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 $K\gg1$ arbitrarily large, we classify solutions of the following conformally invariant system with mixed order and exponentially increasing nonlinearity in $\mathbb{ R}^{ 3}$: $$ \begin{ cases} \ (-\Delta)^{ \frac{ 1}{ 2}} u=v^{ 4} ,&x\in \mathbb{ R}^{ 3},\\ \ -\Delta v=e^{ pw} ,&x\in \mathbb{ R}^{ 3},\\ \ (-\Delta)^{ \frac{ 3}{ 2}} w=u^{ 3} ,&x\in \mathbb{ R}^{ 3}, \end{ cases} $$ where $p>0$, $w(x)=o(|x|^{ 2})$ at $\infty$ and $u,v\geq0$ satisfies the finite total curvature condition $\int_{ \mathbb{ R}^{ 3}}u^{ 3}(x)\mathrm{ d}x<+\infty$. Moreover, under the extremely mild assumption that \emph{ either} $u(x)$ or $v(x)=O(|x|^{ K})$ as $|x|\rightarrow+\infty$ for some $K\gg1$ arbitrarily large \emph{ or} $\int_{ \mathbb{ R}^{ 4}}e^{ \Lambda pw(y)}\mathrm{ d}y<+\infty$ for some $\Lambda\geq1$, we also prove classification of solutions to the conformally invariant system with mixed order and exponentially increasing nonlinearity in $\mathbb{ R}^{ 4}$: \begin{ align*} \begin{ cases} \ (-\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{ cases} \end{ align*} where $p>0$, and $w(x)=o(|x|^{ 2})$ at $\infty$ and $u,v\geq0$ satisfies the finite total curvature condition $\int_{ \mathbb{ R}^{ 4}}v^{ 4}(x)\mathrm{ d}x<+\infty$. 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.
Categories: math.AP