In this paper, we are concerned with the following Schrödinger system
\(\begin{cases}-\Delta u_1 = \lambda_1 |u_1|^{p(r)-2} u_1 + \beta |u_2|^{\frac{p(r)}{2}} |u_1|^{\frac{p(r)}{2}-2} u_1, & x \in \mathbb{R}^N, \\ -\Delta u_2 = \lambda_2 |u_2|^{p(r)-2} u_2 + \beta |u_1|^{\frac{p(r)}{2}} |u_2|^{\frac{p(r)}{2}-2} u_2, & x \in \mathbb{R}^N,\end{cases}\)
where N ≥ 3, λ1, λ2, β > 0, \(p(r)={2N\over N-2}+f(r)\) with f ∈ C([0, +∞), [0, +∞)). Under some suitable assumptions on f, we prove that the system admits a positive solution if β > 0 for N ≥ 5 or β ∈ (0, β0] ∪ [β1, +∞) for N = 3,4, where 0 < β0 < β1 are some constants. In particular, the system is slightly supercritical when f ≢ 0. Delicate analysis near the origin and infinity will be involved.
