  We investigate the Cauchy problem for the focusing inhomogeneous nonlinear
Schr\"odinger equation $i \partial_t u + \Delta u = - |x|^b |u|^{p-1} u$ in the
radial Sobolev space $H^1_{\text{rad}}(\mathbb{R}^N)$, where $b>0$ and $p>1$.
We show the global existence and energy scattering in the inter-critical
regime, i.e., $p>\frac{N+4+2b}{N}$ and $p<\frac{N+2+2b}{N-2}$ if $N\geq 3$. We
also obtain blowing-up solutions for the mass-critical and mass-supercritical
nonlinearities. The main difficulty, coming from the spatial growing
nonlinearity, is overcome by refined Gagliardo-Nirenberg type inequalities. Our
proofs are based on improved Gagliardo-Nirenberg inequalities, the
Morawetz-Sobolev approach of Dodson and Murphy, radial Sobolev embeddings, and
localized virial estimates.
