  Firstly, we introduce a new notion called induced upper metric mean dimension
with potential, which naturally generalizes the definition of upper metric mean
dimension with potential given by Tsukamoto to more general cases, then we
establish variational principles for it in terms of upper and lower rate
distortion dimensions and show there exists a Bowen's equation between induced
upper metric mean dimension with potential and upper metric mean dimension with
potential.
  Secondly, we continue to introduce two new notions, called BS metric mean
dimension and Packing BS metric mean dimension on arbitrary subsets, to
establish Bowen's equations for Bowen upper metric mean dimension and Packing
upper metric mean dimension with potential on subsets. Besides, we also obtain
two variational principles for BS metric mean dimension and Packing BS metric
mean dimension on subsets.
  Finally, the special interest about the Bowen upper metric mean dimension of
the set of generic points of ergodic measures are also involved.
