A Liouville theorem for Möbius invariant equations

Li, Yanyan; Lu, Han; Lu, Siyuan

doi:10.48550/arxiv.2012.15720

Cited by 1 publication

(4 citation statements)

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“…By the conformal invariance property of A u , if λ(A u ) ∈ ∂Γ, then λ(A u x,λ ) ∈ ∂Γ. See our previous paper [52] for details.…”

Section: Previous Resultsmentioning

confidence: 99%

“…Proposition 2.2. ( [52]) Let Γ = Γ p for some 1 < p ≤ 2, and let u be a viscosity supersolution of (2.1) in R 2 \ B r 0 for some r 0 > 0. Then there exists K 0 > 0, such that inf ∂Br u(r) + 4 ln r is monotonically nondecreasing in r for r > K 0 .…”

Section: Previous Resultsmentioning

confidence: 99%

“…By the Liouville theorem in our earlier paper [52], we have ũ * = U a * ,x * for some a * > 0 and x * ∈ R 2 . Since ũ * (0) = lim ũi (0) = c and ũ * ≤ C 1 + c, we have, for some constant C depending only on C 1 ,…”

Section: It Follows Thatmentioning

confidence: 87%

“…Equations f (λ(A u )) = 1, which we call Möbius invariant equations, is naturally associated with A u . A Liouville type theorem for the Möbius invariant equations was established in our previous paper [52].…”

Section: Letmentioning

confidence: 94%

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On the $σ_2$-Nirenberg problem on $\mathbb{S}^2$

Li,

Lu,

2021

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We establish theorems on the existence and compactness of solutions to the σ2-Nirenberg problem on the standard sphere S 2 . A first significant ingredient, a Liouville type theorem for the associated fully nonlinear Möbius invariant elliptic equations, was established in an earlier paper of ours. Our proof of the existence and compactness results requires a number of additional crucial ingredients which we prove in this paper: A Liouville type theorem for the associated fully nonlinear Möbius invariant degenerate elliptic equations, a priori estimates of first and second order derivatives of solutions to the σ2-Nirenberg problem, and a Bôcher type theorem for the associated fully nonlinear Möbius invariant elliptic equations. Given these results, we are able to complete a fine analysis of a sequence of blow-up solutions to the σ2-Nirenberg problem. In particular, we prove that there can be at most one blow-up point for such a blow-up sequence of solutions. This, together with a Kazdan-Warner type identity, allows us to prove L ∞ a priori estimates for solutions of the σ2-Nirenberg problem under some simple generic hypothesis. The higher derivative estimates then follow from classical estimates of Nirenberg and Schauder. In turn, the existence of solutions to the σ2-Nirenberg problem is obtained by an application of the by now standard degree theory for second order fully nonlinear elliptic operators.

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“…By the conformal invariance property of A u , if λ(A u ) ∈ ∂Γ, then λ(A u x,λ ) ∈ ∂Γ. See our previous paper [52] for details.…”

Section: Previous Resultsmentioning

confidence: 99%

Section: Previous Resultsmentioning

confidence: 99%

Section: It Follows Thatmentioning

confidence: 87%

Section: Letmentioning

confidence: 94%

See 2 more Smart Citations

On the $σ_2$-Nirenberg problem on $\mathbb{S}^2$

Li,

Lu,

2021

Preprint

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A Liouville theorem for Möbius invariant equations

Abstract: In this paper we classify Möbius invariant differential operators of second order in two dimensional Euclidean space, and establish a Liouville type theorem for general Möbius invariant elliptic equations.

Cited by 1 publication

References 43 publications

On the $σ_2$-Nirenberg problem on $\mathbb{S}^2$

On the $σ_2$-Nirenberg problem on $\mathbb{S}^2$

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