Boundedness to a logistic chemotaxis system with singular sensitivity

In this paper, we study the parabolic-elliptic Keller-Segel system with singular sensitivity and logistic-type source: $u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+ru-\mu u^k$, $0=\Delta v-v+u$ under the non-flux boundary conditions in a smooth bounded convex domain $\Omega\subset\mathbb{R}^n$, $\chi,r,\mu>0$, $k>1$ and $n\ge 2$. It is shown that the system possesses a globally bounded classical solution if $k>\frac{3n-2}{n}$, and $r>\frac{\chi^2}{4}$ for $0<\chi\le 2$, or $r> \chi-1$ for $\chi>2$. In addition, under the same condition for $r,\chi$, the system admits a global generalized solution when $k\in(2-\frac{1}{n},\frac{3n-2}{n}]$, moreover this global generalized solution should be globally bounded provided $\frac{r}{\mu}$ and the initial data $u_0$ suitably small.