Mixed order conformally invariant system with exponential growth and nonlocal nonlinear terms in critical dimensions

In this paper, under the extremely mild assumption $u(x)=O(\mathrm{\mid }x{\mathrm{\mid }}^K)$ as $\mathrm{\mid }x\mathrm{\mid }\to +\mathrm{\infty }$ for some $K\gg 1$ arbitrarily large, we classify solutions of the following mixed order conformally invariant system with exponentially increasing and nonlocal nonlinearities in ${\mathbb{R}}^n$: $$\{\begin{array}{rl}{\displaystyle (-\mathrm{\Delta }{)}^{\frac{1}{2}}u}& {\displaystyle =e^{pv}}\\ {\displaystyle (-\mathrm{\Delta }{)}^{\frac{n}{2}}v}& {\displaystyle =(\frac{1}{\mathrm{\mid }x{\mathrm{\mid }}^2}\ast u^2)u^2}\end{array}\text{in}{\mathbb{R}}^n,$$ where $n=3,4$, $p>0$, $u⩾0$, $v(x)=o(\mathrm{\mid }x{\mathrm{\mid }}^2)$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$ and $u$ satisfies the finite total mass condition. The finite total mass condition can be deduced from either $u\in L^{\frac{2n}{n-1}}({\mathbb{R}}^n)$ or $u\in {\dot{H}}^{\frac{1}{2}}({\mathbb{R}}^n)$. This system is closely related to the conformally invariant equations $(-\mathrm{\Delta }{)}^{\frac{1}{2}}u=(\frac{1}{\mathrm{\mid }x{\mathrm{\mid }}^2}\ast u^2)u$ and $(-\mathrm{\Delta }{)}^{\frac{n}{2}}u=(n-1)!e^{nu}$ in ${\mathbb{R}}^n$ with $n=3,4$, which have been quite extensively studied.