# American Institute of Mathematical Sciences

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 & Applied Analysis, 2012, 11 (3) : 1003-1011. doi: 10.3934/cpaa.2012.11.1003
##### 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.  doi: 10.1016/j.crma.2006.01.024.  Google Scholar [2] J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space,, J. Differential Equations, 163 (2000), 41.  doi: 10.1006/jdeq.1999.3701.  Google Scholar [3] T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems,, J. Part. Diff. Eqns., 19 (2006), 200.   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.  doi: 10.1007/s11784-007-0033-6.  Google Scholar [5] E. N. Dancer and J. Wei, Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction,, Trans. Amer. Math. Soc., 361 (2009), 1189.  doi: 10.1090/S0002-9947-08-04735-1.  Google Scholar [6] N. Ikoma, Uniqueness of positive solutions for a nonlinear elliptic system,, NoDEA, 16 (2009), 555.  doi: 10.1007/s00030-009-0017-x.  Google Scholar [7] X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations,, Adv. Diff. Eqns., 5 (2000), 899.   Google Scholar [8] M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbbR^n$,, Arch. Rat. Mech. Anal., 105 (1989), 243.  doi: 10.1007/BF00251502.  Google Scholar [9] 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.  doi: 10.1007/s00220-005-1313-x.  Google Scholar [10] T. C. Lin and J. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials,, J. Diff. Eqns., 229 (2006), 538.  doi: 10.1016/j.jde.2005.12.011.  Google Scholar [11] O. Lopes, Uniqueness of a symmetric positive solutions to an ODE system,, Elect. J. Diff. Eqns., 162 (2009), 1.   Google Scholar [12] B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations,, Comm. Math. Physics, 271 (2007), 199.  doi: 10.1007/s00220-006-0179-x.  Google Scholar

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##### 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.  doi: 10.1016/j.crma.2006.01.024.  Google Scholar [2] J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space,, J. Differential Equations, 163 (2000), 41.  doi: 10.1006/jdeq.1999.3701.  Google Scholar [3] T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems,, J. Part. Diff. Eqns., 19 (2006), 200.   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.  doi: 10.1007/s11784-007-0033-6.  Google Scholar [5] E. N. Dancer and J. Wei, Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction,, Trans. Amer. Math. Soc., 361 (2009), 1189.  doi: 10.1090/S0002-9947-08-04735-1.  Google Scholar [6] N. Ikoma, Uniqueness of positive solutions for a nonlinear elliptic system,, NoDEA, 16 (2009), 555.  doi: 10.1007/s00030-009-0017-x.  Google Scholar [7] X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations,, Adv. Diff. Eqns., 5 (2000), 899.   Google Scholar [8] M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbbR^n$,, Arch. Rat. Mech. Anal., 105 (1989), 243.  doi: 10.1007/BF00251502.  Google Scholar [9] 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.  doi: 10.1007/s00220-005-1313-x.  Google Scholar [10] T. C. Lin and J. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials,, J. Diff. Eqns., 229 (2006), 538.  doi: 10.1016/j.jde.2005.12.011.  Google Scholar [11] O. Lopes, Uniqueness of a symmetric positive solutions to an ODE system,, Elect. J. Diff. Eqns., 162 (2009), 1.   Google Scholar [12] B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations,, Comm. Math. Physics, 271 (2007), 199.  doi: 10.1007/s00220-006-0179-x.  Google Scholar
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