  In this paper, we consider the following attraction repulsion chemotaxis
model with nonlinear signal term:
  \begin{align*}
  &u_{t}=\nabla \cdot(\nabla u-\xi_{1} u \nabla v +\xi_{2} u \nabla w), \quad
  &0=\Delta v -\lambda_{1}v +f_{1}(u), \quad
  &0=\Delta w -\lambda_{2}w +f_{2}(u), \quad x \in \mathbb{R}^{N}, t>0,
  \end{align*}
  where $\xi_{1},\xi_{2},\lambda_{1},\lambda_{2}$ are for some positive
constants, and \begin{equation*}
  f_{1} \in C^{1}([0,\infty)) \; \text{satisfying} \; 0 \leqslant f_{1}(s)
\leqslant c_{1}s^{l}, \; \forall s \geqslant 0 \ \text{and} \ l> 0,
\end{equation*} \begin{equation*}
  f_{2} \in C^{1}([0,\infty))\; \text{satisfying} \; 0 \leqslant f_{2}(s)
\leqslant c_{2}s^{m}, \; \forall s \geqslant 0 \ \text{and} \ m> 0.
\end{equation*}
  We will show that this problem has a unique global bounded solution when $
l>\frac{2}{N}, l<m \ \text{with} \ m \geqslant 1$, or $l=m<\frac{2}{N}$.
