In this paper, 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 mixed order conformally invariant system with exponentially increasing and nonlocal nonlinearities in $\mathbb{ R}^{ n}$: $$ \left\{ \begin{ aligned} (-\Delta)^{ \frac{ 1}{ 2}}u & = e^{ pv} \\ (-\Delta)^{ \frac{ n}{ 2}}v & = \left(\frac{ 1}{ |x|^2}*u^2\right)u^2 \end{ aligned} \right. \quad \text{ in}\; \mathbb{ R}^n, $$ where $n=3,\,4$, $p>0$, $u\geqslant0$, $v(x)=o(|x|^2)$ as $|x|\to\infty$ 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 $(-\Delta)^{ \frac{ 1}{ 2}}u=\left(\frac{ 1}{ |x|^{ 2}}*u^2\right)u$ and $(-\Delta)^{ \frac{ n}{ 2}}u=(n-1)!e^{ nu}$ in $\mathbb{ R}^{ n}$ with $n=3,4$, which have been quite extensively studied.
Categories: math.AP