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

### Ali Hyder

Universität Basel, Switzerland

## Abstract

We study the existence of solutions to the problem

$(-\Delta)^{\frac{n}{2}}u=Qe^{nu} \quad \mathrm{in }\mathbb R^n, \quad V: = \int_{\mathbb R^n}e^{nu} dx < \infty,$

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 $V \le |S^4|$.

## Cite this article

Ali Hyder, Existence of entire solutions to a fractional Liouville equation in $\mathbb R^n$. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 27 (2016), no. 1, pp. 1–14

DOI 10.4171/RLM/718