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

Grigor'yan–Sun in [6] (with p=2) and Sun in [10] (with p>1) proved that if supr≫1⁡vol(B(x0,r))rpσp−σ−1(ln⁡r)p−1p−σ−1<∞ then the only non-negative weak solution of Δpu+uσ≤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 Δpu+f(u,∇u)≤0. Our construction is based on a perturbation of the fundamental solution to the p-Laplace equation, and we believe that the ideas introduced here are applicable to other nonlinear differential inequalities on manifolds.