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June  2021, 14(6): 1857-1870. doi: 10.3934/dcdss.2020461

Structure of positive solutions to a class of Schrödinger systems

Xiyou Cheng 1, and  Zhitao Zhang 2,

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

Received  June 2020 Revised  September 2020 Published  June 2021 Early access  November 2020

Fund Project: Supported in part by the NSFC (11771428, 11926335)

This paper is devoted to dealing with the existence and uniqueness of positive solutions for the following coupled nonlinear Schrödinger systems with multi-parameters
$ \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*} $
on the range of
$ \lambda $
 and the different coupling constants
$ \beta_1, \beta_2 $
, where
$ \Omega \subset \mathbb{R}^N $
$ (N \geqslant 1) $
 is a bounded smooth domain,
$ \lambda > 0 $
 and
$ \mu_1 \leqslant \mu_2 $
. Under some conditions, we establish some interesting positive solutions structure theorems in the
$ \beta_1 \beta_2 $
-plane, especially we obtain the new structure theorems for the cases that
$ \mu_1 $
 and
$ \mu_2 $
 have different signs or they are negative. In addition, we get interesting uniqueness results via synchronized solutions techniques.
Keywords: Schrödinger system, Positive solution, Existence and uniqueness, Synchronized solution.
Mathematics Subject Classification: 35J10, 35J57, 35J25.
Citation: Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1857-1870. doi: 10.3934/dcdss.2020461
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.  Google Scholar

[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.  Google Scholar

[3]

T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Part. Differ. Equ., 19 (2006), 200-207.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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D. Gilberg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[16]

P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.  doi: 10.1137/1024101.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

P. N. Srikanth, Uniqueness of solutions of nonlinear Dirichlet problems, Differ. Integral Equ., 6 (1993), 663-670.   Google Scholar

[30]

R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Part. Differ. Equ., 19 (2006), 200-207.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

D. Gilberg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.  doi: 10.1137/1024101.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

P. N. Srikanth, Uniqueness of solutions of nonlinear Dirichlet problems, Differ. Integral Equ., 6 (1993), 663-670.   Google Scholar

[30]

R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  structure (Ⅰ) of solutions for (1), with $\lambda > \lambda_0$
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Figure 2.  structure (Ⅰ) of solutions for (1), with $0 < \lambda < \lambda_0$
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Figure 3.  structure (Ⅰ) of solutions for (1), with $\lambda = \lambda_0$
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Figure 4.  structure (Ⅱ) of solutions for (1), with $\lambda > \lambda_0$
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Figure 5.  structure (Ⅱ) of solutions for (1), with $0 < \lambda < \lambda_0$
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Figure 6.  structure (Ⅱ) of solutions for (1), with $\lambda = \lambda_0$
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