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

Abstract

We study the existence of solutions to the problem (-Delta)(n/2)u=Qe(nu) in R-n; V:= integral(Rn) e(nu) dx < infinity, 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 <=vertical bar S-3 vertical bar, and to the case n = 4, Q = 6, where C-S. Lin showed that V <=vertical bar S-4 vertical bar.