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

Abstract

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.

A constructive approach to positive solutions of Δpu + f(u,∇u)≤0 on Riemannian manifolds

“…r Q−1 ln q r 1 n−1 for large enough r. (6) In view of the desired construction, the geodesic ball B r = B(0, r) on M coincides with the Euclidean ball {x : |x| ≤ r}. Letting S (r) = |∂B r | be the surface area of B r in M, we use (6) to achieve S (r) = ω n ψ n−1 (r).…”

“…Here, it is perhaps appropriate to mention that with general nonlinearity f (u, ∇u) the sharpness argument in [6,10] seems unsuitable to be conducted.…”

“…This article stems from an essential understanding of Grigor'yan-Sun's work [6] (with p = 2) and Sun's follow-up [10] (with p > 1) on how the volume condition 1.1. Grigor'yan-Sun's result for Δu + u σ ≤ 0.…”

“…Quite remarkably, the sharpness arguments provided in [6,9,10] are based on an existence theory of some ODEs with polynomial non-linearity u σ and the eigenvalue theory of Laplace equations. But, there is no evidence showing that their arguments can be used to find out an explicit positive solution even locally, such as near the origin or infinity.…”

“…

Grigor'yan-Sun in [6] (with p = 2) and Sun in [10] then the only non-negative weak solution of Δ p u + u σ ≤ 0 on a complete Riemannian manifold is identically 0; moreover, the powers of r and ln r are sharp. In this note, we present a constructive approach to the sharpness, which is flexible enough to treat the sharpness for Δ p u + f (u, ∇u) ≤ 0.

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