  We show that the Dirichlet problem at infinity is unsolvable for the
p-Laplace equation for any nonconstant continuous boundary data, for certain
range of p>n, 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 ${\mathbb R}^2$ with
positive Gauss curvature such that every positive p-harmonic function must be
constant for $p\geq 4$.
