In this paper, we study a parabolic-elliptic cross-diffusion system with flux limitation \(\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta u- \nabla \cdot (S(u)f(|\nabla v|^{2})\nabla v),&x\in \Omega ,t>0, \\&0=\Delta v-M+u,&x\in \Omega ,t>0, \\ \end{aligned} \right. \end{aligned}\) under homogeneous Neumann boundary conditions within a smooth, bounded domain \(\Omega \subset {\mathbb {R}}^N\) , where \(m\in {\mathbb {R}}\) , \(M:=\frac{1}{|\Omega |} \int _{\Omega } u_0(x) d x\) , \(f\left( |\nabla v|^2\right) =(1+|\nabla v|^2)^{-\alpha }, \alpha \in {\mathbb {R}}\) . It is shown that in the case of \(\begin{aligned} 0\le S(u)\le u^m\;\;\;\;for\;\; all \;\;u\ge 0, \end{aligned}\) when \(N\ge 2, 0<m\le 1\) and \(\begin{aligned} \alpha >\frac{mN-2}{2N-2}, \end{aligned}\) the solution is global and bounded in time for all nonnegative initial data. Furthermore, when \(S(u)=u^{m}\) and \(\Omega \subset {\mathbb {R}}^N\) \((N\ge 3)\) is a ball, if \(1<m<\frac{4}{3}\) and \(\frac{4N-3mN+2m}{4(N-1)(m-1)}<\alpha <\frac{m}{(N-1)(2-m)}\) , there exist some initial data \(u_{0}\) such that the solution u(x, t) blows up in finite time in the \(L^{\infty }\) -norm sense.
