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On Nonnegative Solutions of the Inequality Δ u + uσ ≤ 0 on Riemannian Manifolds

We study the uniqueness of a nonnegative solution of the differential inequality (*)Δu+uσ≤0 on a complete Riemannian manifold, where σ > 1 is a parameter. We prove that if, for some x0 ∊ M and all large enough r volB(x0,r)≤Crplnqr, where p=2σσ−1,q=1σ−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.