Structure of conformal metrics on $\mathbb{R}^n$ with constant $Q$-curvature

TL;DR

This paper classifies solutions to a nonlocal conformal geometry equation involving the fractional Laplacian in inity, revealing their structure and behavior in relation to constant Q-curvature metrics.

Contribution

It provides a complete classification of solutions to a nonlocal PDE related to conformal metrics with constant Q-curvature on inity.

Findings

Solutions are classified by their asymptotic behavior at infinity.

The work extends previous classifications in even dimensions and dimension three.

It characterizes all solutions in terms of geometric and analytic properties.

Abstract

In this article we study the nonlocal equation \begin{align} (-\Delta)^{\frac{n}{2}}u=(n-1)!e^{nu}\quad \text{in $\mathbb{R}^n$}, \quad\int_{\mathbb{R}^n}e^{nu}dx<\infty, \notag \end{align} which arises in the conformal geometry. Inspired by the previous work of C. S. Lin and L. Martinazzi in even dimension and T. Jin, A. Maalaoui, L. Martinazzi, J. Xiong in dimension three we classify all solutions to the above equation in terms of their behavior at infinity.