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

Abstract

In this paper, without any assumption on $v$ and under extremely mild assumption $u(x)=O(|x{|}^K)$ at $\infty$ for some $K\gg 1$ 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} (-Δ)^{\frac{1}{2}}u(x)=e^{pv(x)}, \qquad x\in\mathbb{R}^{2}, \\ -Δv(x)=u^{4}(x), \qquad x\in\mathbb{R}^{2}, \end{cases}\end{equation*} where $p\in (0,+\infty )$, $u\ge 0$ 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 $n\ge 3$ are also included.

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