fitzpatrick, I. Massabò and J. Pejsachowicz,
Global several-parameter bifurcation and continuation theorems: a unified approach via completmenting maps, Math. Ann., 263 (1983), 61-73.
doi: 10.1007/BF01457084. |
[21] |
B. Gidas and J. Spruck,
Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.
doi: 10.1002/cpa.3160340406. |
[22] |
N. Ikoma,
Uniqueness of positive solutions for a nonlinear elliptic system, Nonlinear Differ. Equ. Appl., 16 (2009), 555-567.
doi: 10.1007/s00030-009-0017-x. |
[23] |
M. K. Kwong,
Uniqueness of positive solutions of $\Delta u + u + u^p$ in $ \mathbb{R} ^N$, Arch. Rat. Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[24] |
K. Li and Z. Zhang, Existence of solutions for a Schrödinger system with linear and nonlinear couplings, J. Math. Phys., 57 (2016), 081504, 17 pp.
doi: 10.1063/1.4960046. |
[25] |
T.-C. Lin and J. Wei,
Ground state of N coupled nonlinear Schrödinger equations in $\mathbb{R}^n$, n ≤ 3, Comm. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[26] |
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. |
[27] |
M. Mitchell, Z. Chen, M. Shih and M. Segev, Self-trapping of partially incoherent light, Phys. Rev. Lett., 77 (1996), 490-493. Google Scholar |
[28] |
A. Pomponio,
Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258-281.
doi: 10.1016/j.jde.2005.09.002. |
[29] |
P. H. Rabinowitz,
Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[30] |
C. Rüegg, N. Cavadini, A. Furrer, H.-U. Gdel, K. Krmer, H. Mutka, A. Wildes, K. Habicht and P. Vorderwisch, Bose-Einstein condensation of the triplet states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65. Google Scholar |
[31] |
B. Sirakov,
Least energy solitary waves for a system of nonlinear Schrödinger equations, Comm. Math. Physics, 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[32] |
W. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[33] |
R. Tian and Z. Zhang,
Existence and bifurcation of solutions for a double coupled system of Schrödinger equation, Sci. China Math., 58 (2015), 1607-1620.
doi: 10.1007/s11425-015-5028-y. |
[34] |
E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-721. Google Scholar |
[35] |
J. Wei and W. Yao,
Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011.
doi: 10.3934/cpaa.2012.11.1003. |
show all references
References:
[1] |
N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 1-4. Google Scholar |
[2] |
J. C. Alexander and S. S. Antman,
Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems, Arch. Ration. Mech. Anal., 76 (1981), 339-354.
doi: 10.1007/BF00249970. |
[3] |
A. Ambrosetti and E. Colorado,
Bound and ground states of coupled nonlinear Schrödinger equations, C.R. Math. Acad. Sci. Paris, 342 (2006), 453-458.
doi: 10.1016/j.crma.2006.01.024. |
[4] |
A. Ambrosetti and E. Colorado,
Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.
doi: 10.1112/jlms/jdl020. |
[5] |
A. Ambrosetti, E. Colorado and D. Ruiz,
Standing waves multi-bump solitions to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var., 30 (2007), 85-112.
doi: 10.1007/s00526-006-0079-0. |
[6] |
T. Bartsch, E. N. Dancer and Z. Wang,
A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var., 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[7] |
T. Bartsch and Z.-Q. Wang,
Note on ground states of nonlinear Schrödinger systems, J. Partial Differ. Equ., 19 (2006), 200-207.
|
[8] |
T. Bartsch, Z.-Q. Wang and J. Wei,
Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.
doi: 10.1007/s11784-007-0033-6. |
[9] |
J. Belmonte-Beitia, V. M. Prez-Garca and P. J. Torres,
Solitary waves for linearly coupled nonlinear Schrödinger equations with inhomogeneous coefficients, J. Nonlinear Sci., 19 (2009), 437-451.
doi: 10.1007/s00332-008-9037-7. |
[10] |
J. Busca and B. Sirakov,
Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56.
doi: 10.1006/jdeq.1999.3701. |
[11] |
Z. Chen and W. Zou,
An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var., 48 (2013), 695-711.
doi: 10.1007/s00526-012-0568-2. |
[12] |
E. N. Dancer,
Boundary-value problems for ordinary differential equations on infinite intervals, Proc. Lond. Math. Soc., 30 (1975), 76-94.
doi: 10.1112/plms/s3-30.1.76. |
[13] |
E. N. Dancer, K. Wang and Z. Zhang,
The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture, J. Funct. Anal., 262 (2012), 1087-1131.
doi: 10.1016/j.jfa.2011.10.013. |
[14] |
E. N. Dancer, K. Wang and Z. Zhang,
Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species, J. Differential Equations, 251 (2011), 2737-2769.
doi: 10.1016/j.jde.2011.06.015. |
[15] |
E. N. Dancer and J. Wei,
Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction, Trans. Amer. Math. Soc., 361 (2009), 1189-1208.
doi: 10.1090/S0002-9947-08-04735-1. |
[16] |
E. N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non. Linéaire, 27 (2010), 953-969.
doi: 10.1016/j.anihpc.2010.01.009. |
[17] |
B. Deconinck, P. G. Kevrekidis, H. E. Nistazakis and D. J. Frantzeskakis, Linearly coupled Bose-Einstein condensates: from Rabi oscillations and quasiperiodic solutions to oscillating domain walls and spiral waves, Phys. Rev. A, 70 (2004), 705-706. Google Scholar |
[18] |
N. Dunford and J. T. Schwartz, Linear Operators. Part II. Spectral Theory. Selfadjoint Operators in Hilbert Space, Wiley, New York, 1988. |
[19] |
B. D. Esry, C. H. Greene, J. P. Burke Jr and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597. Google Scholar |
[20] |
P. M. Fitzpatrick, I. Massabò and J. Pejsachowicz,
Global several-parameter bifurcation and continuation theorems: a unified approach via completmenting maps, Math. Ann., 263 (1983), 61-73.
doi: 10.1007/BF01457084. |
[21] |
B. Gidas and J. Spruck,
Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.
doi: 10.1002/cpa.3160340406. |
[22] |
N. Ikoma,
Uniqueness of positive solutions for a nonlinear elliptic system, Nonlinear Differ. Equ. Appl., 16 (2009), 555-567.
doi: 10.1007/s00030-009-0017-x. |
[23] |
M. K. Kwong,
Uniqueness of positive solutions of $\Delta u + u + u^p$ in $ \mathbb{R} ^N$, Arch. Rat. Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[24] |
K. Li and Z. Zhang, Existence of solutions for a Schrödinger system with linear and nonlinear couplings, J. Math. Phys., 57 (2016), 081504, 17 pp.
doi: 10.1063/1.4960046. |
[25] |
T.-C. Lin and J. Wei,
Ground state of N coupled nonlinear Schrödinger equations in $\mathbb{R}^n$, n ≤ 3, Comm. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[26] |
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. |
[27] |
M. Mitchell, Z. Chen, M. Shih and M. Segev, Self-trapping of partially incoherent light, Phys. Rev. Lett., 77 (1996), 490-493. Google Scholar |
[28] |
A. Pomponio,
Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258-281.
doi: 10.1016/j.jde.2005.09.002. |
[29] |
P. H. Rabinowitz,
Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[30] |
C. Rüegg, N. Cavadini, A. Furrer, H.-U. Gdel, K. Krmer, H. Mutka, A. Wildes, K. Habicht and P. Vorderwisch, Bose-Einstein condensation of the triplet states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65. Google Scholar |
[31] |
B. Sirakov,
Least energy solitary waves for a system of nonlinear Schrödinger equations, Comm. Math. Physics, 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[32] |
W. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[33] |
R. Tian and Z. Zhang,
Existence and bifurcation of solutions for a double coupled system of Schrödinger equation, Sci. China Math., 58 (2015), 1607-1620.
doi: 10.1007/s11425-015-5028-y. |
[34] |
E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-721. Google Scholar |
[35] |
J. Wei and W. Yao,
Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011.
doi: 10.3934/cpaa.2012.11.1003. |



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