Mixed order conformally invariant system with exponential growth and nonlocal nonlinear terms in critical dimensions

TL;DR

This paper classifies solutions to a conformally invariant system with exponential growth and nonlocal nonlinearities in critical dimensions, under mild decay assumptions, extending understanding of related fractional PDEs.

Contribution

It provides a classification of solutions for a mixed order conformally invariant system with exponential and nonlocal nonlinearities in dimensions 3 and 4, under mild decay conditions.

Findings

Solutions are classified under mild decay assumptions.

The system relates to well-studied conformally invariant equations.

Finite total mass condition is derived from integrability or Sobolev space membership.

Abstract

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.