This paper deals with the following parabolic-elliptic chemotaxis system with singular sensitivity and nonlinear signal production, 0.1 \(\begin{aligned} \left\{ \begin{array}{llll} u_t=\Delta u-\chi \nabla \cdot (\frac{u}{v^\gamma }\nabla v)+\lambda u-\mu u^k,\quad & x\in \Omega ,\quad t>0,\\ 0=\Delta v-v+u^\alpha ,\quad & x\in \Omega ,\quad t>0,\\ \end{array} \right. \end{aligned}\) under homogeneous Neumann boundary conditions and suitable initial value conditions where \(\Omega \subset \mathbb {R}^n\) ( \(n\ge 1\) , \(R>0\) ), \(\chi ,\gamma ,\lambda ,\mu >0\) , \(0<\alpha \le 1\) and \(k>1\) . It is proved that the following results: If \(n=1\) , \(\gamma \in \left( \frac{\alpha }{2},1\right] \) , \(k>1\) and \(\alpha \in (0,1]\) , it is proved that model (0.1) possesses a globally bounded classical solution. If \(n\ge 2\) , \(\gamma \in (0,1]\) , \(\alpha \in \left( \max \left\{ 0,1-\frac{k}{n}\right\} ,1\right) \) and \(k>\frac{3n}{2+n}\) , it is shown that model (0.1) has a global classical solution. Especially, for \(\gamma =1\) , the solution is globally bounded when \(\lambda >\frac{\chi ^2}{4}\) with \(0<\chi \le 2\) or \(\lambda >\chi -1\) with \(\chi >2\) , and which exponentially converges to \(\left( \left( \frac{\lambda }{\mu }\right) ^{\frac{1}{k-1}}, \left( \frac{\lambda }{\mu }\right) ^{\frac{\alpha }{k-1}}\right) \) as \(t\rightarrow \infty \) with \(\mu \) suitably large.
