  We prove that every entire solution of the minimal graph equation that is
bounded from below and has at most linear growth must be constant on a complete
Riemannian manifold $M$ with only one end if $M$ has asymptotically
non-negative sectional curvature. On the other hand, we prove the existence of
bounded non-constant minimal graphic and $p$-harmonic functions on rotationally
symmetric Cartan-Hadamard manifolds under optimal assumptions on the sectional
curvatures.
