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On Nonnegative Solutions of the Inequality Δ u + u σ ≤ 0 on Riemannian Manifolds
Author(s) -
Grigor'yan Alexander,
Sun Yuhua
Publication year - 2014
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21493
Subject(s) - mathematics , uniqueness , riemannian manifold , geodesic , ball (mathematics) , combinatorics , manifold (fluid mechanics) , pure mathematics , mathematical analysis , mechanical engineering , engineering
We study the uniqueness of a nonnegative solution of the differential inequality( * )Δ u + u σ≤ 0 on a complete Riemannian manifold, where σ > 1 is a parameter. We prove that if, for some x 0 ∊ M and all large enough rvol B ( x 0 , r ) ≤ C r pln q r , where p = 2 σ σ − 1 , q = 1 σ − 1, and B ( x , r ) is a geodesic ball, then the only nonnegative solution of (*) is identically 0. We also show the sharpness of the above values of the exponents p , q . © 2014 Wiley Periodicals, Inc.