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

Abstract

We study the existence of solutions to the problemwhere 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 |.

Local and Nonlocal Singular Liouville Equations in Euclidean Spaces

“…Using a variational argument as in [12] one should be able to find solutions to (3) of the form u = v + p with the polynomial p prescribed. For instance one could try to prove that for n ≥ 3, α > −1, 0 < Λ < Λ 1 min{1, 1 + α} and a given polynomial p with deg(p) ≤ n − 1 and satisfying…”

Large blow-up sets for the prescribed Q-curvature equation in the Euclidean space

“…The simplicity of the proof of Theorem 1 comes at the cost of not being able to prescribe the total Q-curvature of the metric g u k := e 2u k |dx| 2 , which will necessarily go to zero, together with the volume of g u k . Resting on variational methods from [15] going back to [6], we can extend Theorem 1 to the case in which we prescribe both the blow-up set S ϕ and the total curvature of the metrics g u k . This time, though, we will have to restrict to non-negative functions V k .…”

“…A slightly different version of the following proposition appears in [15]. For the sake of completeness we give a sketch of the proof.…”

Conformally Euclidean metrics on ℝn with arbitrary total Q-curvature

“…A partial converse to Theorem A has been proven in dimension 4 by Wei-Ye [21] and extended by Hyder-Martinazzi [12] for n ≥ 4 even and Hyder [11] for n ≥ 3.…”