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Global attractor for a one dimensional weakly damped half-wave equation
Structure of positive solutions to a class of Schrödinger systems
1. | School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China |
2. | Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China |
$ \begin{equation*} \begin{cases} &- \varDelta u = \lambda u - \mu_1 u^3 + \beta_1 uv^2,\quad \rm {in}\ \Omega,\\ &- \varDelta v = \lambda v - \mu_2 v^3 + \beta_2 u^2v,\quad \rm {in}\ \Omega,\\ &u, v > 0\quad \rm {in}\ \Omega,\quad u, v = 0\quad \rm {on}\ \partial \Omega, \end{cases} \end{equation*} $ |
$ \lambda $ |
$ \beta_1, \beta_2 $ |
$ \Omega \subset \mathbb{R}^N $ |
$ (N \geqslant 1) $ |
$ \lambda > 0 $ |
$ \mu_1 \leqslant \mu_2 $ |
$ \beta_1 \beta_2 $ |
$ \mu_1 $ |
$ \mu_2 $ |
References:
[1] |
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. |
[2] |
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. |
[3] |
T. Bartsch and Z.-Q. Wang,
Note on ground states of nonlinear Schrödinger systems, J. Part. Differ. Equ., 19 (2006), 200-207.
|
[4] |
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. |
[5] |
H. Berestycki,
Le nombre de solutions de certains problèmes semi-linéaires elliptiques, J. Funct. Anal., 40 (1981), 1-29.
doi: 10.1016/0022-1236(81)90069-0. |
[6] |
H. Brézis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[7] |
S.-M. Chang, C.-S. Lin, T.-C. Lin and W.-W. Lin,
Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates, Phys. D, 196 (2004), 341-361.
doi: 10.1016/j.physd.2004.06.002. |
[8] |
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. |
[9] |
E. N. Dancer, K. Wang and Z. Zhang,
Uniform Hölder estiamte for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species, J. Differ. Equ., 251 (2011), 2737-2769.
doi: 10.1016/j.jde.2011.06.015. |
[10] |
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. |
[11] |
E. N. Dancer, K. Wang and Z. Zhang,
Addendum to "The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture", J. Funct. Anal., 264 (2013), 1125-1129.
doi: 10.1016/j.jfa.2012.10.009. |
[12] |
B. D. Esry, C. H. Greene, J. P. Jr. Burke and J. L. Bohn,
Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.
doi: 10.1103/PhysRevLett.78.3594. |
[13] |
D. Gilberg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001. |
[14] |
Z. Guo and J. Wei,
Global solution branch and Morse index estimates of a semilinear elliptic equation with super-critical exponent, Trans. Am. Math. Soc., 363 (2011), 4777-4799.
doi: 10.1090/S0002-9947-2011-05292-X. |
[15] |
M. K. Kwong,
Uniqueness of positive solutions of $\Delta u - u + u^p = 0$ in $\mathbb{R}^N$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[16] |
P. L. Lions,
On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.
doi: 10.1137/1024101. |
[17] |
T.-C. Lin and J. Wei,
Ground state of $N$ coupled nonlinear Schrödinger equations in $\mathbb{R}^n$, $n \leqslant 3$, Commun. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[18] |
T.-C. Lin and J. Wei,
Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations, Phys. D, 220 (2006), 99-115.
doi: 10.1016/j.physd.2006.07.009. |
[19] |
T.-C. Lin and J. Wei,
Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differ. Equ., 229 (2006), 538-569.
doi: 10.1016/j.jde.2005.12.011. |
[20] |
Z. Liu and Z.-Q. Wang,
Multiple bound states of nonlinear Schrödinger systems, Commun. Math. Phys., 282 (2008), 721-731.
doi: 10.1007/s00220-008-0546-x. |
[21] |
Z. Liu and Z.-Q. Wang,
Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud., 10 (2010), 175-193.
doi: 10.1515/ans-2010-0109. |
[22] |
W. Long and S. Peng,
Segregated vector solutions for a class of Bose-Einstein systems, J. Differ. Equ., 257 (2014), 207-230.
doi: 10.1016/j.jde.2014.03.019. |
[23] |
L. A. Maia, E. Montefusco and B. Pellacci,
Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differ. Equ., 299 (2006), 743-767.
doi: 10.1016/j.jde.2006.07.002. |
[24] |
R. Mandel,
Minimal energy solutions for cooperative nonlinear Schrödinger systems, Nonlinear Differ. Equ. Appl., 22 (2015), 239-262.
doi: 10.1007/s00030-014-0281-2. |
[25] |
B. Noris, H. Tavares, S. Terracini and G. Verzini,
Convergence of minimax and continuation of critical points for singularly perturbed systems, J. Eur. Math. Soc., 14 (2012), 1245-1273.
doi: 10.4171/JEMS/332. |
[26] |
A. S. Parkins and D. F. Walls,
The physics of trapped dilute-gas Bose-Einstein condensates, Phys. Rep., 303 (1998), 1-80.
doi: 10.1016/S0370-1573(98)00014-3. |
[27] |
S. Peng and Z.-Q. Wang,
Segregated and synchronized vector solutions for nonlinear Schröinger systems, Arch. Ration. Mech. Anal., 208 (2013), 305-339.
doi: 10.1007/s00205-012-0598-0. |
[28] |
B. Sirakov,
Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^n$, Commun. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[29] |
P. N. Srikanth,
Uniqueness of solutions of nonlinear Dirichlet problems, Differ. Integral Equ., 6 (1993), 663-670.
|
[30] |
R. Tian and Z.-Q. Wang,
Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.
|
[31] |
R. Tian and Z.-Q. Wang,
Bifurcation results on positive solutions of an indefinite nonlinear elliptic system, Discr. Continu. Dynamic Syst. Ser. A, 33 (2013), 335-344.
doi: 10.3934/dcds.2013.33.335. |
[32] |
R. Tian and Z.-Q. Wang,
Bifurcation results on positive solutions for an indefinite nonlinear elliptic system Ⅱ, Adv. Nonlinear Stud., 13 (2013), 245-262.
doi: 10.1515/ans-2013-0115. |
[33] |
K. Wang and Z. Zhang, Some new results in competing systems with many species,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 739–761.
doi: 10.1016/j.anihpc.2009.11.004. |
[34] |
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. |
[35] |
L. Zhang,
Uniqueness of positive solutions of $\Delta u + u^p + u = 0$ in a finite ball, Commun. Part. Differ. Equ., 17 (1992), 1141-1164.
doi: 10.1080/03605309208820880. |
[36] |
Z. Zhang and W. Wang,
Structure of positive solutions to a schrodinger system, J. Fixed Point Theory Appl., 19 (2017), 877-887.
doi: 10.1007/s11784-016-0383-z. |
show all references
References:
[1] |
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. |
[2] |
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. |
[3] |
T. Bartsch and Z.-Q. Wang,
Note on ground states of nonlinear Schrödinger systems, J. Part. Differ. Equ., 19 (2006), 200-207.
|
[4] |
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. |
[5] |
H. Berestycki,
Le nombre de solutions de certains problèmes semi-linéaires elliptiques, J. Funct. Anal., 40 (1981), 1-29.
doi: 10.1016/0022-1236(81)90069-0. |
[6] |
H. Brézis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[7] |
S.-M. Chang, C.-S. Lin, T.-C. Lin and W.-W. Lin,
Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates, Phys. D, 196 (2004), 341-361.
doi: 10.1016/j.physd.2004.06.002. |
[8] |
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. |
[9] |
E. N. Dancer, K. Wang and Z. Zhang,
Uniform Hölder estiamte for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species, J. Differ. Equ., 251 (2011), 2737-2769.
doi: 10.1016/j.jde.2011.06.015. |
[10] |
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. |
[11] |
E. N. Dancer, K. Wang and Z. Zhang,
Addendum to "The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture", J. Funct. Anal., 264 (2013), 1125-1129.
doi: 10.1016/j.jfa.2012.10.009. |
[12] |
B. D. Esry, C. H. Greene, J. P. Jr. Burke and J. L. Bohn,
Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.
doi: 10.1103/PhysRevLett.78.3594. |
[13] |
D. Gilberg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001. |
[14] |
Z. Guo and J. Wei,
Global solution branch and Morse index estimates of a semilinear elliptic equation with super-critical exponent, Trans. Am. Math. Soc., 363 (2011), 4777-4799.
doi: 10.1090/S0002-9947-2011-05292-X. |
[15] |
M. K. Kwong,
Uniqueness of positive solutions of $\Delta u - u + u^p = 0$ in $\mathbb{R}^N$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[16] |
P. L. Lions,
On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.
doi: 10.1137/1024101. |
[17] |
T.-C. Lin and J. Wei,
Ground state of $N$ coupled nonlinear Schrödinger equations in $\mathbb{R}^n$, $n \leqslant 3$, Commun. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[18] |
T.-C. Lin and J. Wei,
Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations, Phys. D, 220 (2006), 99-115.
doi: 10.1016/j.physd.2006.07.009. |
[19] |
T.-C. Lin and J. Wei,
Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differ. Equ., 229 (2006), 538-569.
doi: 10.1016/j.jde.2005.12.011. |
[20] |
Z. Liu and Z.-Q. Wang,
Multiple bound states of nonlinear Schrödinger systems, Commun. Math. Phys., 282 (2008), 721-731.
doi: 10.1007/s00220-008-0546-x. |
[21] |
Z. Liu and Z.-Q. Wang,
Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud., 10 (2010), 175-193.
doi: 10.1515/ans-2010-0109. |
[22] |
W. Long and S. Peng,
Segregated vector solutions for a class of Bose-Einstein systems, J. Differ. Equ., 257 (2014), 207-230.
doi: 10.1016/j.jde.2014.03.019. |
[23] |
L. A. Maia, E. Montefusco and B. Pellacci,
Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differ. Equ., 299 (2006), 743-767.
doi: 10.1016/j.jde.2006.07.002. |
[24] |
R. Mandel,
Minimal energy solutions for cooperative nonlinear Schrödinger systems, Nonlinear Differ. Equ. Appl., 22 (2015), 239-262.
doi: 10.1007/s00030-014-0281-2. |
[25] |
B. Noris, H. Tavares, S. Terracini and G. Verzini,
Convergence of minimax and continuation of critical points for singularly perturbed systems, J. Eur. Math. Soc., 14 (2012), 1245-1273.
doi: 10.4171/JEMS/332. |
[26] |
A. S. Parkins and D. F. Walls,
The physics of trapped dilute-gas Bose-Einstein condensates, Phys. Rep., 303 (1998), 1-80.
doi: 10.1016/S0370-1573(98)00014-3. |
[27] |
S. Peng and Z.-Q. Wang,
Segregated and synchronized vector solutions for nonlinear Schröinger systems, Arch. Ration. Mech. Anal., 208 (2013), 305-339.
doi: 10.1007/s00205-012-0598-0. |
[28] |
B. Sirakov,
Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^n$, Commun. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[29] |
P. N. Srikanth,
Uniqueness of solutions of nonlinear Dirichlet problems, Differ. Integral Equ., 6 (1993), 663-670.
|
[30] |
R. Tian and Z.-Q. Wang,
Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.
|
[31] |
R. Tian and Z.-Q. Wang,
Bifurcation results on positive solutions of an indefinite nonlinear elliptic system, Discr. Continu. Dynamic Syst. Ser. A, 33 (2013), 335-344.
doi: 10.3934/dcds.2013.33.335. |
[32] |
R. Tian and Z.-Q. Wang,
Bifurcation results on positive solutions for an indefinite nonlinear elliptic system Ⅱ, Adv. Nonlinear Stud., 13 (2013), 245-262.
doi: 10.1515/ans-2013-0115. |
[33] |
K. Wang and Z. Zhang, Some new results in competing systems with many species,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 739–761.
doi: 10.1016/j.anihpc.2009.11.004. |
[34] |
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. |
[35] |
L. Zhang,
Uniqueness of positive solutions of $\Delta u + u^p + u = 0$ in a finite ball, Commun. Part. Differ. Equ., 17 (1992), 1141-1164.
doi: 10.1080/03605309208820880. |
[36] |
Z. Zhang and W. Wang,
Structure of positive solutions to a schrodinger system, J. Fixed Point Theory Appl., 19 (2017), 877-887.
doi: 10.1007/s11784-016-0383-z. |






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