Isolated singularities for a semilinear equation for the fractional Laplacian arising in conformal geometry

DelaTorre, Azahara; González, María del Mar

doi:10.4171/rmi/1038

Cited by 34 publications

(42 citation statements)

References 59 publications

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“…s can be understood as a Dirichlet-to-Neumann for an extension problem in the spirit of the construction of the fractional Laplacian by [4,5,17,13]. Without being very precise on the extension manifold X n+1 , with metricḡ * , and the extension variable ρ * , we recall the following proposition in [1]:…”

Section: Conformal Geometry Results and The Asymptotic Behavior For Tmentioning

confidence: 99%

Bound state solutions for the supercritical fractional Schrödinger equation

Chan

González

et al. 2020

Nonlinear Analysis

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We prove the existence of positive solutions for the supercritical nonlinear fractionalfor s ∈ (0, 1), n > 2s. We show that if V (x) = o(|x| −2s ) as |x| → +∞, then for p > n+2s−1 n−2s−1 , this problem admits a continuum of solutions. More generally, for p > n+2s n−2s , conditions for solvability are also provided. This result is the extension of the work by Davila, Del Pino, Musso and Wei to the fractional case. Our main contributions are: the existence of a smooth, radially symmetric, entire solution of (−∆) s w = w p in R n , and the analysis of its properties. The difficulty here is the lack of phase-plane analysis for a nonlocal ODE; instead we use conformal geometry methods together with Schaaf's argument as in the paper by Ao, Chan, DelaTorre, Fontelos, González and Wei on the singular fractional Yamabe problem.

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Section: Conformal Geometry Results and The Asymptotic Behavior For Tmentioning

confidence: 99%

Bound state solutions for the supercritical fractional Schrödinger equation

Chan

González

et al. 2020

Nonlinear Analysis

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“…Remark 4.2. Note that (4.4) is a special case of the Funk-Hecke formula for the m = 0 spherical harmonic (see [69], pages [29][30].…”

Section: New Ode Methods For Non-local Equationsmentioning

confidence: 99%

“…The operator P g0 γ on R × S N −1 is explicit. Indeed, in [29] the authors calculate its principal symbol using the spherical harmonic decomposition for S N −1 . With some abuse of notation, let µ m , m = 0, 1, 2, .…”

Section: 2mentioning

confidence: 99%

“…Singular solutions for the standard fractional Laplacian in R n \ R k can be better understood by considering the conformal metric g k from (3.22), that is the product of a sphere S N −1 and a half-space H k+1 . Inspired by the arguments by DelaTorre and González [29], our point of view is to rewrite the well known extension problem in R n+1 + for the fractional Laplacian in R n due to [19], as a different, but equivalent, extension problem and to consider the corresponding Dirichlet-to-Neumann operator P g k γ , defined in S N −1 × H k+1 . Here R n+1 + is replaced by anti-de Sitter (AdS) space, but the arguments run in parallel.…”

Section: Introductionmentioning

confidence: 99%

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On higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo–Pacard program

Ao¹,

Chan²,

DelaTorre³

et al. 2019

Duke Math. J.

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We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, for which we establish the classical gluing method of Mazzeo and Pacard ([63]) for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of a fractional order ODE, and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. Note, however, that no traditional phase-plane analysis is available here. Instead, first, we provide a rigorous construction of radial fast-decaying solutions by a blow-up argument and a bifurcation method. Second, we use conformal geometry to rewrite this non-local ODE, giving a hint of what a non-local phase-plane analysis should be. Third, for the linear theory, we use complex analysis and some non-Euclidean harmonic analysis to examine a fractional Schrödinger equation with a Hardy type critical potential. We construct its Green's function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a two-dimensional kernel as in the second order case.

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“…The calculation of the scattering operator and the conformal fractional Laplacian in this setting can be found in [10,11] and, in particular:…”

Section: The Model Cylindermentioning

confidence: 99%

Boundary Connected Sum of Escobar Manifolds

González

Sire

2019

J Geom Anal

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Let (X1,ḡ1) and (X2,ḡ2) be two compact Riemannian manifolds with boundary (M1, g1) and (M2, g2) respectively. The Escobar problem consists in prescribing a conformal metric on a compact manifold with boundary with zero scalar curvature in the interior and constant mean curvature of the boundary. The present work is the construction of a connected sum X = X1#X2 by excising half ball near points on the boundary. The resulting metric on X has zero scalar curvature and a CMC boundary. We fully exploit the nonlocal aspect of the problem and use new tools developed in recent years to handle such kinds of issues. Our problem is of course a very well-known problem in geometric analysis and that is why we consider it but the results in the present paper can be extended to other more analytical problems involving connected sums of constant fractional curvatures.

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Isolated singularities for a semilinear equation for the fractional Laplacian arising in conformal geometry

Cited by 34 publications

References 59 publications

Bound state solutions for the supercritical fractional Schrödinger equation

Bound state solutions for the supercritical fractional Schrödinger equation

On higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo–Pacard program

Boundary Connected Sum of Escobar Manifolds

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