Vol. 10, No. 3, 2017

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Conformally Euclidean metrics on $\mathbb R^n$ with arbitrary total $Q$-curvature

Ali Hyder

Vol. 10 (2017), No. 3, 635–652

We study the existence of solution to the problem

(Δ)n2u = Qenu in n ,κ :=nQenudx < ,

where Q 0, κ(0,) and n 3. Using ODE techniques, Martinazzi (for n = 6) and Huang and Ye (for n = 4m + 2) proved the existence of a solution to the above problem with Q  constant > 0 and for every κ (0,). We extend these results in every dimension n 5, thus completely answering the problem opened by Martinazzi. Our approach also extends to the case in which Q is nonconstant, and under some decay assumptions on Q we can also treat the cases n = 3 and n = 4.

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$Q$-curvature, nonlocal equation, conformal geometry
Mathematical Subject Classification 2010
Primary: 35G20, 35R11, 53A30
Received: 5 August 2016
Revised: 8 November 2016
Accepted: 22 January 2017
Published: 17 April 2017
Ali Hyder
Departement Mathematik und Informatik
Universität Basel
CH-4051 Basel