# Existence of entire solutions to a fractional Liouville equation in $R_{n}$

### Ali Hyder

Universität Basel, Switzerland

## Abstract

We study the existence of solutions to the problem

$(−Δ)_{2n}u=Qe_{nu}inR_{n},V:=∫_{R_{n}}e_{nu}dx<∞,$

where $Q=(n−1)!$ or $Q=−(n−1)!$. Extending the works of Wei–Ye and Hyder–Martinazzi to arbitrary odd dimension $n≥3$ 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 $V≤∣S_{3}∣$, and to the case $n=4$, $Q=6$, where C.-S. Lin showed that $V≤∣S_{4}∣$.

## Cite this article

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

DOI 10.4171/RLM/718