In the present study, we consider the following parabolic-elliptic
chemotaxis system: $$\begin{cases} u_t=∇·(γ(v)∇u−u\chi(v)∇v) +λu−\mu u^σ
, x∈Ω, \ t>0, \\ 0=∆v−v+u^κ, x∈Ω, \ t>0,\end{cases}$$ where $Ω⊂\mathbb{R}^n(n≥2)$ is a smooth and bounded domain, $λ>0,$ $\mu>0,$ $σ>1,$ $κ>0.$ Under appropriate assumptions on $γ(v)$ and $\chi(v),$ we obtain the global boundedness of solutions when $κn<2$ or $κn ≥2,$ $σ ≥κ+1,$ which generalize the previous result to the case with nonlinear signal secretion and superlinear logistic
term when $n≥2.$ Moreover, if adding additional conditions $σ≥2κ$ and $\mu$ is
sufficiently large, it is shown that the global solution $(u,v)$ converges to $$\left(\left(\frac{λ}{\mu}\right)^{\frac{1}{σ−1}}, \left(\frac{λ}{\mu}\right)^{\frac{κ}{σ−1}}\right)$$ exponentially as $t→∞.$
