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

Abstract

Grigor'yan–Sun in [6] (with $p=2$) and Sun in [10] (with $p>1$) proved that if

then the only non-negative weak solution of ${\mathrm{\Delta }}_{p}u+u^{\sigma }\le 0$ 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 ${\mathrm{\Delta }}_{p}u+f(u,\nabla u)\le 0$. 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.