Radial stationary solutions to a class of wave system as well as their asymptotical behavior

For a stationary version to a class of wave system $ \left\{ \begin{aligned} & -\left(a_1 + b_1\int_{\mathbb{R}^3} {|\nabla u|^2} dx + c\int_{\mathbb{R}^3} {|\nabla v|^2} dx\right) \Delta u + u = \frac{p}{Q} |u|^{p-2}u|v|^q, \\ & -\left(a_2 + b_2\int_{\mathbb{R}^3} {|\nabla v|^2} dx +c\int_{\mathbb{R}^3} {|\nabla u|^2} dx\right) \Delta v + v = \frac{q}{Q} |u|^p |v|^{q-2}v, \end{aligned} \right. $ $u, v \in H_r^1 (\mathbb{R}^3)$, by establishing a variant variational identity and constraint set, we prove that for $a_s \gt 0$, $b_s \gt 0$, $(s = 1, \ 2)$, $c\geq 0$ and $p \gt 1$, $q \gt 1$ with $Q: = p+q \in (2, 6)$, the system admits a positive radially symmetric ground state solution in $H_r^1 (\mathbb{R}^3)\times H_r^1 (\mathbb{R}^3)$. Moreover, for any fixed $a_1 \gt 0$ and $a_2 \gt 0$, as $b_1^2 + b_2^2 + c^2 \to 0$, this solution converges to a positive radially symmetric solution to $ - a_1 \Delta u + u = \frac{p}{Q} |u|^{p-2}u|v|^q, \quad - a_2 \Delta v + v = \frac{q}{Q} |u|^p |v|^{q-2}v,\quad u,\ v\in H_r^1 (\mathbb{R}^3). $