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Classification results for a sub‐elliptic system involving the Δ λ ‐Laplacian
Author(s) -
Duong Anh Tuan,
Giang Trung Hieu,
Le Phuong,
Vu Thi Hien Anh
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6968
Subject(s) - mathematics , laplace operator , operator (biology) , p laplacian , extension (predicate logic) , class (philosophy) , pure mathematics , type (biology) , semi elliptic operator , elliptic operator , laplace transform , discrete mathematics , mathematical analysis , differential operator , boundary value problem , ecology , biochemistry , chemistry , repressor , artificial intelligence , biology , computer science , transcription factor , programming language , gene
In this paper, we study a system of the form− Δ λ u = v− Δ λ v = u pinℝ N , where p ∈ ℝ , and Δ λ is a sub‐elliptic operator defined byΔ λ =∑ i = 1 N∂x iλ i 2∂x i. Under some general hypotheses of the functions λ i , i = 1,2 , … , N , we first prove that the system has no positive super‐solution when p ≤ 1 . In the case p > 1 , we establish a Liouville type theorem for the class of stable positive solutions. This result is an extension of some result in Hajlaoui et al. ( Pacific J Math . 2014;270(1):79–93) for the case of Laplace operator.