  We investigate a class of nonlinear Schr\"odinger equations with competing
inhomogeneous nonlinearities in the non-radial inter-critical regime, $$ i
\partial_t u +\Delta u =|x|^{-b_1} |u|^{p_1-2} u - |x|^{-b_2} |u|^{p_2-2}u
\quad \mbox{in} \,\, \mathbb{R} \times \mathbb{R}^N, $$ where $N \geq 1$, $b_1,
b_2>0$ and $p_1,p_2>2$.
  First, we establish the existence/nonexistence, symmetry, decay, uniqueness,
non-degeneracy and instability of ground states. Then, we prove the scattering
versus blowup below the ground state energy threshold. Our approach relies on
Tao's scattering criterion and Dodson-Murphy's Virial/Morawetz inequalities. We
also obtain an upper bound of the blow-up rate. The novelty here is that the
equation does not enjoy any scaling invariance due to the presence of competing
nonlinearties and the singular weights prevent the invariance by translation in
the space variable.
  To the best of authors knowledge, this is the first time when inhomegenous
NLS equation with a focusing leading order nonlinearity and a defocusing
perturbation is investigated.
