Concentration phenomena for Liouville's equation in dimension four

Adimurthi,; Robert, Frédéric; Struwe, Michaël

doi:10.4171/jems/44

J. Eur. Math. Soc.

2006

DOI: 10.4171/jems/44

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Concentration phenomena for Liouville's equation in dimension four

Adimurthi Adimurthi¹,

Frédéric Robert²,

Michaël Struwe³

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Cited by 29 publications

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References 19 publications

(23 reference statements)

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“…Over the past three decades, considerable effort has been devoted to classifying all solutions to problem (1.4). This classification has applications in proving quantization, compactness and existence results for solutions to equations locally modeled by (1.4), see for example [2,30,1,26,28,21] and the references therein. When n = 2 it was shown in [5] that every solution to (1.4) is spherical and hence satisfies V = |S 2 |.…”

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confidence: 99%

Classification of solutions to a system of n^{\rm th} order equations on \mathbb R^n

Glück¹

2020

Communications on Pure &Amp; Applied Analysis

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This work concerns the distributional solutions of a conformally invariant system of n th-order elliptic equations on R n having exponential type nonlinearity. The system in question is a natural generalization of the constant Q-curvature equation on R n. Under an L 1-finiteness assumption and some assumptions on the coupling coefficients, an asymptotic estimate for solutions as |x| → ∞ is obtained. Under a growth constraint and further L 1-norm assumptions the method of moving spheres is used to show that, up to an additive polynomial of low degree, each of the unknown functions is a standard bubble with common center and scale parameters.

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Classification of solutions to a system of n^{\rm th} order equations on \mathbb R^n

Glück¹

2020

Communications on Pure &Amp; Applied Analysis

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“Large” conformal metrics of prescribed $$\varvec{Q}$$-curvature in the negative case

Galimberti

2017

Nonlinear Differ. Equ. Appl.

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Given a compact and connected four dimensional smooth Riemannian manifold (M, g0) with kP := M Qg 0 dVg 0 < 0 and a smooth non-constant function f0 with maxp∈M f0(p) = 0, all of whose maximum points are non-degenerate, we assume that the Paneitz operator is nonnegative and with kernel consisting of constants. Then, we are able to prove that for sufficiently small λ > 0 there are at least two distinct conformal metrics g λ = e 2u λ g0 and g λ = e 2u λ g0 of Q-curvature Qg λ = Q g λ = f0 + λ. Moreover, by means of the "monotonicity trick" in a way similar to [9], we obtain crucial estimates for the "large" solutions u λ which enable us to study their "bubbling behavior" as λ ↓ 0.

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A Liouville-type theorem in conformally invariant equations

2023

Math. Ann.

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Concentration phenomena for Liouville's equation in dimension four

Cited by 29 publications

References 19 publications

Classification of solutions to a system of n^{\rm th} order equations on \mathbb R^n

Classification of solutions to a system of n^{\rm th} order equations on \mathbb R^n

“Large” conformal metrics of prescribed $$\varvec{Q}$$-curvature in the negative case

A Liouville-type theorem in conformally invariant equations

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