Liouville theorem for k-curvature equation in half space with fully nonlinear boundary condition

Wei Wei

doi:10.1515/acv-2025-0049

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Liouville theorem for k-curvature equation in half space with fully nonlinear boundary condition

Wei Wei

Published/Copyright: January 14, 2026

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From the journal Advances in Calculus of Variations

Abstract

We establish the Liouville theorem for positive constant σ k -curvature equation in ℝ + n and positive constant boundary ℬ k g curvature equation, where the boundary curvature ℬ k g is discovered by Sophie Chen in [S.-Y. S. Chen, Conformal deformation on manifolds with boundary, Geom. Funct. Anal. 19 2009, 4, 1029–1064] from the natural variational functional for σ k ⁢ ( A g ) .

Keywords: Liouville theorem; fully nonlinear boundary equation

MSC 2020: 53C21; 53C20

Communicated by Verena Bögelein

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12571218

Award Identifier / Grant number: 12201288

Award Identifier / Grant number: BK20220755

Funding statement: The author is partially supported by National Natural Science Foundation of China, Grants No. 12571218, No. 12201288 and No. BK20220755, and by the Alexander von Humboldt Foundation.

A Appendix

In this appendix, we give a corner Hopf Lemma for second order boundary condition by mimicking the proof of Li and Zhang [28] and list several lemmas in [25] for the readers’ convenience.

A.1 Corner Hopf Lemma

Let { A i ⁢ j ⁢ ( x ) } n × n and { C α ⁢ β ⁢ ( x ) } ( n - 1 ) × ( n - 1 ) be two positive function matrices such that there exist positive constants λ 1 , λ 2 , Λ 1 , Λ 2 such that

λ 1 ⁢ δ i ⁢ j ≤ { A i ⁢ j } n × n ≤ Λ 1 ⁢ δ i ⁢ j , λ 2 ⁢ δ α ⁢ β ≤ { C α ⁢ β } ( n - 1 ) × ( n - 1 ) ≤ Λ 2 ⁢ δ α ⁢ β .

For any x ¯ ∈ ∂ ⁡ B λ ∩ ∂ ⁡ ℝ + n , taking Ω = ( ℝ + n \ B λ ) ∩ B 1 ⁢ ( x ¯ ) , σ = x n and ρ = | x | 2 - λ 2 . Let n → be the unit inward normal vector of the surface { x n = 0 } ∩ ∂ Ω . The proof of the following theorem is almost same as Li and Zhang [28] and we omit it.

Lemma A.1.

Let u ∈ C 2 ⁢ ( Ω ¯ ) be a positive function in Ω, u ⁢ ( x ¯ ) = 0 and there exists a positive constant A such that

{ 𝒫 ⁢ u := - A i ⁢ j ⁢ u i ⁢ j + B i ⁢ u i ≥ - A ⁢ u in ⁢ Ω , ℒ ⁢ u := - C α ⁢ β ⁢ ∂ α ⁢ β ⁡ u + D α ⁢ ∂ α ⁡ u - C 0 ⁢ u n ≥ - A ⁢ u on { σ = 0 , ρ > 0 } ,

where C 0 is a positive function and D α are functions. Then

∂ ⁡ u ∂ ⁡ ν ′ ⁢ ( x ¯ ) > 0 ,

where ν ′ is the unit normal vector on { σ = 0 , ρ = 0 } entering { σ = 0 , ρ > 0 } .

A.2 Useful lemmas

For the readers’ convenience, we list some classical lemmas in [24, 25, 28].

Lemma A.2 (Li and Zhang [28]).

Let f ∈ C 1 ⁢ ( R n ) , n ≥ 1 , l > 0 . Suppose that for every x ∈ R n , there exists λ ⁢ ( x ) > 0 such that

( λ ⁢ ( x ) | y - x | ) l ⁢ f ⁢ ( x + λ ⁢ ( x ) 2 ⁢ ( y - x ) | y - x | 2 ) = f ⁢ ( y ) for ⁢ y ∈ ℝ n \ { x } .

Then for some a ≥ 0 , d > 0 , x ¯ ∈ R n ,

f ⁢ ( x ) = ± ( a d 2 + | x - x ¯ | 2 ) l 2 .

Lemma A.3 (Li and Zhang [28]).

Let f ∈ C 1 ⁢ ( R + n ) , n ≥ 2 , ν > 0 . Assume that

( λ | y - x | ) ν ⁢ f ⁢ ( x + λ 2 ⁢ ( y - x ) | y - x | 2 ) ≤ f ⁢ ( y ) for all ⁢ λ > 0 , x ∈ ∂ ⁡ ℝ + n , | y - x | ≥ λ , y ∈ ℝ + n .

Then

f ⁢ ( x ) = f ⁢ ( x ′ , t ) = f ⁢ ( 0 , t ) for all ⁢ x = ( x ′ , t ) ∈ ℝ + n .

Lemma A.2 and Lemma A.3 can be found in the appendix of [28]. The following lemma is about the classification of radial solution, which can be found in [25] for a more general operator including σ k .

Theorem 3 (Li and Li [25]).

For n ≥ 3 , assume that u ∈ C 2 ⁢ ( B 1 n ) is radially symmetric and satisfies

σ k 1 k ⁢ ( A u ) = 1 , A u ∈ Γ k + , u > 0 in ⁢ 𝔹 1 n .

Then

u ⁢ ( x ) ≡ ( a 1 + b ⁢ | x | 2 ) n - 2 2 ⁢ in ⁢ 𝔹 1 n ,

where a > 0 , b ≥ - 1 and σ k 1 k ⁢ ( ( 2 ⁢ b a 2 ) ⁢ I n × n ) = 1 .

Acknowledgements

The author would like to thank Professor X. Z. Chen for enlightening discussions and constant support, and Dr. Biao Ma for his suggestion in writing the Appendix. The author would like to thank the referees for their helpful comments and suggestions, which have improved the presentation and readability of the paper.

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Received: 2025-04-29

Accepted: 2025-12-23

Published Online: 2026-01-14

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https://doi.org/10.1515/acv-2025-0049

Keywords for this article

Liouville theorem; fully nonlinear boundary equation