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This paper deals with the existence of solutions to the following system: $$\left\{\begin{array}{c}-\Delta u+u=\frac{\alpha }{\alpha +\beta }{a\left(x\right)\left|v\right|}^{\beta }{\left|u\right|}^{\alpha -2}u\text{in}{ℝ}^N\hfill \\ -\Delta v+v=\frac{\beta }{\alpha +\beta }{a\left(x\right)\left|u\right|}^{\alpha }{\left|v\right|}^{\beta -2}v\text{in}{ℝ}^N.\hfill \end{array}\right.$$ −Δu+u=αα+βa(x)|v|β|u|α−2u inRN−Δv+v=βα+βa(x)|u|α|v|β−2v inRN. With the help of the Nehari manifold and the linking theorem, we prove the existence of at least two nontrivial solutions. One of them is positive. Our main tools are the concentration-compactness principle and the Ekeland’s variational principle.