Structure of positive solutions to a class of Schrödinger systems

Xiyou Cheng; Zhitao Zhang

doi:10.3934/dcdss.2020461

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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 and Continuous Dynamical Systems - S, 2021, 14 (6) : 1857-1870. doi: 10.3934/dcdss.2020461

References:

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[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.

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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