Nonnegative solutions of a fractional sub-Laplacian differential inequality on Heisenberg group

Y. Liu; Yuzhao Wang; J. Xiao

doi:10.4310/DPDE.2015.v12.n4.a4

Nonnegative solutions of a fractional sub-Laplacian differential inequality on Heisenberg group

Y. Liu
, Yuzhao Wang
, J. Xiao

Mathematics

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

In this paper we study nonnegative solutions of
(†) |g|^γ_Hnu^p≤(−Δ_Hⁿ)α/2u on Hⁿ,
where Hⁿ is the Heisenberg group; |⋅|_Hⁿ is the homogeneous norm; Δ_Hⁿis the sub-Laplacian; (p,α,γ)∈(1,∞)×(0,2)×[0,(p−1)Q); and Q=2n+2 is the homogeneous dimension of Hⁿ. In particular, we prove that any nonnegative solution of (†) is zero if and only if p≤Q+γ/Q−α.

Original language	English
Pages (from-to)	379-403
Number of pages	25
Journal	Dynamics of Partial Differential Equations
Volume	12
Issue number	4
DOIs	https://doi.org/10.4310/DPDE.2015.v12.n4.a4
Publication status	Published - 10 Dec 2015

Keywords

Heisenberg group
nonnegative weak solution
fractional sub-Laplacian

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title = "Nonnegative solutions of a fractional sub-Laplacian differential inequality on Heisenberg group",

abstract = "In this paper we study nonnegative solutions of (†) |g|γHnup≤(−ΔHn)α/2u on Hn, where Hn is the Heisenberg group; |⋅|Hn is the homogeneous norm; ΔHn is the sub-Laplacian; (p,α,γ)∈(1,∞)×(0,2)×[0,(p−1)Q); and Q=2n+2 is the homogeneous dimension of Hn. In particular, we prove that any nonnegative solution of (†) is zero if and only if p≤Q+γ/Q−α.",

keywords = "Heisenberg group, nonnegative weak solution, fractional sub-Laplacian",

author = "Y. Liu and Yuzhao Wang and J. Xiao",

year = "2015",

month = dec,

day = "10",

doi = "10.4310/DPDE.2015.v12.n4.a4",

language = "English",

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pages = "379--403",

journal = "Dynamics of Partial Differential Equations",

issn = "1548-159X",

publisher = "International Press of Boston, Inc.",

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TY - JOUR

T1 - Nonnegative solutions of a fractional sub-Laplacian differential inequality on Heisenberg group

AU - Liu, Y.

AU - Wang, Yuzhao

AU - Xiao, J.

PY - 2015/12/10

Y1 - 2015/12/10

N2 - In this paper we study nonnegative solutions of (†) |g|γHnup≤(−ΔHn)α/2u on Hn, where Hn is the Heisenberg group; |⋅|Hn is the homogeneous norm; ΔHn is the sub-Laplacian; (p,α,γ)∈(1,∞)×(0,2)×[0,(p−1)Q); and Q=2n+2 is the homogeneous dimension of Hn. In particular, we prove that any nonnegative solution of (†) is zero if and only if p≤Q+γ/Q−α.

AB - In this paper we study nonnegative solutions of (†) |g|γHnup≤(−ΔHn)α/2u on Hn, where Hn is the Heisenberg group; |⋅|Hn is the homogeneous norm; ΔHn is the sub-Laplacian; (p,α,γ)∈(1,∞)×(0,2)×[0,(p−1)Q); and Q=2n+2 is the homogeneous dimension of Hn. In particular, we prove that any nonnegative solution of (†) is zero if and only if p≤Q+γ/Q−α.

KW - Heisenberg group

KW - nonnegative weak solution

KW - fractional sub-Laplacian

U2 - 10.4310/DPDE.2015.v12.n4.a4

DO - 10.4310/DPDE.2015.v12.n4.a4

M3 - Article

SN - 1548-159X

VL - 12

SP - 379

EP - 403

JO - Dynamics of Partial Differential Equations

JF - Dynamics of Partial Differential Equations

IS - 4

ER -

Nonnegative solutions of a fractional sub-Laplacian differential inequality on Heisenberg group

Abstract

Keywords

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