Structure of conformal metrics on $\mathbb{R}^n$ with constant $Q$-curvature

Hyder, Ali

doi:10.48550/arxiv.1504.07095

Cited by 9 publications

(17 citation statements)

References 19 publications

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“…The proof will be given at the end of Section 2. Similar higher-dimensional, also in the fractional case have appeared in [6,11,12,15,18,27].…”

supporting

confidence: 64%

The nonlocal Liouville-type equation in $\mathbb{R}$ and conformal immersions of the disk with boundary singularities

Da Lio,

Martinazzi

2016

Preprint

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In this paper we perform a blow-up and quantization analysis of the fractional Liouville equation in dimension 1. More precisely, given a sequence u k : R → R of solutions to (−∆)with K k bounded in L ∞ and e u k bounded in L 1 uniformly with respect to k, we show that up to extracting a subsequence u k can blow-up at (at most) finitely many points B = {a 1 , . . . , a N } and either (i)N j=1 α j δ a j with α j ≥ π for every j. This result, resting on the geometric interpretation and analysis of (1) provided in a recent collaboration of the authors with T. Rivière and on a classical work of Blank about immersions of the disk into the plane, is a fractional counterpart of the celebrated works of Brézis-Merle and Li-Shafrir on the 2-dimensional Liouville equation, but providing sharp quantization estimates (α j = π and α j ≥ π) which are not known in dimension 2 under the weak assumption that (K k ) be bounded in L ∞ and is allowed to change sign.

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“…The proof will be given at the end of Section 2. Similar higher-dimensional, also in the fractional case have appeared in [6,11,12,15,18,27].…”

supporting

confidence: 64%

The nonlocal Liouville-type equation in $\mathbb{R}$ and conformal immersions of the disk with boundary singularities

Da Lio,

Martinazzi

2016

Preprint

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“…(see Theorems 1 and 2 in [18]). This is consistent with a result of F. Robert [23], extended in [20], stating that in a region Ω 0 such that ∆u k L 1 (Ω 0 ) ≤ C, u k has a bubbling behaviour leading to solutions of the form (12). It was open whether there exists a sequence (u k ) of solutions to ( 2)-( 3) on some domain…”

Section: Gluing Open Problemssupporting

confidence: 86%

“…to one of the solutions provided by Theorems 1 and 2? Moreover, as shown by Chang-Chen [6], when m ≥ 2 problem (13) has several solutions which are not of the form (12). Such solutions behave polynomially at infinity, as shown in [17,18] (see also [12,16] for similar results in odd dimension).…”

Section: Gluing Open Problemsmentioning

confidence: 93%

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

Hyder

Iula

Martinazzi

2017

Commun. Contemp. Math.

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Let m ≥ 2 be an integer. For any open domain Ω ⊂ R 2m , non-positive function ϕ ∈ C ∞ (Ω) such that ∆ m ϕ ≡ 0, and bounded sequence (V k ) ⊂ L ∞ (Ω) we prove the existence of a sequence of functions (u k ) ⊂ C 2m−1 (Ω) solving the Liouville equation of order 2m (−∆) m u k = V k e 2mu k in Ω, lim sup k→∞ Ω e 2mu k dx < ∞, and blowing up exactly on the set S ϕ := {x ∈ Ω : ϕ(x) = 0}, i.e. lim k→∞ u k (x) = +∞ for x ∈ S ϕ and lim k→∞ u k (x) = −∞ for x ∈ Ω \ S ϕ , thus showing that a result of Adimurthi, Robert and Struwe is sharp. We extend this result to the boundary of Ω and to the case Ω = R 2m . Several related problems remain open.

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“…Such an estimate is proven for example in [GO,Lemma 1], even though the dependence of the constant on the seminorms in S, is not explicitly written there (but it can be easily derived from the proof.) See also [Hy2,Lemma 2.2]. With the aid of ( 13) one can extend the definition of (−∆) α 2 u as a tempered distribution for any u ∈ L p , 1 ≤ p < ∞.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Sharp exponential integrability for critical Riesz potentials and fractional Laplacians on Rn

Fontana

Morpurgo

2018

Nonlinear Analysis

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We derive sharp Adams inequalities for the Riesz and more general Riesz-like potentials on the whole of R n . As a consequence, we obtain sharp Moser-Trudinger inequalities for the critical Sobolev spaces W α, n α (R n ), 0 < α < n. These inequalities involve fractional Laplacians, higher order gradients, general homogeneous elliptic operators with constant coefficients, and general trace type Borel measures.

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Structure of conformal metrics on $\mathbb{R}^n$ with constant $Q$-curvature

Cited by 9 publications

References 19 publications

The nonlocal Liouville-type equation in $\mathbb{R}$ and conformal immersions of the disk with boundary singularities

The nonlocal Liouville-type equation in $\mathbb{R}$ and conformal immersions of the disk with boundary singularities

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

Sharp exponential integrability for critical Riesz potentials and fractional Laplacians on Rn

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