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 $V \le |S^4|$.

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

Ali Hyder

Abstract

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