LIOUVILLE TYPE THEOREMS FOR THE p-HARMONIC FUNCTIONS ON CERTAIN MANIFOLDS

摘要

We show that for a certain range of p > n, the Dirichlet problem at infinity is unsolvable for the p-Laplace equation for any nonconstant continuous boundary data on an n-dimensional Cartan-Hadamard manifold constructed from a complete noncompact shrinking gradient Ricci soliton. Using the steady gradient Ricci soliton, we find an incomplete Riemannian metric on R-2 with positive Gauss curvature such that every positive pharmonic function must be constant for p >= 4.