-
Previous Article
Existence of intermediate weak solution to the equations of multi-dimensional chemotaxis systems
- DCDS Home
- This Issue
-
Next Article
Viscosity dominated limit of global solutions to a hyperbolic equation in MEMS
A perturbation result for system of Schrödinger equations of Bose-Einstein condensates in $\mathbb{R}^3$
1. | School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, China |
2. | Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190 |
References:
[1] |
A. Ambrosetti, On Schrödinger-Poisson systems, Milan journal of mathematics, 76 (2008), 257-274.
doi: 10.1007/s00032-008-0094-z. |
[2] |
A. Ambrosetti, J. Garcia Azorero and I. Peral, Remarks on a class of semilinear elliptic equations on $\mathbb R^n$, via perturbation methods, Advanced Nonlinear Studies, 1 (2001), 1-13. |
[3] |
A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbb R^n$, Progress in Mathematics, 240, Birkhäuser Verlag, Basel, 2006. |
[4] |
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. Var. Partial Differential Equations, 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[5] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[6] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations, II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375.
doi: 10.1007/BF00250556. |
[7] |
E. N. Dancer, K. L. Wang and Z. T. 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. |
[8] |
E. N. Dancer, K. L. Wang and Z. T. 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. |
[9] |
E. N. Dancer, K. L. Wang and Z. T. Zhang, Addendum to "The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture'' [J. Funct. Anal., 262 (2012), 1087-1131] [MR2863857], J. Funct. Anal., 264 (2013), 1125-1129.
doi: 10.1016/j.jfa.2011.10.013. |
[10] |
E. N. Dancer and J. C. Wei, Spike solutions in coupled nonlinear chrödinger equations with attractive interaction, Trans. Amer. Math. Soc., 361 (2009), 1189-1208.
doi: 10.1090/S0002-9947-08-04735-1. |
[11] |
E. N. Dancer, J. C. Wei and W. Tobias, 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. |
[12] |
E. N. Dancer and T. Weth, Liouville-type results for non-cooperative elliptic systems in a half-space, J. Lond. Math. Soc., 86 (2012), 111-128.
doi: 10.1112/jlms/jdr080. |
[13] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order, Reprint of the 1998 ed., Springer-Verlag, Berlin, 2001. |
[14] |
M. K. Kwong, Uniqueness of positive radial solutions for $\Delta u- u + u^p= 0$ in $\mathbbR^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[15] |
Y. Sato and Z. Q. Wang, On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. PoincaréAnal. Non Linéaire, 30 (2013), 1-22.
doi: 10.1016/j.anihpc.2012.05.002. |
[16] |
H. Tavares and S. Terracini, Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems, Ann. Inst. H. PoincaréAnal. Non Linéaire, 29 (2012), 279-300.
doi: 10.1016/j.anihpc.2011.10.006. |
[17] |
S. Terracini and G. Verzini, Multipulse Phases in k-Mixtures of Bose-Einstein Condensates, Arch. Rational Mech. Anal., 194 (2009), 717-741.
doi: 10.1007/s00205-008-0172-y. |
[18] |
J. C. Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Differential Equations, 129 (1996), 315-333.
doi: 10.1006/jdeq.1996.0120. |
show all references
References:
[1] |
A. Ambrosetti, On Schrödinger-Poisson systems, Milan journal of mathematics, 76 (2008), 257-274.
doi: 10.1007/s00032-008-0094-z. |
[2] |
A. Ambrosetti, J. Garcia Azorero and I. Peral, Remarks on a class of semilinear elliptic equations on $\mathbb R^n$, via perturbation methods, Advanced Nonlinear Studies, 1 (2001), 1-13. |
[3] |
A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbb R^n$, Progress in Mathematics, 240, Birkhäuser Verlag, Basel, 2006. |
[4] |
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. Var. Partial Differential Equations, 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[5] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[6] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations, II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375.
doi: 10.1007/BF00250556. |
[7] |
E. N. Dancer, K. L. Wang and Z. T. 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. |
[8] |
E. N. Dancer, K. L. Wang and Z. T. 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. |
[9] |
E. N. Dancer, K. L. Wang and Z. T. Zhang, Addendum to "The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture'' [J. Funct. Anal., 262 (2012), 1087-1131] [MR2863857], J. Funct. Anal., 264 (2013), 1125-1129.
doi: 10.1016/j.jfa.2011.10.013. |
[10] |
E. N. Dancer and J. C. Wei, Spike solutions in coupled nonlinear chrödinger equations with attractive interaction, Trans. Amer. Math. Soc., 361 (2009), 1189-1208.
doi: 10.1090/S0002-9947-08-04735-1. |
[11] |
E. N. Dancer, J. C. Wei and W. Tobias, 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. |
[12] |
E. N. Dancer and T. Weth, Liouville-type results for non-cooperative elliptic systems in a half-space, J. Lond. Math. Soc., 86 (2012), 111-128.
doi: 10.1112/jlms/jdr080. |
[13] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order, Reprint of the 1998 ed., Springer-Verlag, Berlin, 2001. |
[14] |
M. K. Kwong, Uniqueness of positive radial solutions for $\Delta u- u + u^p= 0$ in $\mathbbR^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[15] |
Y. Sato and Z. Q. Wang, On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. PoincaréAnal. Non Linéaire, 30 (2013), 1-22.
doi: 10.1016/j.anihpc.2012.05.002. |
[16] |
H. Tavares and S. Terracini, Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems, Ann. Inst. H. PoincaréAnal. Non Linéaire, 29 (2012), 279-300.
doi: 10.1016/j.anihpc.2011.10.006. |
[17] |
S. Terracini and G. Verzini, Multipulse Phases in k-Mixtures of Bose-Einstein Condensates, Arch. Rational Mech. Anal., 194 (2009), 717-741.
doi: 10.1007/s00205-008-0172-y. |
[18] |
J. C. Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Differential Equations, 129 (1996), 315-333.
doi: 10.1006/jdeq.1996.0120. |
[1] |
Chuangye Liu, Zhi-Qiang Wang. Synchronization of positive solutions for coupled Schrödinger equations. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2795-2808. doi: 10.3934/dcds.2018118 |
[2] |
Zhanping Liang, Yuanmin Song, Fuyi Li. Positive ground state solutions of a quadratically coupled schrödinger system. Communications on Pure and Applied Analysis, 2017, 16 (3) : 999-1012. doi: 10.3934/cpaa.2017048 |
[3] |
Juncheng Wei, Wei Yao. Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1003-1011. doi: 10.3934/cpaa.2012.11.1003 |
[4] |
Hongyu Ye. Positive solutions for critically coupled Schrödinger systems with attractive interactions. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 485-507. doi: 10.3934/dcds.2018022 |
[5] |
Jiabao Su, Rushun Tian, Zhi-Qiang Wang. Positive solutions of doubly coupled multicomponent nonlinear Schrödinger systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2143-2161. doi: 10.3934/dcdss.2019138 |
[6] |
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 |
[7] |
Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure and Applied Analysis, 2021, 20 (2) : 867-884. doi: 10.3934/cpaa.2020294 |
[8] |
Guowei Dai, Rushun Tian, Zhitao Zhang. Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger Systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 1905-1927. doi: 10.3934/dcdss.2019125 |
[9] |
Claudianor O. Alves, Chao Ji. Multiple positive solutions for a Schrödinger logarithmic equation. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2671-2685. doi: 10.3934/dcds.2020145 |
[10] |
Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1857-1870. doi: 10.3934/dcdss.2020461 |
[11] |
Yang Han. On the cauchy problem for the coupled Klein Gordon Schrödinger system with rough data. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 233-242. doi: 10.3934/dcds.2005.12.233 |
[12] |
Shuangjie Peng, Huirong Pi. Spike vector solutions for some coupled nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2205-2227. doi: 10.3934/dcds.2016.36.2205 |
[13] |
Chuangye Liu, Rushun Tian. Normalized solutions for 3-coupled nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5115-5130. doi: 10.3934/cpaa.2020229 |
[14] |
Seunghyeok Kim. On vector solutions for coupled nonlinear Schrödinger equations with critical exponents. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1259-1277. doi: 10.3934/cpaa.2013.12.1259 |
[15] |
Lushun Wang, Minbo Yang, Yu Zheng. Infinitely many segregated solutions for coupled nonlinear Schrödinger systems. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 6069-6102. doi: 10.3934/dcds.2019265 |
[16] |
Zhongwei Tang. Segregated peak solutions of coupled Schrödinger systems with Neumann boundary conditions. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 5299-5323. doi: 10.3934/dcds.2014.34.5299 |
[17] |
Mohammad Ali Husaini, Chuangye Liu. Synchronized and ground-state solutions to a coupled Schrödinger system. Communications on Pure and Applied Analysis, 2022, 21 (2) : 639-667. doi: 10.3934/cpaa.2021192 |
[18] |
Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107 |
[19] |
Xiang-Dong Fang. Positive solutions for quasilinear Schrödinger equations in $\mathbb{R}^N$. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1603-1615. doi: 10.3934/cpaa.2017077 |
[20] |
Haidong Liu, Zhaoli Liu. Positive solutions of a nonlinear Schrödinger system with nonconstant potentials. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1431-1464. doi: 10.3934/dcds.2016.36.1431 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]