Existence of entire solutions to a fractional Liouville equation in R^n

We study the existence of solutions to the problem
\[
(−∆)^{n/2} u = Qe^{nu} in \mathbb{R}^n, V := \int_{\mathbb{R}^n} e^{nu} dx < ∞,
\]
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|.