    May  2012, 11(3): 1003-1011. doi: 10.3934/cpaa.2012.11.1003

## Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations

 1 Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong 2 Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong

Received  October 2010 Revised  April 2011 Published  December 2011

We study the uniqueness of positive solutions of the following coupled nonlinear Schrödinger equations: \begin{eqnarray*} \Delta u_1-\lambda_1 u_1+\mu_1u_1^3+\beta u_1u_2^2=0\quad in\quad R^N,\\ \Delta u_2-\lambda_2u_2+\mu_2u_2^3+\beta u_1^2u_2=0\quad in\quad R^N, \\ u_1>0, u_2>0, u_1, u_2 \in H^1 (R^N), \end{eqnarray*} where $N\leq3$, $\lambda_1,\lambda_2,\mu_1,\mu_2$ are positive constants and $\beta\geq 0$ is a coupling constant. We prove first the uniqueness of positive solution for sufficiently small $\beta > 0$. Secondly, assuming that $\lambda_1=\lambda_2$, we show that $u_1=u_2\sqrt{\beta-\mu_1}/\sqrt{\beta-\mu_2}$ when $\beta > \max\{\mu_1,\mu_2\}$ and thus obtain the uniqueness of positive solution using the corresponding result of scalar equation. Finally, for $N=1$ and $\lambda_1=\lambda_2$, we prove the uniqueness of positive solution when $0\leq \beta\notin [\min\{\mu_1,\mu_2\},\max\{\mu_1,\mu_2\}]$ and thus give a complete classification of positive solutions.
Citation: Juncheng Wei, Wei Yao. Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1003-1011. doi: 10.3934/cpaa.2012.11.1003
##### References:
  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.   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.   T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Part. Diff. Eqns., 19 (2006), 200-207.  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.   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.   N. Ikoma, Uniqueness of positive solutions for a nonlinear elliptic system, NoDEA, 16 (2009), 555-567. doi: 10.1007/s00030-009-0017-x.   X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Diff. Eqns., 5 (2000), 899-928.  M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbbR^n$, Arch. Rat. Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.   T. C. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n\leq 3$, Communications in Mathematical Physics, 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.   T. C. Lin and J. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Diff. Eqns., 229 (2006), 538-569. doi: 10.1016/j.jde.2005.12.011.   O. Lopes, Uniqueness of a symmetric positive solutions to an ODE system, Elect. J. Diff. Eqns., 162 (2009), 1-8.  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.   show all references

##### References:
  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.   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.   T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Part. Diff. Eqns., 19 (2006), 200-207.  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.   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.   N. Ikoma, Uniqueness of positive solutions for a nonlinear elliptic system, NoDEA, 16 (2009), 555-567. doi: 10.1007/s00030-009-0017-x.   X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Diff. Eqns., 5 (2000), 899-928.  M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbbR^n$, Arch. Rat. Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.   T. C. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n\leq 3$, Communications in Mathematical Physics, 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.   T. C. Lin and J. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Diff. Eqns., 229 (2006), 538-569. doi: 10.1016/j.jde.2005.12.011.   O. Lopes, Uniqueness of a symmetric positive solutions to an ODE system, Elect. J. Diff. Eqns., 162 (2009), 1-8.  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.   Shuangjie Peng, Huirong Pi. Spike vector solutions for some coupled nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2205-2227. doi: 10.3934/dcds.2016.36.2205  Chuangye Liu, Rushun Tian. Normalized solutions for 3-coupled nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5115-5130. doi: 10.3934/cpaa.2020229  Santosh Bhattarai. Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1789-1811. doi: 10.3934/dcds.2016.36.1789  Seunghyeok Kim. On vector solutions for coupled nonlinear Schrödinger equations with critical exponents. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1259-1277. doi: 10.3934/cpaa.2013.12.1259  M. D. Todorov, C. I. Christov. Conservative numerical scheme in complex arithmetic for coupled nonlinear Schrödinger equations. Conference Publications, 2007, 2007 (Special) : 982-992. doi: 10.3934/proc.2007.2007.982  Tai-Chia Lin, Tsung-Fang Wu. Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2911-2938. doi: 10.3934/dcds.2013.33.2911  Chuangye Liu, Zhi-Qiang Wang. Synchronization of positive solutions for coupled Schrödinger equations. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2795-2808. doi: 10.3934/dcds.2018118  Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure and Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260  Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations and Control Theory, 2022, 11 (2) : 559-581. doi: 10.3934/eect.2021013  Razvan Mosincat, Haewon Yoon. Unconditional uniqueness for the derivative nonlinear Schrödinger equation on the real line. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 47-80. doi: 10.3934/dcds.2020003  Alp Eden, Elİf Kuz. Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1803-1823. doi: 10.3934/cpaa.2009.8.1803  Alessio Pomponio, Simone Secchi. A note on coupled nonlinear Schrödinger systems under the effect of general nonlinearities. Communications on Pure and Applied Analysis, 2010, 9 (3) : 741-750. doi: 10.3934/cpaa.2010.9.741  Jiabao Su, Rushun Tian, Zhi-Qiang Wang. Positive solutions of doubly coupled multicomponent nonlinear Schrödinger systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2143-2161. doi: 10.3934/dcdss.2019138  Chuangye Liu, Zhi-Qiang Wang. A complete classification of ground-states for a coupled nonlinear Schrödinger system. Communications on Pure and Applied Analysis, 2017, 16 (1) : 115-130. doi: 10.3934/cpaa.2017005  Lushun Wang, Minbo Yang, Yu Zheng. Infinitely many segregated solutions for coupled nonlinear Schrödinger systems. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 6069-6102. doi: 10.3934/dcds.2019265  W. Josh Sonnier, C. I. Christov. Repelling soliton collisions in coupled Schrödinger equations with negative cross modulation. Conference Publications, 2009, 2009 (Special) : 708-718. doi: 10.3934/proc.2009.2009.708  Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure and Applied Analysis, 2021, 20 (2) : 867-884. doi: 10.3934/cpaa.2020294  Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations and Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337  Chenjie Fan, Zehua Zhao. Decay estimates for nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3973-3984. doi: 10.3934/dcds.2021024  Alexander Pankov. Nonlinear Schrödinger Equations on Periodic Metric Graphs. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 697-714. doi: 10.3934/dcds.2018030

2020 Impact Factor: 1.916