The asymptotic Dirichlet problem for harmonic functions on a noncompact complete Riemannian manifold has a long history. It is to find the harmonic function satisfying the given Dirichlet boundary condition at infinity. By now, it is well understood [A, AS, Ch, S], when M is a Cartan-Hadamard manifold with sectional curvature $-b^2 leq K_M leq -a^2 < 0$. (By a Cartan-Hadamard manifold, we mean a complete simply connected manifold of non-positive sectional curvature.)