Existence and asymptotics for solutions of a non-local Q-curvature equation in dimension three

Abstract

We study conformal metrics on $R^3$, i.e., metrics of the form $g_u=e^{2u}|dx{|}^2$, which have constant $Q$-curvature and finite volume. This is equivalent to studying the non-local equation $$(-\Delta {)}^{\frac{3}{2}}u=2e^{3u}$$ in $R^3$ $$V:={\int }_{{\mathbb{R}}^3}{e}^{3u}dx<\infty ,$$ where $V$ is the volume of $g_u$. Adapting a technique of A. Chang and W-X. Chen to the non-local framework, we show the existence of a large class of such metrics, particularly for $V\le 2{\pi }^2=|S^3|$. Inspired by previous works of C-S. Lin and L. Martinazzi, who treated the analogue cases in even dimensions, we classify such metrics based on their behavior at infinity.

22 pages

References in corpus (6)

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• A threshold phenomenon for embeddings of $H_0^m$ into Orlicz spaces
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Cited by in corpus (7)

• Fractional Adams-Moser-Trudinger type inequalities
• Blow-up analysis of a nonlocal Liouville-type equation
• Conformally Euclidean metrics on ${\mathbb{R}}^n$ with arbitrary total $Q$-curvature
• Conformal metrics in ${\mathbb{R}}^{2m}$ with constant $Q$-curvature and arbitrary volume
• Blow-up behaviour of a fractional Adams-Moser-Trudinger type inequality in odd dimension
• Large blow-up sets for the prescribed Q-curvature equation in the Euclidean space
• Sign-changing solutions for the one-dimensional non-local sinh-Poisson equation