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Maximum and comparison principles to Lane‐Emden systems
Author(s) -
Leite Edir Junior Ferreira,
Montenegro Marcos
Publication year - 2020
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms.12256
Subject(s) - maximum principle , uniqueness , mathematics , bounded function , eigenvalues and eigenvectors , measure (data warehouse) , homogeneous , order (exchange) , mathematical analysis , pure mathematics , combinatorics , mathematical optimization , physics , computer science , optimal control , finance , quantum mechanics , database , economics
This paper focuses on maximum and comparison principles related to the Lane–Emden problem− L 1 u = λ ρ ( x )| v | α − 1 vinΩ ,− L 2 v = μ τ ( x )| u | β − 1 uinΩ ,u = v = 0on∂ Ω ,where α , β > 0 , α β = 1 , Ω is a smooth bounded open subset of R n with n ⩾ 1 , ρ and τ are positive functions on Ω and L 1 and L 2 are second‐order uniformly elliptic linear operators in Ω . We characterize the couples( λ , μ ) ∈ R 2such that the weak maximum principle associated to this problem holds in Ω . Moreover, weak and strong versions of maximum and comparison principles are also proved. As applications, we establish existence and uniqueness of solution for the non‐homogeneous counterpart of the above problem as well as Aleksandrov–Bakelman–Pucci‐type estimates. Lower bounds for principal eigenvalues of Lane–Emden systems in terms of the measure of Ω are also derived.