Classification of solutions to $3$-D and $4$-D mixed order conformally invariant systems with critical and exponential growth

Abstract

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.