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

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 $\mathbb{R}^{n}$: $$ \left\{ \begin{aligned} (-Δ)^{\frac{1}{2}}u & = e^{pv} \\ (-Δ)^{\frac{n}{2}}v & = \left(\frac{1}{|x|^2}*u^2\right)u^2 \end{aligned} \right. \quad \text{in}\; \mathbb{R}^n, $$ 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 $(-Δ)^{\frac{1}{2}}u=\left(\frac{1}{|x|^{2}}*u^2\right)u$ and $(-Δ)^{\frac{n}{2}}u=(n-1)!e^{nu}$ in $\mathbb{R}^{n}$ with $n=3,4$, which have been quite extensively studied.

Theorems & Definitions (44)

• Theorem 1.1
• Remark 1.2
• Lemma 2.1: Maximum Principle CLLS
• Lemma 2.2: Liouville Theorem BKN
• Lemma 2.3: Green's function on the ball K
• Lemma 2.4: $\exp^L+L\ln{L}$ Inequality DQ2
• Lemma 2.5: Lin
• Lemma 2.6: Hardy-Littlewood-Sobolev Inequality FLLieb
• Lemma 2.7: Calculus Lemma LZ1
• Remark 2.8