Communications on Pure and Applied Analysis (CPAA)

Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations

Pages: 1003 - 1011, Volume 11, Issue 3, May 2012      doi:10.3934/cpaa.2012.11.1003

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Juncheng Wei - Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China (email)
Wei Yao - Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong (email)

Abstract: 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.

Keywords:  Coupled nonlinear Schrödinger equations, uniqueness.
Mathematics Subject Classification:  Primary: 35B40, 35B05; Secondary: 35J55.

Received: October 2010;      Revised: April 2011;      Available Online: December 2011.

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