Summary: The chemotaxis system \[ \begin{cases} u_t =\Delta u-\chi \nabla \cdot (u|\nabla v|^{p-2}\nabla v)+\lambda u-\mu u^{\kappa}, \\ 0=\Delta v+u-h(u,v) \end{cases} \tag{\(*\)} \] is considered in a smoothly bounded domain \(\Omega \subset \mathbb{R}^n\) (\(n \in \mathbb{N}\)), where \(\chi >0\), \(p >1\), \(\lambda \geq 0\), \(\mu >0\), \(\kappa >1\), and \(h=v\) or \(h=\frac{1}{|\Omega|} \int_{\Omega} u\). It is firstly proved that if \(n=1\) and \(p>1\) is arbitrary, or \(n \geq 2\) and \(p \in (1, \frac{n}{n-1})\), then for all continuous initial data a corresponding no-flux type initial-boundary value problem for \((\ast)\) admits a globally defined and bounded weak solution. Secondly, it is shown that if \(n \geq 2\), \(\Omega = B_R (0) \subset \mathbb{R}^n\) is a ball with some \(R>0\), \(p > \frac{n}{n-1}\) and \(\kappa >1\) is small enough, then one can find a nonnegative radially symmetric function \(u_0\) and a weak solution of \((\ast)\) with initial datum \(u_0\) which blows up in finite time.