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On periodic solutions in the Whitney's inverted pendulum problem
Positive solutions of doubly coupled multicomponent nonlinear Schrödinger systems
1. | School of Mathematical Science, Capital Normal University, Beijing 10048, China |
2. | Center for Applied Mathematics, Tianjin University, Tianjin 300072, China |
3. | Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA |
$ \begin{equation*} \left\{\begin{array}{ll} -\Delta u_j + \lambda_ju_j+ \sum_{k\neq j}\gamma_{jk}u_k = \mu_ju_j^3+ u_j\sum_{k\neq j}\beta_{jk}u_k^2,\\ u_j(x)\geq0\ \ \hbox{and}\ \ u_j\in H_0^1(\Omega), \end{array} \right. \end{equation*} $ |
$ \Omega\subset \mathbb{R} ^N $ |
$ N = 2,3 $ |
$ \lambda_j, \gamma_{jk} = \gamma_{kj}, \mu_j, \beta_{jk} = \beta_{kj} $ |
$ j, k = 1, 2, ..., n $ |
$ n\geq 2 $ |
$ \lambda_j\equiv\lambda, \gamma_{jk}\equiv\gamma, \mu_j\equiv\mu, \beta_{jk}\equiv\beta $ |
References:
[1] |
A. Ambrosetti, G. Cerami and D. Ruiz,
Solitons of linearly coupled systems of semilinear non-autonomous equations on $ \mathbb{R} ^n$, J. Funct. Anal., 254 (2008), 2816-2845.
doi: 10.1016/j.jfa.2007.11.013. |
[2] |
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. |
[3] |
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. |
[4] |
T. Bartsch,
Bifurcation in a multicomponent system of nonlinear Schrödinger equations, J. Fixed Point Theory Appl., 13 (2013), 37-50.
doi: 10.1007/s11784-013-0109-4. |
[5] |
T. Bartsch, E. N. Dancer and Z.-Q. Wang,
A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Vari. Part. Diff. Equ., 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[6] |
T. Bartsch, R. Tian and Z.-Q. Wang,
Bifurcations for a coupled Schr dinger system with multiple components,, Angew. Math. Phys., 66 (2015), 2109-2123.
doi: 10.1007/s00033-015-0498-x. |
[7] |
T. Bartsch and Z.-Q. Wang,
Note on ground states of nonlinear Schrödinger systems, J. Part. Diff. 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] |
G. Dai, R. Tian and Z. Zhang, Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger systems., Preprint. |
[10] |
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. |
[11] |
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), 063605. |
[12] |
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.
|
[13] |
P. M. Fitzpatrick, I. Massabò and J. Pejsachowicz,
Global several-parameter bifurcation and continuation thereoms: a Unified approach via complementing maps, Math. Ann., 263 (1983), 61-73.
doi: 10.1007/BF01457084. |
[14] |
K. Li and Z. Zhang,
Existence of solutions for a Schrödinger system with linear and nonlinear couplings, J. Math. Phys., 57 (2016), 081504, 17pp.
doi: 10.1063/1.4960046. |
[15] |
T. Lin and J. Wei,
Ground state of $N$ Coupled Nonlinear Schrödinger equations in $ \mathbb{R} ^n, n\leq3 $, Commun. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[16] |
T. Lin and J. Wei,
Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations, Physics D: Nonlinear Phenomena, 220 (2006), 99-115.
doi: 10.1016/j.physd.2006.07.009. |
[17] |
Z. Liu and Z.-Q. Wang,
Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phy., 282 (2008), 721-731.
doi: 10.1007/s00220-008-0546-x. |
[18] |
Z. Liu and Z.-Q. Wang,
Ground states and bound states of a nonlinear Schrödinger system, Advanced Nonlinear Studies, 10 (2010), 175-193.
doi: 10.1515/ans-2010-0109. |
[19] |
L. A. Maia, E. Montefusco and B. Pellacci,
Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Diff. Equ., 299 (2006), 743-767.
doi: 10.1016/j.jde.2006.07.002. |
[20] |
J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian System, Spinger-Verlag, New York, 1989.
doi: 10.1007/978-1-4757-2061-7. |
[21] |
M. Mitchell, Z. Chen, M. Shih and M. Segev,
Self-trapping of partially spatially incoherent light, Phys. Rev. Lett., 77 (1996), 490-493.
|
[22] |
Ch. Rüegg, N. Cavadini, A. Furrer, H.-U. Güdel, K. Krämer, H. Mutka, A. Wildes, K. Habicht and P. Vorderwischu,
Bose-Einstein condensation of the triplet states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65.
|
[23] |
B. Sirakov,
Least energy solitary waves for a system of nonlinear Schrödinger equations in $ \mathbb{R} ^n$, Comm. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[24] |
R. Tian and Z.-Q. Wang,
Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.
|
[25] |
R. Tian and Z.-Q. Wang,
Bifurcation results on positive solutions of an indefinite nonlinear elliptic system, Discrete Contin. Dyn. Syst. - Series A, 33 (2013), 335-344.
doi: 10.3934/dcds.2013.33.335. |
[26] |
R. Tian and Z.-Q. Wang,
Bifurcation results on positive solutions of an indefinite nonlinear elliptic system Ⅱ, Adv. Nonlinear Stud., 13 (2013), 245-262.
doi: 10.1515/ans-2013-0115. |
[27] |
R. Tian and Z.-T. Zhang,
Existence and bifurcation of solutions for a double coupled system of Schrödinger equations, Sci. China Math., 58 (2015), 1607-1620.
doi: 10.1007/s11425-015-5028-y. |
[28] |
Z.-Q. Wang,
A Zp index theory, Acta Mathematica Sinica, New Series, 6 (1990), 18-23.
doi: 10.1007/BF02108859. |
[29] |
J. Wei and T. Weth,
Nonradial symmetric bound states for a system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl., 18 (2007), 279-293.
doi: 10.4171/RLM/495. |
show all references
References:
[1] |
A. Ambrosetti, G. Cerami and D. Ruiz,
Solitons of linearly coupled systems of semilinear non-autonomous equations on $ \mathbb{R} ^n$, J. Funct. Anal., 254 (2008), 2816-2845.
doi: 10.1016/j.jfa.2007.11.013. |
[2] |
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. |
[3] |
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. |
[4] |
T. Bartsch,
Bifurcation in a multicomponent system of nonlinear Schrödinger equations, J. Fixed Point Theory Appl., 13 (2013), 37-50.
doi: 10.1007/s11784-013-0109-4. |
[5] |
T. Bartsch, E. N. Dancer and Z.-Q. Wang,
A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Vari. Part. Diff. Equ., 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[6] |
T. Bartsch, R. Tian and Z.-Q. Wang,
Bifurcations for a coupled Schr dinger system with multiple components,, Angew. Math. Phys., 66 (2015), 2109-2123.
doi: 10.1007/s00033-015-0498-x. |
[7] |
T. Bartsch and Z.-Q. Wang,
Note on ground states of nonlinear Schrödinger systems, J. Part. Diff. 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] |
G. Dai, R. Tian and Z. Zhang, Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger systems., Preprint. |
[10] |
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. |
[11] |
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), 063605. |
[12] |
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.
|
[13] |
P. M. Fitzpatrick, I. Massabò and J. Pejsachowicz,
Global several-parameter bifurcation and continuation thereoms: a Unified approach via complementing maps, Math. Ann., 263 (1983), 61-73.
doi: 10.1007/BF01457084. |
[14] |
K. Li and Z. Zhang,
Existence of solutions for a Schrödinger system with linear and nonlinear couplings, J. Math. Phys., 57 (2016), 081504, 17pp.
doi: 10.1063/1.4960046. |
[15] |
T. Lin and J. Wei,
Ground state of $N$ Coupled Nonlinear Schrödinger equations in $ \mathbb{R} ^n, n\leq3 $, Commun. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[16] |
T. Lin and J. Wei,
Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations, Physics D: Nonlinear Phenomena, 220 (2006), 99-115.
doi: 10.1016/j.physd.2006.07.009. |
[17] |
Z. Liu and Z.-Q. Wang,
Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phy., 282 (2008), 721-731.
doi: 10.1007/s00220-008-0546-x. |
[18] |
Z. Liu and Z.-Q. Wang,
Ground states and bound states of a nonlinear Schrödinger system, Advanced Nonlinear Studies, 10 (2010), 175-193.
doi: 10.1515/ans-2010-0109. |
[19] |
L. A. Maia, E. Montefusco and B. Pellacci,
Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Diff. Equ., 299 (2006), 743-767.
doi: 10.1016/j.jde.2006.07.002. |
[20] |
J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian System, Spinger-Verlag, New York, 1989.
doi: 10.1007/978-1-4757-2061-7. |
[21] |
M. Mitchell, Z. Chen, M. Shih and M. Segev,
Self-trapping of partially spatially incoherent light, Phys. Rev. Lett., 77 (1996), 490-493.
|
[22] |
Ch. Rüegg, N. Cavadini, A. Furrer, H.-U. Güdel, K. Krämer, H. Mutka, A. Wildes, K. Habicht and P. Vorderwischu,
Bose-Einstein condensation of the triplet states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65.
|
[23] |
B. Sirakov,
Least energy solitary waves for a system of nonlinear Schrödinger equations in $ \mathbb{R} ^n$, Comm. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[24] |
R. Tian and Z.-Q. Wang,
Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.
|
[25] |
R. Tian and Z.-Q. Wang,
Bifurcation results on positive solutions of an indefinite nonlinear elliptic system, Discrete Contin. Dyn. Syst. - Series A, 33 (2013), 335-344.
doi: 10.3934/dcds.2013.33.335. |
[26] |
R. Tian and Z.-Q. Wang,
Bifurcation results on positive solutions of an indefinite nonlinear elliptic system Ⅱ, Adv. Nonlinear Stud., 13 (2013), 245-262.
doi: 10.1515/ans-2013-0115. |
[27] |
R. Tian and Z.-T. Zhang,
Existence and bifurcation of solutions for a double coupled system of Schrödinger equations, Sci. China Math., 58 (2015), 1607-1620.
doi: 10.1007/s11425-015-5028-y. |
[28] |
Z.-Q. Wang,
A Zp index theory, Acta Mathematica Sinica, New Series, 6 (1990), 18-23.
doi: 10.1007/BF02108859. |
[29] |
J. Wei and T. Weth,
Nonradial symmetric bound states for a system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl., 18 (2007), 279-293.
doi: 10.4171/RLM/495. |
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