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

By Tianling Jin, Ali Maalaoui, L. Martinazzi, Jingang Xiong. 2013

53 citations recorded by the source corpus.

We study conformal metrics on $${\mathbb {R}}^{3}$$R3, i.e., metrics of the form $$g_u=e^{2u}|dx|^2$$gu=e2u|dx|2, which have constant $$Q$$Q-curvature and finite volume. This is equivalent to studying the non-local equation $$\begin{aligned} (-\Delta )^\frac{3}{2} u = 2 e^{3u}\text { in }{\mathbb {R}}^{3},\quad V:=\int _{{\mathbb {R}}^{3}}e^{3u}\,dx<\infty , \end{aligned}$$(-Δ)32u=2e3uinR3,V:=∫R3e3udx<∞,where $$V$$V is the volume of $$g_u$$gu. 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 $$V\le 2\pi ^2=|{\mathbb {S}}^3|$$V≤2π2=|S3|. 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2b0c51fd1dfdf034ecc9defa8bd16ec3b3adc19c

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Blow-up analysis of a nonlocal Liouville-type equation (2015, result): “This is identical to the proof of Lemma 14 in [24].”
Fractional Adams–Moser–Trudinger type inequalities (2015, background): “For existence results for (26) in the case Ω = Rn, n odd, we refer to [20] and [25], when n is even we refer to [10], [11], [22], [35], [48].”
Asymptotic method of moving planes for fractional parabolic equations (2020, background): “During the last decade, elliptic nonlocal equations, especially those involving fractional Laplacians, have been studied by more and more scholars and a series of results have been obtained, such as [9] for symmetry and nonexistence of solutions, [12, 16] for Liouville Theorems with indefinite nonli”