Classification of solutions to conformally invariant systems with mixed order and exponentially increasing or nonlocal nonlinearity

In this paper, without any assumption on $v$ and under extremely mild assumption $u(x)=O(|x|^{K})$ at $\infty$ for some $K\gg1$ arbitrarily large, we prove classification of solutions to the following conformally invariant system with mixed order and exponentially increasing nonlinearity in $\mathbb{R}^{2}$: \begin{equation*}\\\begin{cases} (-\Delta)^{\frac{1}{2}}u(x)=e^{pv(x)}, \qquad x\in\mathbb{R}^{2}, \\ -\Delta v(x)=u^{4}(x), \qquad x\in\mathbb{R}^{2}, \end{cases}\end{equation*} where $p\in(0,+\infty)$, $u\geq 0$ and satisfies the finite total curvature condition $\int_{\mathbb{R}^{2}}u^{4}(x)\mathrm{d}x<+\infty$. In order to show integral representation formula and crucial asymptotic property for $v$, we derive and use an $\exp^{L}+L\ln L$ inequality, which is itself of independent interest. When $p=\frac{3}{2}$, the system is closely related to single conformally invariant equations $(-\Delta)^{\frac{1}{2}}u=u^{3}$ and $-\Delta v=e^{2v}$ on $\mathbb{R}^{2}$, which have been quite extensively studied (cf. \cite{BF,C,CY,CL,CLL,CLZ} etc). We also derive classification results for nonnegative solutions to conformally invariant system with mixed order and Hartree type nonlocal nonlinearity in $\mathbb{R}^{3}$. Extensions to mixed order conformally invariant systems in $\mathbb{R}^{n}$ with general dimensions $n\geq3$ are also included.