On Nonnegative Solutions of the Inequality Δ  u  +  u  σ  ≤ 0 on Riemannian Manifolds | Zendy

Grigor'yan Alexander | Zendy; Sun Yuhua | Zendy

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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.

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