  In this paper, we consider the nonlinear Schr\"odinger equation with a real
valued potential V=V(x). We study global behavior of solutions to the equation
with a data below the ground state under some conditions for the potential V
and prove a scattering result and a blowing-up result in mass-supercritical and
energy-subcritical. Our proof of the blowing-up or growing-up result without
radially symmetric assumption is based on the argument by Du-Wu-Zhang in [6].
We can exclude the possibility of the growing-up result by the argument in
[23], [15], and [10] if "the data and the potential are radially symmetric" or
"the data has finite variance".
