Uniqueness of positive solutions for a class of Schrodinger systems with saturable nonlinearity

Abstract

This paper is devoted to the study of the nonexistence and the uniqueness of positive solutions for a class of the following nonlinear coupled Schrodinger systems with saturable nonlinearity \begin{equation} \begin{cases} -\Delta u_{1}+\lambda _{1}u_{1}=\dfrac {u_{1}(\mu _1 u_{1}^2+\beta u_{2}^2)}{1+s(\mu _1 u_{1}^2+\beta u_{2}^2)} &\mbox {in } \mathbb {R}^N, \\-\Delta u_{2}+\lambda _{2}u_{2}=\dfrac {u_{2}(\mu _2 u_{2}^2+\beta u_{1}^2)}{1+s(\mu _2 u_{2}^2+\beta u_{1}^2)} &\mbox {in } \mathbb {R}^N, \\ u_{1}, u_{2}\in H^1(\mathbb {R}^N),\ u_{1}>0,\ u_{2}>0 &\mbox {in } \mathbb {R}^N, \end{cases} \end{equation} where $\lambda _{j}, \mu _{j}, j=1, 2$, are positive constants, $s$ is a positive parameter and $\beta $ is a positive coupling parameter. Moreover, we will show that any positive solution is a priori bounded.