Classification of solutions for some mixed order elliptic system

In this paper, we classify the solution of following mixed-order conformally invariant system with coupled nonlinearity in $ \mathbb{R}^4 $: \begin{equation} \left\{ \begin{split} & -\Delta u(x) = u^{p_1}(x) e^{q_1v(x)}, \quad x\in \mathbb{R}^4, \\ (-\Delta)^2 v(x) u^{p_2}(x) e^{q_2v(x)}, \end{split} \right. \end{equation}\;\;\;\;\;\;(1) where 0\leq p_1 < 1 $, p_2 >0 q_1 > 0 q_2 \geq u>0 and satisfies$ \begin{align*} \int_{\mathbb{R}^4} e^{q_1v(x)} dx \infty, e^{q_2 v(x)} \infty. \end{align*} $Under additional assumptions (H1) or (H2), study asymptotic behavior solutions to (1) establish equivalent integral formula for (1). By using method moving spheres, obtain classification results