Grigor'yan–Sun in  (with p=2) and Sun in  (with p>1) proved that if supr≫1vol(B(x0,r))rpσp−σ−1(lnr)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 lnr 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.
|Journal||l' Institut Henri Poincare. Annales (C). Analyse Non Lineaire|
|Early online date||22 Jul 2015|
|Publication status||Published - Nov 2016|
- non-negative solution
- volume growth consideration
- complete Riemannian manifold