Existence of entire solutions to a fractional Liouville equation in Rⁿ

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We study the existence of solutions to the problem (-Δ)ⁿ/²u = Qenu Rⁿ, V:= ∫ᵣₙenudx < ∞, 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≤ |S³|, and to the case n=4, Q=6, where C-S. Lin showed that V≤ |S⁴|.