Existence of solutions for a semilinear elliptic system

This paper deals with the existence of solutions to the following system: $$\left\{\begin{array}{l} -\Delta u+u=\frac{\alpha}{\alpha+\beta}a(x)|v|^{\beta} |u|^{\alpha-2}u\quad\mbox{ in }\mathbb{R}^N\\ [0.2cm] -\Delta v+v=\frac{\beta}{\alpha+\beta}a(x)|u|^{\alpha} |v|^{\beta-2}v\quad\mbox{ in }\mathbb{R}^N. \end{array}\right. $