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

TL;DR

This paper classifies solutions to a conformally invariant mixed-order PDE system with exponential and nonlocal nonlinearities in various dimensions, establishing integral formulas and asymptotic properties under mild growth conditions.

Contribution

It provides the first classification results for solutions to such complex systems with mixed order and exponential growth, including new inequalities and extensions to higher dimensions.

Findings

Classification of solutions in  with mild growth assumptions

Development of an L+LL inequality of independent interest

Extension of results to  and higher dimensions

Abstract

In this paper, without any assumption on $v$ and under extremely mild assumption $u(x)=O(|x|^{K})$ at $\infty$ for some $K\gg1$ 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} (-\Delta)^{\frac{1}{2}}u(x)=e^{pv(x)}, \qquad x\in\mathbb{R}^{2}, \\ -\Delta v(x)=u^{4}(x), \qquad x\in\mathbb{R}^{2}, \end{cases}\end{equation*} where $p\in(0,+\infty)$, $u\geq 0$ 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 $n\geq3$ are also included.