Conformal metrics on $\mathbb{R}^{2m}$ with constant Q-curvature,  prescribed volume and asymptotic behavior

Hyder, Ali; Martinazzi, Luca

doi:10.3934/dcds.2015.35.283

Cited by 24 publications

(32 citation statements)

References 15 publications

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“…Proof of (13) From Corollary 1.2 we have Ṡsph := S ∞ \ S ϕ ⊂ S sph , and hence, Ṡsph is either empty or finite. If the set Ṡsph is empty then (13) follows trivially as…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

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Concentration phenomena to a higher order Liouville equation

Hyder¹

2020

Preprint

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We study blow-up and quantization phenomena for a sequence of solutions (u k ) to the prescribed Q-curvature problemunder natural assumptions on Q k . It is well-known that, up to a subsequence, either (u k ) is bounded in a suitable norm, or there exists β k → ∞ such that u k = β k (ϕ + o(1)) in Ω \ (S 1 ∪ S ϕ ) for some non-trivial non-positive n-harmonic function ϕ and for a finite set S 1 , where S ϕ is the zero set of ϕ. We prove quantization of the total curvature Ω Q k e 2nu k dx on the region Ω ⋐ (Ω \ S ϕ ). We also consider a non-local case in dimension three.

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“…Proof of (13) From Corollary 1.2 we have Ṡsph := S ∞ \ S ϕ ⊂ S sph , and hence, Ṡsph is either empty or finite. If the set Ṡsph is empty then (13) follows trivially as…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

“…Proof of (15) Since (15) follows from (13) if N ℓ > 1, we only need to consider the case N ℓ = 1. Let x 0 = x (ℓ) ∈ Ṡsph be such that ∇ 2 ϕ(x (ℓ) ) > 0.…”

Section: This Proves (43)mentioning

confidence: 99%

Concentration phenomena to a higher order Liouville equation

Hyder¹

2020

Preprint

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“…Due to its geometric meaning, the constant Q-curvature case for equation ( 3) has been extensively studied (see e.g. [4,5,6,14,16,17,20] and the references within). The problem which we address, namely the case of non-normal metrics with sign-changing prescribed Q-curvature having power-like growth, is (to the best of our knowledge) still unexplored.…”

Section: Introductionmentioning

confidence: 99%

Existence and asymptotic behavior of non-normal conformal metrics on $\mathbb{R}^4$ with sign-changing $Q$-curvature

Bernardini¹

2021

Preprint

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We consider the following prescribed Q-curvature problem(1)We show that for every polynomial P of degree 2 such that lim |x|→+∞ P = −∞, and for every Λ ∈ (0, Λ sph ), there exists at least one solution to problem (1) which assume the form u = w + P , where w behaves logarithmically at infinity. Conversely, we prove that all solutions to (1) have the form v + P , whereand P is a polynomial of degree at most 2 bounded from above. Moreover, if u is a solution to (1), it has the following asymptotic behavior u(x) = − Λ 8π 2 log |x| + P + o(log |x|), as |x| → +∞. As a consequence, we give a geometric characterization of solutions in terms of the scalar curvature at infinity of the associated conformal metric e 2u |dx| 2 .

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“…with conformal metric written asg = e 2w g. In this case the conformal factor is always positive and the progress on this problem was obtained earlier compared to the higherdimensional case, see [3,8,10,23,33,36]. For prescribed variable curvature, we also mention the papers [7,15,21,37].…”

Section: Introductionmentioning

confidence: 99%

On Conformal Metrics with Constant $Q$-Curvature

Malchiodi¹

2019

ATA

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We review some recent results in the literature concerning existence of conformal metrics with constant Q-curvature. The problem is rather similar to the classical Yamabe problem: however it is characterized by a fourth-order operator that might lack in general a maximum principle. For several years existence of geometrically admissible solutions was known only in particular cases. Recently, there has been instead progress in this direction for some general classes of conformal metrics.

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Conformal metrics on $\mathbb{R}^{2m}$ with constant Q-curvature, prescribed volume and asymptotic behavior

Cited by 24 publications

References 15 publications

Concentration phenomena to a higher order Liouville equation

Concentration phenomena to a higher order Liouville equation

Existence and asymptotic behavior of non-normal conformal metrics on $\mathbb{R}^4$ with sign-changing $Q$-curvature

On Conformal Metrics with Constant $Q$-Curvature

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