November  2019, 12(7): 1905-1927. doi: 10.3934/dcdss.2019125

Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger Systems

1. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

2. 

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

3. 

HLM, CEMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

4. 

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author: Zhitao Zhang

Received  September 2017 Revised  April 2018 Published  December 2018

Fund Project: Dedicated to the 70th birthday of Professor E. N. Dancer. Research supported by NSFC(11771428, 11871129, 11601353, 11325107) and Beijing Natural Science Foundation(1174013).

In this paper, we consider the following coupled elliptic system
$ \begin{equation} \left\{ \begin{array}{ll} -\Delta u+\lambda_1 u = \mu_1 u^3+\beta uv^2-\gamma v &\text{in } \mathbb{R}^N, \\ -\Delta v+\lambda_2 v = \mu_2 v^3+\beta vu^2-\gamma u &\text{in } \mathbb{R}^N, \\ u(x), v(x)\rightarrow 0 \text{ as } \vert x\vert\rightarrow+\infty. \end{array} \right.\nonumber \end{equation} $
Under symmetric assumptions
$ \lambda_1 = \lambda_2, \mu_1 = \mu_2 $
, we determine the number of
$ \gamma $
-bifurcations for each
$ \beta\in(-1, +\infty) $
, and study the behavior of global
$ \gamma $
-bifurcation branches in
$ [-1, 0]\times H_r^1\left( \mathbb{R} ^N\right)\times H_r^1\left( \mathbb{R} ^N\right) $
. Moreover, several results for
$ \gamma = 0 $
, such as priori bounds, are of independent interests, which are improvements of corresponding theorems in [6] and [35].
Citation: Guowei Dai, Rushun Tian, Zhitao Zhang. Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger Systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1905-1927. doi: 10.3934/dcdss.2019125
References:
[1]

N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 1-4.   Google Scholar

[2]

J. C. Alexander and S. S. Antman, Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems, Arch. Ration. Mech. Anal., 76 (1981), 339-354.  doi: 10.1007/BF00249970.  Google Scholar

[3]

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

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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

[5]

A. AmbrosettiE. Colorado and D. Ruiz, Standing waves multi-bump solitions to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var., 30 (2007), 85-112.  doi: 10.1007/s00526-006-0079-0.  Google Scholar

[6]

T. BartschE. N. Dancer and Z. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

[7]

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

[8]

T. BartschZ.-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

[9]

J. Belmonte-BeitiaV. M. Prez-Garca and P. J. Torres, Solitary waves for linearly coupled nonlinear Schrödinger equations with inhomogeneous coefficients, J. Nonlinear Sci., 19 (2009), 437-451.  doi: 10.1007/s00332-008-9037-7.  Google Scholar

[10]

J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56.  doi: 10.1006/jdeq.1999.3701.  Google Scholar

[11]

Z. Chen and W. Zou, An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var., 48 (2013), 695-711.  doi: 10.1007/s00526-012-0568-2.  Google Scholar

[12]

E. N. Dancer, Boundary-value problems for ordinary differential equations on infinite intervals, Proc. Lond. Math. Soc., 30 (1975), 76-94.  doi: 10.1112/plms/s3-30.1.76.  Google Scholar

[13]

E. N. DancerK. 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

[14]

E. N. DancerK. Wang and Z. 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.  Google Scholar

[15]

E. N. Dancer and J. Wei, Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction, Trans. Amer. Math. Soc., 361 (2009), 1189-1208.  doi: 10.1090/S0002-9947-08-04735-1.  Google Scholar

[16]

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

[17]

B. DeconinckP. G. KevrekidisH. 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), 705-706.   Google Scholar

[18]

N. Dunford and J. T. Schwartz, Linear Operators. Part II. Spectral Theory. Selfadjoint Operators in Hilbert Space, Wiley, New York, 1988.  Google Scholar

[19]

B. D. EsryC. H. GreeneJ. P. Burke Jr and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.   Google Scholar

[20]

P. M. FitzpatrickI. Massabò and J. Pejsachowicz, Global several-parameter bifurcation and continuation theorems: a unified approach via completmenting maps, Math. Ann., 263 (1983), 61-73.  doi: 10.1007/BF01457084.  Google Scholar

[21]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar

[22]

N. Ikoma, Uniqueness of positive solutions for a nonlinear elliptic system, Nonlinear Differ. Equ. Appl., 16 (2009), 555-567.  doi: 10.1007/s00030-009-0017-x.  Google Scholar

[23]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u + u + u^p$ in $ \mathbb{R} ^N$, Arch. Rat. Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

[24]

K. Li and Z. Zhang, Existence of solutions for a Schrödinger system with linear and nonlinear couplings, J. Math. Phys., 57 (2016), 081504, 17 pp. doi: 10.1063/1.4960046.  Google Scholar

[25]

T.-C. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbb{R}^n$, n ≤ 3, Comm. Math. Phys., 255 (2005), 629-653.  doi: 10.1007/s00220-005-1313-x.  Google Scholar

[26]

T.-C. Lin and J. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differential Equations, 229 (2006), 538-569.  doi: 10.1016/j.jde.2005.12.011.  Google Scholar

[27]

M. MitchellZ. ChenM. Shih and M. Segev, Self-trapping of partially incoherent light, Phys. Rev. Lett., 77 (1996), 490-493.   Google Scholar

[28]

A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258-281.  doi: 10.1016/j.jde.2005.09.002.  Google Scholar

[29]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[30]

C. RüeggN. CavadiniA. FurrerH.-U. GdelK. KrmerH. MutkaA. WildesK. Habicht and P. Vorderwisch, Bose-Einstein condensation of the triplet states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65.   Google Scholar

[31]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations, Comm. Math. Physics, 271 (2007), 199-221.  doi: 10.1007/s00220-006-0179-x.  Google Scholar

[32]

W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar

[33]

R. Tian and Z. Zhang, Existence and bifurcation of solutions for a double coupled system of Schrödinger equation, Sci. China Math., 58 (2015), 1607-1620.  doi: 10.1007/s11425-015-5028-y.  Google Scholar

[34]

E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-721.   Google Scholar

[35]

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

show all references

References:
[1]

N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 1-4.   Google Scholar

[2]

J. C. Alexander and S. S. Antman, Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems, Arch. Ration. Mech. Anal., 76 (1981), 339-354.  doi: 10.1007/BF00249970.  Google Scholar

[3]

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

[4]

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

[5]

A. AmbrosettiE. Colorado and D. Ruiz, Standing waves multi-bump solitions to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var., 30 (2007), 85-112.  doi: 10.1007/s00526-006-0079-0.  Google Scholar

[6]

T. BartschE. N. Dancer and Z. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

[7]

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

[8]

T. BartschZ.-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

[9]

J. Belmonte-BeitiaV. M. Prez-Garca and P. J. Torres, Solitary waves for linearly coupled nonlinear Schrödinger equations with inhomogeneous coefficients, J. Nonlinear Sci., 19 (2009), 437-451.  doi: 10.1007/s00332-008-9037-7.  Google Scholar

[10]

J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56.  doi: 10.1006/jdeq.1999.3701.  Google Scholar

[11]

Z. Chen and W. Zou, An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var., 48 (2013), 695-711.  doi: 10.1007/s00526-012-0568-2.  Google Scholar

[12]

E. N. Dancer, Boundary-value problems for ordinary differential equations on infinite intervals, Proc. Lond. Math. Soc., 30 (1975), 76-94.  doi: 10.1112/plms/s3-30.1.76.  Google Scholar

[13]

E. N. DancerK. 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

[14]

E. N. DancerK. Wang and Z. 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.  Google Scholar

[15]

E. N. Dancer and J. Wei, Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction, Trans. Amer. Math. Soc., 361 (2009), 1189-1208.  doi: 10.1090/S0002-9947-08-04735-1.  Google Scholar

[16]

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

[17]

B. DeconinckP. G. KevrekidisH. 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), 705-706.   Google Scholar

[18]

N. Dunford and J. T. Schwartz, Linear Operators. Part II. Spectral Theory. Selfadjoint Operators in Hilbert Space, Wiley, New York, 1988.  Google Scholar

[19]

B. D. EsryC. H. GreeneJ. P. Burke Jr and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.   Google Scholar

[20]

P. M. FitzpatrickI. Massabò and J. Pejsachowicz, Global several-parameter bifurcation and continuation theorems: a unified approach via completmenting maps, Math. Ann., 263 (1983), 61-73.  doi: 10.1007/BF01457084.  Google Scholar

[21]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar

[22]

N. Ikoma, Uniqueness of positive solutions for a nonlinear elliptic system, Nonlinear Differ. Equ. Appl., 16 (2009), 555-567.  doi: 10.1007/s00030-009-0017-x.  Google Scholar

[23]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u + u + u^p$ in $ \mathbb{R} ^N$, Arch. Rat. Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

[24]

K. Li and Z. Zhang, Existence of solutions for a Schrödinger system with linear and nonlinear couplings, J. Math. Phys., 57 (2016), 081504, 17 pp. doi: 10.1063/1.4960046.  Google Scholar

[25]

T.-C. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbb{R}^n$, n ≤ 3, Comm. Math. Phys., 255 (2005), 629-653.  doi: 10.1007/s00220-005-1313-x.  Google Scholar

[26]

T.-C. Lin and J. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differential Equations, 229 (2006), 538-569.  doi: 10.1016/j.jde.2005.12.011.  Google Scholar

[27]

M. MitchellZ. ChenM. Shih and M. Segev, Self-trapping of partially incoherent light, Phys. Rev. Lett., 77 (1996), 490-493.   Google Scholar

[28]

A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258-281.  doi: 10.1016/j.jde.2005.09.002.  Google Scholar

[29]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[30]

C. RüeggN. CavadiniA. FurrerH.-U. GdelK. KrmerH. MutkaA. WildesK. Habicht and P. Vorderwisch, Bose-Einstein condensation of the triplet states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65.   Google Scholar

[31]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations, Comm. Math. Physics, 271 (2007), 199-221.  doi: 10.1007/s00220-006-0179-x.  Google Scholar

[32]

W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar

[33]

R. Tian and Z. Zhang, Existence and bifurcation of solutions for a double coupled system of Schrödinger equation, Sci. China Math., 58 (2015), 1607-1620.  doi: 10.1007/s11425-015-5028-y.  Google Scholar

[34]

E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-721.   Google Scholar

[35]

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

Figure 1.  Schematic diagrams of global $ \gamma $-bifurcations for $ \beta\in[\beta_{k+1}, \beta_k) $
Figure 2.  Schematic diagrams of β-bifurcation branches
Figure 3.  Global bifurcation diagram in the symmetric case µ1 = µ2 = 1
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