Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations
Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations
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DOI：
10.3934/cpaa.2012.11.1003
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发表时间：
2011-12
影响因子：
1
通讯作者：
Juncheng Wei;W. Yao
中科院分区：
文献类型：
--
作者：
Juncheng Wei;W. Yao
我们研究以下耦合非线性薛定谔方程的正解的唯一性： \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*} 其中 $N\leq3$、$\lambda_1、\lambda_2、\mu_1、\mu_2$ 是正常数，$\beta\geq 0$ 是耦合常数。我们首先证明对于足够小的 $\beta > 0$ 正解的唯一性。其次，假设$\lambda_1=\lambda_2$，当$\beta > \max\{\mu_1,\mu_2\}$时，我们证明$u_1=u_2\sqrt{\beta-\mu_1}/\sqrt{\beta-\mu_2}$，从而利用标量方程的相应结果得到正解的唯一性。最后，对于$N=1$和$\lambda_1=\lambda_2$，我们证明当$0\leq \beta\notin [\min\{\mu_1,\mu_2\},\max\{\mu_1,\mu_2\}]$时正解的唯一性，从而给出正解的完整分类。
We study the uniqueness of positive solutions of the following coupled nonlinear Schrodinger 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.