Coupled Elliptic systems with sublinear growth

Consider the coupled elliptic system $\left(\begin{array}{ccc}-\Delta u+u={\rho }_1\left(x\right)u^{p_1}+\lambda v& \text{in}& R^N\\ -\Delta v+v={\rho }_2\left(x\right)v^{p_2}+\lambda u& \text{in}& R^N,\\ u\left(x\right),v\left(x\right)\to 0& \text{as}& \left|x\right|\to \infty .\end{array}\right)$We observe that in 2008, A. Ambrosetti, G. Cerami and D. Ruiz proved the existence of positive bound and ground states in the case $\lambda \in (0,1)$, $p_1=p=p_2$, $1<p<2^{\ast }-1,$ ${\rho }_1\left(x\right)$ and ${\rho }_2\left(x\right)$ tends to one at infinity. In this work we complement their result, because we show that the previous system has no solutions when $0<p_1,p_2<1$, as well as we establish sharp hypotheses on the powers $0<p_1,p_2$ the parameter $\lambda$ and the weights ${\rho }_1\left(x\right)$, ${\rho }_2\left(x\right)$ that will allow us to obtain the existence and uniqueness of a positive bounded solution.