Classification of solutions for some mixed order elliptic system

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