In this paper, without any assumption on v and under the extremely mild assumption \(u(x)= O(|x|^{K})\) as \(|x|\rightarrow +\infty\) for some \(K\gg 1\) arbitrarily large, we classify solutions of the following conformally invariant system with mixed order and exponentially increasing nonlinearity in \(\mathbb {R}^{3}\) : \({\left\{ \begin{array}{ll} \ (-\Delta )^{\frac{1}{2}} u=v^{4} ,& x\in \mathbb {R}^{3},\\ \ -\Delta v=e^{pw} ,& x\in \mathbb {R}^{3},\\ \ (-\Delta )^{\frac{3}{2}} w=u^{3} ,& x\in \mathbb {R}^{3}, \end{array}\right. }\) where \(p>0\) , \(u,v\ge 0\) , \(w(x)=o(|x|^{2})\) at \(\infty\) and u satisfies the finite total curvature condition \(\int _{\mathbb {R}^{3}}u^{3}(x)\textrm{d}x<+\infty\) . Moreover, under extremely mild assumption which is much weaker than \(v(x)=O(|x|^{K})\) as \(|x|\rightarrow +\infty\) for some \(K\gg 1\) arbitrarily large, we also prove classification of solutions to the conformally invariant system with mixed order and exponentially increasing nonlinearity in \(\mathbb {R}^{4}\) : \(\begin{aligned} {\left\{ \begin{array}{ll} \ (-\Delta )^{\frac{1}{2}} u=e^{pw} ,& x\in \mathbb {R}^{4},\\ \ -\Delta v=u^2 ,& x\in \mathbb {R}^{4},\\ \ (-\Delta )^{2} w=v^{4} ,& x\in \mathbb {R}^{4}, \end{array}\right. } \end{aligned}\) where \(p>0\) , \(u,v\ge 0\) , \(w(x)=o(|x|^{2})\) at \(\infty\) and v satisfies the finite total curvature condition \(\int _{\mathbb {R}^{4}}v^{4}(x)\textrm{d}x<+\infty\) . The key ingredients are deriving the integral representation formulae and crucial asymptotic behaviors of solutions (u, v, w) and calculating the explicit value of the total curvature. These systems are closely related to conformally invariant equations \((-\Delta )^{\frac{1}{2}}u=u^{\frac{n+1}{n-1}}\) , \(-\Delta v=v^{\frac{n+2}{n-2}}\) in \(\mathbb {R}^{n}\) with \(n=3,4\) , \((-\Delta )^{\frac{3}{2}}w=2e^{3w}\) in \(\mathbb {R}^{3}\) and \((-\Delta )^{2}w=6e^{4w}\) in \(\mathbb {R}^{4}\) , which have been quite extensively studied (c.f. [6, 14, 20, 23, 52, 63, 66] etc). Our results indicate that solutions to the mixed order Liouville type systems with exponential growth in \(\mathbb {R}^{n}\) can be classified under almost the same assumptions as the single n-th order Liouville equation.
