doi: 10.3934/dcdss.2020461

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

Received  June 2020 Revised  September 2020 Published  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.
Citation: Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 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. 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

[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. ChangC.-S. LinT.-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. DancerK. 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. 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

[11]

E. N. DancerK. 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. EsryC. H. GreeneJ. 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. MaiaE. 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. NorisH. TavaresS. 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. 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

[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. ChangC.-S. LinT.-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. DancerK. 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. 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

[11]

E. N. DancerK. 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. EsryC. H. GreeneJ. 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. MaiaE. 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. NorisH. TavaresS. 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 $\lambda > \lambda_0$
Figure 2.  structure (Ⅰ) of solutions for (1), with $0 < \lambda < \lambda_0$
Figure 3.  structure (Ⅰ) of solutions for (1), with $\lambda = \lambda_0$
Figure 4.  structure (Ⅱ) of solutions for (1), with $\lambda > \lambda_0$
Figure 5.  structure (Ⅱ) of solutions for (1), with $0 < \lambda < \lambda_0$
Figure 6.  structure (Ⅱ) of solutions for (1), with $\lambda = \lambda_0$
[1]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[2]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392

[3]

Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020294

[4]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[5]

Yan'e Wang, Nana Tian, Hua Nie. Positive solution branches of two-species competition model in open advective environments. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021006

[6]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033

[7]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

[8]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002

[9]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[10]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[11]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[12]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[13]

Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020405

[14]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[15]

Masaru Hamano, Satoshi Masaki. A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1415-1447. doi: 10.3934/dcds.2020323

[16]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011

[17]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002

[18]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115

[19]

Feimin Zhong, Jinxing Xie, Yuwei Shen. Bargaining in a multi-echelon supply chain with power structure: KS solution vs. Nash solution. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020172

[20]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (18)
  • HTML views (71)
  • Cited by (0)

Other articles
by authors

[Back to Top]