Existence of entire solutions to a fractional Liouville equation in $\mathbb{R}^n$

TL;DR

This paper proves the existence of solutions to a fractional Liouville equation in odd-dimensional Euclidean space, showing that both the asymptotic behavior of solutions and the total volume can be prescribed, with some cases allowing arbitrary volume.

Contribution

It extends previous results to all odd dimensions $n \\geq 3$, demonstrating the simultaneous prescription of asymptotics and volume, including cases where volume can be arbitrarily chosen.

Findings

Solutions exist with prescribed asymptotic behavior.

For $Q=-(n-1)!$, volume can be any positive number.

Contrasts with known bounds in specific low-dimensional cases.

Abstract

We study the existence of solutions to the problem $$ (-\Delta)^{\frac{n}{2}}u = Qe^{nu}\quad\text{in }\mathbb{R}^n, \quad V := \int_{\mathbb{R}^n}e^{nu}dx < \infty,$$ where $Q=(n-1)!$ or $Q=-(n-1)!$. Extending the works of Wei-Ye and Hyder-Martinazzi to arbitrary odd dimension $n\geq 3$ we show that to a certain extent the asymptotic behavior of $u$ and the constant $V$ can be prescribed simultaneously. Furthermore if $Q=-(n-1)!$ then $V$ can be chosen to be any positive number. This is in contrast to the case $n=3$, $Q=2$, where Jin-Maalaoui-Martinazzi-Xiong showed that necessarily $V\le |S^3|$, and to the case $n=4$, $Q=6$, where C-S. Lin showed that $V\le |S^4|$.